Accord.Statistics
Base class for binary classifiers.
The data type for the input data. Default is double[].
Initializes a new instance of the class.
Computes class-label decisions for the given .
The input vectors that should be classified as
any of the possible classes.
The location where to store the class-labels.
A set of class-labels that best describe the
vectors according to this classifier.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Views this instance as a multi-class classifier,
giving access to more advanced methods, such as the prediction
of integer labels.
This instance seen as an .
Views this instance as a multi-class classifier,
giving access to more advanced methods, such as the prediction
of integer labels.
This instance seen as an .
Views this instance as a multi-label classifier,
giving access to more advanced methods, such as the prediction
of one-hot vectors.
This instance seen as an .
Base class for
generative binary classifiers.
The data type for the input data. Default is double[].
Predicts a class label vector for the given input vectors, returning the
log-likelihood that the input vector belongs to its predicted class.
The input vector.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
System.Double[].
Computes a numerical score measuring the association between
the given vector and each class.
The input vector.
An array where the result will be stored,
avoiding unnecessary memory allocations.
System.Double[].
Predicts a class label vector for the given input vector, returning the
log-likelihood that the input vector belongs to its predicted class.
The input vector.
Computes the log-likelihood that the given input
vector belongs to each of the possible classes.
The input vector.
System.Double[].
Predicts a class label vector for the given input vector, returning the
log-likelihood that the input vector belongs to its predicted class.
The input vector.
Computes the log-likelihoods that the given input
vectors belongs to each of the possible classes.
A set of input vectors.
System.Double[][].
Computes the log-likelihood that the given input
vector belongs to each of the possible classes.
The input vector.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
System.Double[].
Computes the log-likelihoods that the given input
vectors belongs to each of the possible classes.
A set of input vectors.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
System.Double[][].
Predicts a class label vector for the given input vector, returning the
log-likelihood that the input vector belongs to its predicted class.
The input vector.
The class label predicted by the classifier.
Predicts a class label vector for the given input vector, returning the
log-likelihood that the input vector belongs to its predicted class.
The input vector.
The class label predicted by the classifier.
System.Double.
Predicts a class label vector for the given input vector, returning the
log-likelihood that the input vector belongs to its predicted class.
The input vector.
The class label predicted by the classifier.
System.Double.
Predicts a class label vector for the given input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
System.Double[].
Predicts a class label vector for the given input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
System.Double[].
Predicts a class label vector for the given input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
System.Double[].
Predicts a class label vector for the given input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
The input vector.
The class label predicted by the classifier.
Predicts a class label vector for the given input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
The input vector.
The class label predicted by the classifier.
System.Double[].
Predicts a class label vector for the given input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
The input vector.
The class label predicted by the classifier.
System.Double[].
Predicts a class label vector for the given input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for the given input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the distances will be stored,
avoiding unnecessary memory allocations.
System.Double[].
Predicts a class label vector for the given input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the distances will be stored,
avoiding unnecessary memory allocations.
System.Double[].
Predicts a class label vector for the given input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
The input vector.
The class label predicted by the classifier.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for the given input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
The input vector.
The class label predicted by the classifier.
An array where the distances will be stored,
avoiding unnecessary memory allocations.
System.Double[].
Predicts a class label vector for the given input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
The input vector.
The class label predicted by the classifier.
An array where the distances will be stored,
avoiding unnecessary memory allocations.
System.Double[].
Predicts a class label for each input vector, returning the
log-likelihood that each vector belongs to its predicted class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
System.Double[].
Predicts a class label for each input vector, returning the
log-likelihood that each vector belongs to its predicted class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
System.Double[].
Predicts a class label for each input vector, returning the
log-likelihood that each vector belongs to its predicted class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
Predicts a class label vector for each input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
System.Double[][].
Predicts a class label vector for each input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
System.Double[][].
Predicts a class label vector for each input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
Predicts a class label vector for each input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
Predicts a class label vector for each input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
System.Double[][].
Predicts a class label for each input vector, returning the
log-likelihood that each vector belongs to its predicted class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
Predicts a class label for each input vector, returning the
log-likelihood that each vector belongs to its predicted class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
System.Double[].
Predicts a class label for each input vector, returning the
log-likelihood that each vector belongs to its predicted class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
System.Double[].
Predicts a class label vector for each input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for each input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
System.Double[][].
Predicts a class label vector for each input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
System.Double[][].
Predicts a class label vector for each input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for each input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
System.Double[][].
Predicts a class label vector for each input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
System.Double[][].
Predicts a class label for the given input vector, returning the
probability that the input vector belongs to its predicted class.
The input vector.
Computes the probabilities that the given input
vector belongs to each of the possible classes.
The input vector.
System.Double[].
Predicts a class label for the given input vector, returning the
probability that the input vector belongs to its predicted class.
The input vector.
Computes the probabilities that the given input
vectors belongs to each of the possible classes.
A set of input vectors.
System.Double[][].
Computes the probabilities that the given input
vector belongs to each of the possible classes.
The input vector.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
System.Double[].
Predicts a class label for the given input vector, returning the
probability that the input vector belongs to its predicted class.
The input vector.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Computes the probabilities that the given input
vectors belongs to each of the possible classes.
A set of input vectors.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
System.Double[][].
Predicts a class label for the given input vector, returning the
probability that the input vector belongs to its predicted class.
The input vector.
The class label predicted by the classifier.
Predicts a class label for the given input vector, returning the
probability that the input vector belongs to its predicted class.
The input vector.
The class label predicted by the classifier.
System.Double.
Predicts a class label for the given input vector, returning the
probability that the input vector belongs to its predicted class.
The input vector.
The class label predicted by the classifier.
System.Double.
Predicts a class label vector for the given input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
Predicts a class label vector for the given input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
System.Double[].
Predicts a class label vector for the given input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
System.Double[].
Predicts a class label vector for the given input vector, returning the
probabilities of the input vector belonging to each possible class.
The input vector.
The class label predicted by the classifier.
Predicts a class label vector for the given input vector, returning the
probabilities of the input vector belonging to each possible class.
The input vector.
The class label predicted by the classifier.
System.Double[].
Probabilitieses the specified input.
The input.
The decision.
System.Double[].
Predicts a class label vector for the given input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for the given input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the distances will be stored,
avoiding unnecessary memory allocations.
System.Double[].
Predicts a class label vector for the given input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the distances will be stored,
avoiding unnecessary memory allocations.
System.Double[].
Predicts a class label vector for the given input vector, returning the
probabilities of the input vector belonging to each possible class.
The input vector.
The class label predicted by the classifier.
An array where the distances will be stored,
avoiding unnecessary memory allocations.
System.Double[].
Predicts a class label vector for the given input vector, returning the
probabilities of the input vector belonging to each possible class.
The input vector.
The class label predicted by the classifier.
An array where the distances will be stored,
avoiding unnecessary memory allocations.
System.Double[].
Predicts a class label vector for the given input vector, returning the
probabilities of the input vector belonging to each possible class.
The input vector.
The class label predicted by the classifier.
An array where the distances will be stored,
avoiding unnecessary memory allocations.
System.Double[].
Predicts a class label for each input vector, returning the
probability that each vector belongs to its predicted class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
System.Double[].
Predicts a class label for each input vector, returning the
probability that each vector belongs to its predicted class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
System.Double[].
Predicts a class label for each input vector, returning the
probability that each vector belongs to its predicted class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
Predicts a class label vector for each input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
System.Double[][].
Predicts a class label vector for each input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
System.Double[][].
Predicts a class label vector for each input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
Predicts a class label vector for each input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
System.Double[][].
Predicts a class label vector for each input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
System.Double[][].
Predicts a class label vector for each input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
System.Double[][].
Predicts a class label for each input vector, returning the
probability that each vector belongs to its predicted class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Predicts a class label for each input vector, returning the
probability that each vector belongs to its predicted class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
System.Double[].
Predicts a class label for each input vector, returning the
probability that each vector belongs to its predicted class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
System.Double[].
Predicts a class label vector for each input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
System.Double[][].
Predicts a class label vector for each input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
System.Double[][].
Predicts a class label vector for each input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
System.Double[][].
Predicts a class label vector for each input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for each input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
System.Double[][].
Predicts a class label vector for each input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
System.Double[][].
Applies the transformation to a set of input vectors,
producing an associated set of output vectors.
The input data to which
the transformation should be applied.
The location to where to store the
result of this transformation.
The output generated by applying this
transformation to the given input.
Applies the transformation to a set of input vectors,
producing an associated set of output vectors.
The input data to which
the transformation should be applied.
The location to where to store the
result of this transformation.
The output generated by applying this
transformation to the given input.
Applies the transformation to a set of input vectors,
producing an associated set of output vectors.
The input data to which
the transformation should be applied.
The location to where to store the
result of this transformation.
The output generated by applying this
transformation to the given input.
Views this instance as a multi-class generative classifier,
giving access to more advanced methods, such as the prediction
of integer labels.
This instance seen as an .
Views this instance as a multi-class generative classifier,
giving access to more advanced methods, such as the prediction
of integer labels.
This instance seen as an .
Views this instance as a multi-label generative classifier,
giving access to more advanced methods, such as the prediction
of one-hot vectors.
This instance seen as an .
Base class for
score-based binary classifiers.
The data type for the input data. Default is double[].
Computes a numerical score measuring the association between
the given vector and each class.
The input vector.
An array where the result will be stored,
avoiding unnecessary memory allocations.
Computes a class-label decision for a given .
The input vector that should be classified into
one of the possible classes.
The location where to store the class-labels.
A class-label that best described according
to this classifier.
Computes a class-label decision for a given .
The input vector that should be classified into
one of the possible classes.
A class-label that best described according
to this classifier.
Computes a numerical score measuring the association between
the given vector and its most strongly
associated class (as predicted by the classifier).
The input vector.
Computes a numerical score measuring the association between
the given vector and each class.
The input vector.
Computes a numerical score measuring the association between
the given vector and its most strongly
associated class (as predicted by the classifier).
The input vector.
Computes a numerical score measuring the association between
the given vector and each class.
The input vector.
Computes a numerical score measuring the association between
the given vector and each class.
The input vector.
An array where the result will be stored,
avoiding unnecessary memory allocations.
Computes a numerical score measuring the association between
the given vector and each class.
The input vector.
An array where the result will be stored,
avoiding unnecessary memory allocations.
Predicts a class label for the input vector, returning a
numerical score measuring the strength of association of the
input vector to its most strongly related class.
The input vector.
The class label predicted by the classifier.
Predicts a class label vector for the given input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
The input vector.
The class label predicted by the classifier.
Predicts a class label vector for the given input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
The input vector.
The class label predicted by the classifier.
An array where the result will be stored,
avoiding unnecessary memory allocations.
Predicts a class label for each input vector, returning a
numerical score measuring the strength of association of the
input vector to the most strongly related class.
A set of input vectors.
The class labels predicted for each input
vector, as predicted by the classifier.
Predicts a class label vector for each input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
Predicts a class label for each input vector, returning a
numerical score measuring the strength of association of the
input vector to the most strongly related class.
A set of input vectors.
The class labels predicted for each input
vector, as predicted by the classifier.
An array where the result will be stored,
avoiding unnecessary memory allocations.
Predicts a class label for each input vector, returning a
numerical score measuring the strength of association of the
input vector to the most strongly related class.
A set of input vectors.
The class labels predicted for each input
vector, as predicted by the classifier.
An array where the distances will be stored,
avoiding unnecessary memory allocations.
System.Double[].
Predicts a class label for each input vector, returning a
numerical score measuring the strength of association of the
input vector to the most strongly related class.
A set of input vectors.
The class labels predicted for each input
vector, as predicted by the classifier.
An array where the distances will be stored,
avoiding unnecessary memory allocations.
System.Double[].
Predicts a class label vector for each input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the result will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for each input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the distances will be stored,
avoiding unnecessary memory allocations.
System.Double[][].
Predicts a class label vector for each input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the distances will be stored,
avoiding unnecessary memory allocations.
System.Double[][].
Applies the transformation to a set of input vectors,
producing an associated set of output vectors.
The input data to which
the transformation should be applied.
The location to where to store the
result of this transformation.
The output generated by applying this
transformation to the given input.
Applies the transformation to a set of input vectors,
producing an associated set of output vectors.
The input data to which
the transformation should be applied.
The location to where to store the
result of this transformation.
The output generated by applying this
transformation to the given input.
Applies the transformation to a set of input vectors,
producing an associated set of output vectors.
The input data to which
the transformation should be applied.
The location to where to store the
result of this transformation.
The output generated by applying this
transformation to the given input.
Views this instance as a multi-class distance classifier,
giving access to more advanced methods, such as the prediction
of integer labels.
This instance seen as an .
Views this instance as a multi-class distance classifier,
giving access to more advanced methods, such as the prediction
of integer labels.
This instance seen as an .
Views this instance as a multi-label distance classifier,
giving access to more advanced methods, such as the prediction
of one-hot vectors.
This instance seen as an .
Base class for multi-class classifiers.
Computes a class-label decision for a given .
The input vector that should be classified into
one of the possible classes.
A class-label that best described according
to this classifier.
Computes a class-label decision for a given .
The input vector that should be classified into
one of the possible classes.
A class-label that best described according
to this classifier.
Computes a class-label decision for a given .
The input vector that should be classified into
one of the possible classes.
A class-label that best described according
to this classifier.
Computes a class-label decision for a given .
The input vector that should be classified into
one of the possible classes.
A class-label that best described according
to this classifier.
Computes a class-label decision for a given .
The input vector that should be classified into
one of the possible classes.
The location where to store the class-labels.
A class-label that best described according
to this classifier.
Computes class-label decisions for the given .
The input vectors that should be classified as
any of the possible classes.
The location where to store the class-labels.
A set of class-labels that best describe the
vectors according to this classifier.
Computes class-label decisions for the given .
The input vectors that should be classified as
any of the possible classes.
The location where to store the class-labels.
A set of class-labels that best describe the
vectors according to this classifier.
Computes class-label decisions for the given .
The input vectors that should be classified as
any of the possible classes.
The location where to store the class-labels.
A set of class-labels that best describe the
vectors according to this classifier.
Computes class-label decisions for the given .
The input vectors that should be classified as
any of the possible classes.
The location where to store the class-labels.
A set of class-labels that best describe the
vectors according to this classifier.
Computes a class-label decision for a given .
The input vector that should be classified into
one of the possible classes.
The location where to store the class-labels.
A class-label that best described according
to this classifier.
Computes a class-label decision for a given .
The input vector that should be classified into
one of the possible classes.
The location where to store the class-labels.
A class-label that best described according
to this classifier.
Computes a class-label decision for a given .
The input vector that should be classified into
one of the possible classes.
The location where to store the class-labels.
A class-label that best described according
to this classifier.
Computes a class-label decision for a given .
The input vector that should be classified into
one of the possible classes.
The location where to store the class-labels.
A class-label that best described according
to this classifier.
Computes a class-label decision for a given .
The input vector that should be classified into
one of the possible classes.
The location where to store the class-labels.
A class-label that best described according
to this classifier.
Computes a class-label decision for a given .
The input vector that should be classified into
one of the possible classes.
The location where to store the class-labels.
A class-label that best described according
to this classifier.
Computes a class-label decision for a given .
The input vector that should be classified into
one of the possible classes.
The location where to store the class-labels.
A class-label that best described according
to this classifier.
Computes a class-label decision for a given .
The input vector that should be classified into
one of the possible classes.
The location where to store the class-labels.
A class-label that best described according
to this classifier.
Computes a class-label decision for a given .
The input vector that should be classified into
one of the possible classes.
The location where to store the class-labels.
A class-label that best described according
to this classifier.
Computes class-label decisions for the given .
The input vectors that should be classified as
any of the possible classes.
The location where to store the class-labels.
A set of class-labels that best describe the
vectors according to this classifier.
Computes class-label decisions for the given .
The input vectors that should be classified as
any of the possible classes.
The location where to store the class-labels.
A set of class-labels that best describe the
vectors according to this classifier.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Base class for multi-class classifiers.
The data type for the input data. Default is double[].
Computes class-label decisions for the given .
The input vectors that should be classified as
any of the possible classes.
The location where to store the class-labels.
A set of class-labels that best describe the
vectors according to this classifier.
Computes class-label decisions for the given .
The input vectors that should be classified as
any of the possible classes.
The location where to store the class-labels.
A set of class-labels that best describe the
vectors according to this classifier.
Computes class-label decisions for the given .
The input vectors that should be classified as
any of the possible classes.
The location where to store the class-labels.
A set of class-labels that best describe the
vectors according to this classifier.
Computes a class-label decision for a given .
The input vector that should be classified into
one of the possible classes.
The location where to store the class-labels.
A class-label that best described according
to this classifier.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Views this instance as a multi-label classifier,
giving access to more advanced methods, such as the prediction
of one-hot vectors.
This instance seen as an .
Base class for generative multi-class classifiers.
The data type for the input data. Default is double[].
Computes a numerical score measuring the association between
the given vector and a given
.
The input vector.
The index of the class whose score will be computed.
System.Double.
Computes the log-likelihood that the given input vector
belongs to the specified .
The input vector.
The index of the class whose score will be computed.
Predicts a class label vector for the given input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Computes the log-likelihood that the given input
vector belongs to each of the possible classes.
The input vector.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Computes the probability that the given input vector
belongs to the specified .
The input vector.
The index of the class whose score will be computed.
Computes the probability that the given input vector
belongs to the specified .
The input vector.
The index of the class whose score will be computed.
Computes the probability that the given input vector
belongs to the specified .
The input vector.
The index of the class whose score will be computed.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Computes the probability that the given input vector
belongs to the specified .
The input vector.
The index of the class whose score will be computed.
Computes the probability that the given input vector
belongs to the specified .
The input vector.
The index of the class whose score will be computed.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Computes the probabilities that the given input
vector belongs to each of the possible classes.
The input vector.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for the given input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
Computes the log-likelihood that the given input vector
belongs to the specified .
The input vector.
The index of the class whose score will be computed.
Computes the log-likelihood that the given input vector
belongs to the specified .
The input vector.
The index of the class whose score will be computed.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
Computes the log-likelihood that the given input vector
belongs to the specified .
The input vector.
The index of the class whose score will be computed.
Computes the log-likelihood that the given input vector
belongs to the specified .
The input vector.
The index of the class whose score will be computed.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
Computes the log-likelihood that the given input vector
belongs to the specified .
The input vector.
The index of the class whose score will be computed.
Computes the log-likelihood that the given input vector
belongs to the specified .
The input vector.
The index of the class whose score will be computed.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
Computes the log-likelihood that the given input
vector belongs to its most plausible class.
The input vector.
Computes the log-likelihood that the given input
vectors belongs to each of the possible classes.
The input vector.
Computes the log-likelihood that the given input
vectors belongs to each of the possible classes.
The input vector.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
Computes the log-likelihood that the given input
vector belongs to each of the possible classes.
The input vector.
Computes the log-likelihood that the given input
vectors belongs to each of the possible classes.
The input vector.
Computes the log-likelihood that the given input
vector belongs to each of the possible classes.
The input vector.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for the given input vector, returning the
log-likelihood that the input vector belongs to its predicted class.
The input vector.
The class label predicted by the classifier.
Predicts a class label vector for the given input vector, returning the
log-likelihood that the input vector belongs to its predicted class.
The input vector.
The class label predicted by the classifier.
Predicts a class label vector for the given input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
Predicts a class label vector for the given input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
Predicts a class label vector for the given input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
Predicts a class label for each input vector, returning the
log-likelihood that each vector belongs to its predicted class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
Predicts a class label for each input vector, returning the
log-likelihood that each vector belongs to its predicted class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
Predicts a class label vector for each input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
Predicts a class label vector for each input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
Predicts a class label for each input vector, returning the
log-likelihood that each vector belongs to its predicted class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
Predicts a class label for each input vector, returning the
log-likelihood that each vector belongs to its predicted class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for each input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for each input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
Predicts a class label for the given input vector, returning the
probability that the input vector belongs to its predicted class.
The input vector.
Predicts a class label for the given input vector, returning the
probability that the input vector belongs to its predicted class.
The input vector.
Predicts a class label for the given input vector, returning the
probability that the input vector belongs to its predicted class.
The input vector.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Computes the probabilities that the given input
vector belongs to each of the possible classes.
The input vector.
Computes the probabilities that the given input
vector belongs to each of the possible classes.
The input vector.
Computes the probabilities that the given input
vector belongs to each of the possible classes.
The input vector.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Predicts a class label for the given input vector, returning the
probability that the input vector belongs to its predicted class.
The input vector.
The class label predicted by the classifier.
Predicts a class label for the given input vector, returning the
probability that the input vector belongs to its predicted class.
The input vector.
The class label predicted by the classifier.
Predicts a class label vector for the given input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
Predicts a class label vector for the given input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
Predicts a class label vector for the given input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Predicts a class label for each input vector, returning the
probability that each vector belongs to its predicted class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
Predicts a class label for each input vector, returning the
probability that each vector belongs to its predicted class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
Predicts a class label vector for each input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
Predicts a class label vector for each input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
Predicts a class label for each input vector, returning the
probability that each vector belongs to its predicted class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Predicts a class label for each input vector, returning the
probability that each vector belongs to its predicted class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for each input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for each input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Views this instance as a multi-label generative classifier,
giving access to more advanced methods, such as the prediction
of one-hot vectors.
This instance seen as an .
Views this instance as a multi-class generative classifier.
This instance seen as an .
Views this instance as a multi-class generative classifier.
This instance seen as an .
Base class for
score-based multi-class classifiers.
The data type for the input data. Default is double[].
Computes a numerical score measuring the association between
the given vector and each class.
The input vector.
An array where the result will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for the given input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
The input vector.
The class label predicted by the classifier.
An array where the scores will be stored,
avoiding unnecessary memory allocations.
Computes a numerical score measuring the association between
the given vector and a given
.
The input vector.
The index of the class whose score will be computed.
Computes a class-label decision for a given .
The input vector that should be classified into
one of the possible classes.
A class-label that best described according
to this classifier.
Computes a numerical score measuring the association between
the given vector and a given
.
The input vector.
The index of the class whose score will be computed.
Computes a numerical score measuring the association between
the given vector and a given
.
The input vector.
The index of the class whose score will be computed.
An array where the scores will be stored,
avoiding unnecessary memory allocations.
Computes a numerical score measuring the association between
the given vector and a given
.
The input vector.
The index of the class whose score will be computed.
Computes a numerical score measuring the association between
the given vector and a given
.
The input vector.
The index of the class whose score will be computed.
An array where the scores will be stored,
avoiding unnecessary memory allocations.
Computes a numerical score measuring the association between
the given vector and its most strongly
associated class (as predicted by the classifier).
The input vector.
Computes a numerical score measuring the association between
the given vector and its most strongly
associated class (as predicted by the classifier).
The input vector.
Computes a numerical score measuring the association between
the given vector and its most strongly
associated class (as predicted by the classifier).
The input vector.
An array where the scores will be stored,
avoiding unnecessary memory allocations.
Computes a numerical score measuring the association between
the given vector and each class.
The input vector.
Computes a numerical score measuring the association between
the given vector and each class.
The input vector.
Computes a numerical score measuring the association between
the given vector and each class.
The input vector.
An array where the scores will be stored,
avoiding unnecessary memory allocations.
Predicts a class label for the input vector, returning a
numerical score measuring the strength of association of the
input vector to its most strongly related class.
The input vector.
The class label predicted by the classifier.
Predicts a class label for the input vector, returning a
numerical score measuring the strength of association of the
input vector to its most strongly related class.
The input vector.
The class label predicted by the classifier.
Predicts a class label vector for the given input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
The input vector.
The class label predicted by the classifier.
Predicts a class label vector for the given input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
The input vector.
The class label predicted by the classifier.
Predicts a class label vector for the given input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
The input vector.
The class label predicted by the classifier.
An array where the scores will be stored,
avoiding unnecessary memory allocations.
Predicts a class label for each input vector, returning a
numerical score measuring the strength of association of the
input vector to the most strongly related class.
A set of input vectors.
The class labels predicted for each input
vector, as predicted by the classifier.
Predicts a class label for each input vector, returning a
numerical score measuring the strength of association of the
input vector to the most strongly related class.
A set of input vectors.
The class labels predicted for each input
vector, as predicted by the classifier.
Predicts a class label vector for each input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
Predicts a class label vector for each input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
Predicts a class label for each input vector, returning a
numerical score measuring the strength of association of the
input vector to the most strongly related class.
A set of input vectors.
The class labels predicted for each input
vector, as predicted by the classifier.
An array where the scores will be stored,
avoiding unnecessary memory allocations.
Predicts a class label for each input vector, returning a
numerical score measuring the strength of association of the
input vector to the most strongly related class.
A set of input vectors.
The class labels predicted for each input
vector, as predicted by the classifier.
An array where the scores will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for each input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the scores will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for each input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the scores will be stored,
avoiding unnecessary memory allocations.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Views this instance as a multi-label distance classifier,
giving access to more advanced methods, such as the prediction
of one-hot vectors.
This instance seen as an .
Views this instance as a multi-class generative classifier.
This instance seen as an .
Views this instance as a multi-class generative classifier.
This instance seen as an .
Base class for
multi-label classifiers.
The data type for the input data. Default is double[].
Computes whether a class label applies to an vector.
The input vectors that should be classified as
any of the possible classes.
The class label index to be tested.
A boolean value indicating whether the given
class label applies to the vector.
Computes a class-label decision for a given .
The input vector that should be classified into
one of the possible classes.
A class-label that best described according
to this classifier.
Computes class-label decisions for the given .
The input vectors that should be classified as
any of the possible classes.
The location where to store the class-labels.
A set of class-labels that best describe the
vectors according to this classifier.
Computes class-label decisions for the given .
The input vectors that should be classified as
any of the possible classes.
The location where to store the class-labels.
A set of class-labels that best describe the
vectors according to this classifier.
Computes class-label decisions for the given .
The input vectors that should be classified as
any of the possible classes.
The location where to store the class-labels.
A set of class-labels that best describe the
vectors according to this classifier.
Computes a class-label decision for a given .
The input vector that should be classified into
one of the possible classes.
The location where to store the class-labels.
A class-label that best described according
to this classifier.
Computes a class-label decision for a given .
The input vector that should be classified into
one of the possible classes.
The location where to store the class-labels.
A class-label that best described according
to this classifier.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Base class for
generative multi-label classifiers.
The data type for the input data. Default is double[].
Computes a log-likelihood measuring the association between
the given vector and a given
.
The input vector.
The index of the class whose score will be computed.
The class label associated with the input
vector, as predicted by the classifier.
Predicts a class label vector for the given input vector, returning the
probabilities of the input vector belonging to each possible class.
The input vector.
The class label predicted by the classifier.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for the given input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
The input vector.
The class label predicted by the classifier.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
Computes the log-likelihood that the given input vector
belongs to the specified .
The input vector.
The index of the class whose score will be computed.
Computes the log-likelihood that the given input
vector belongs to each of the possible classes.
The input vector.
Computes the log-likelihood that the given input
vector belongs to each of the possible classes.
The input vector.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
Computes the log-likelihood that the given input
vector belongs to each of the possible classes.
The input vector.
Computes the log-likelihood that the given input
vector belongs to each of the possible classes.
The input vector.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
Computes the probability that the given input vector
belongs to the specified .
The input vector.
The index of the class whose score will be computed.
Computes the probability that the given input vector
belongs to the specified .
The input vector.
The index of the class whose score will be computed.
Computes the probability that the given input vector
belongs to the specified .
The input vector.
The index of the class whose score will be computed.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Computes the probability that the given input vector
belongs to the specified .
The input vector.
The index of the class whose score will be computed.
Computes the probability that the given input vector
belongs to the specified .
The input vector.
The index of the class whose score will be computed.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Computes the probabilities that the given input
vector belongs to each of the possible classes.
The input vector.
Computes the probabilities that the given input
vector belongs to each of the possible classes.
The input vector.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Computes the probabilities that the given input
vector belongs to each of the possible classes.
The input vector.
Computes the probabilities that the given input
vector belongs to each of the possible classes.
The input vector.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Computes the log-likelihood that the given input vector
belongs to the specified .
The input vector.
The index of the class whose score will be computed.
Computes the log-likelihood that the given input vector
belongs to the specified .
The input vector.
The index of the class whose score will be computed.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
Computes the log-likelihood that the given input vector
belongs to the specified .
The input vector.
The index of the class whose score will be computed.
Computes the log-likelihood that the given input vector
belongs to the specified .
The input vector.
The index of the class whose score will be computed.
An array where the log-likelihoods will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for the given input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
Predicts a class label vector for the given input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
Predicts a class label vector for the given input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
Predicts a class label vector for the given input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
The input vector.
The class label predicted by the classifier.
Predicts a class label vector for the given input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
The input vector.
The class label predicted by the classifier.
Predicts a class label vector for the given input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for the given input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for the given input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
The input vector.
The class label predicted by the classifier.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for the given input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
The input vector.
The class label predicted by the classifier.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for each input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
Predicts a class label vector for each input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
Predicts a class label vector for each input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
Predicts a class label vector for each input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
Predicts a class label vector for each input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
Predicts a class label vector for each input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for each input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for each input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for each input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for each input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for the given input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
Predicts a class label vector for the given input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
Predicts a class label vector for the given input vector, returning the
probabilities of the input vector belonging to each possible class.
The input vector.
The class label predicted by the classifier.
Predicts a class label vector for the given input vector, returning the
probabilities of the input vector belonging to each possible class.
The input vector.
The class label predicted by the classifier.
Predicts a class label vector for the given input vector, returning the
probabilities of the input vector belonging to each possible class.
The input vector.
The class label predicted by the classifier.
Predicts a class label vector for the given input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for the given input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for the given input vector, returning the
probabilities of the input vector belonging to each possible class.
The input vector.
The class label predicted by the classifier.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for the given input vector, returning the
probabilities of the input vector belonging to each possible class.
The input vector.
The class label predicted by the classifier.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for each input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
Predicts a class label vector for each input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
Predicts a class label vector for each input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
Predicts a class label vector for each input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
Predicts a class label vector for each input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
Predicts a class label vector for each input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for each input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for each input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for each input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for each input vector, returning the
probabilities of the input vector belonging to each possible class.
A set of input vectors.
The labels predicted by the classifier.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Views this instance as a multi-class generative classifier.
This instance seen as an .
Views this instance as a multi-class generative classifier.
This instance seen as an .
Base class for
score-based multi-label classifiers.
The data type for the input data. Default is double[].
Computes a numerical score measuring the association between
the given vector and a given
.
The input vector.
The index of the class whose score will be computed.
The class label associated with the input
vector, as predicted by the classifier.
Computes a numerical score measuring the association between
the given vector and a given
.
The input vector.
The index of the class whose score will be computed.
Predicts a class label vector for the given input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the scores will be stored,
avoiding unnecessary memory allocations.
Computes a numerical score measuring the association between
the given vector and a given
.
The input vector.
The index of the class whose score will be computed.
Computes a numerical score measuring the association between
the given vector and a given
.
The input vector.
The index of the class whose score will be computed.
An array where the scores will be stored,
avoiding unnecessary memory allocations.
Computes a numerical score measuring the association between
the given vector and a given
.
The input vector.
The index of the class whose score will be computed.
Computes a numerical score measuring the association between
the given vector and a given
.
The input vector.
The index of the class whose score will be computed.
An array where the scores will be stored,
avoiding unnecessary memory allocations.
Computes a numerical score measuring the association between
the given vector and each class.
The input vector.
Computes a numerical score measuring the association between
the given vector and each class.
The input vector.
Computes a numerical score measuring the association between
the given vector and each class.
The input vector.
An array where the scores will be stored,
avoiding unnecessary memory allocations.
Computes a numerical score measuring the association between
the given vector and each class.
The input vector.
An array where the scores will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for the given input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
Predicts a class label vector for the given input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
Predicts a class label vector for the given input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
Predicts a class label vector for the given input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the scores will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for the given input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the scores will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for each input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
Predicts a class label vector for each input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
Predicts a class label vector for each input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
Predicts a class label vector for each input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the scores will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for each input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the scores will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for each input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the scores will be stored,
avoiding unnecessary memory allocations.
Computes a class-label decision for a given .
The input vector that should be classified into
one of the possible classes.
An array where the scores will be stored,
avoiding unnecessary memory allocations.
A class-label that best described according
to this classifier.
Predicts a class label vector for the given input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
The input vector.
The class label predicted by the classifier.
Predicts a class label vector for the given input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
The input vector.
The class label predicted by the classifier.
An array where the scores will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for each input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
Predicts a class label vector for each input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the scores will be stored,
avoiding unnecessary memory allocations.
Computes a class-label decision for a given .
The input vector that should be classified into
one of the possible classes.
An array where the scores will be stored,
avoiding unnecessary memory allocations.
A class-label that best described according
to this classifier.
Predicts a class label vector for the given input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
The input vector.
The class label predicted by the classifier.
Predicts a class label vector for the given input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
The input vector.
The class label predicted by the classifier.
An array where the scores will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for each input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
Predicts a class label vector for each input vector, returning a
numerical score measuring the strength of association of the input vector
to each of the possible classes.
A set of input vectors.
The class labels associated with each input
vector, as predicted by the classifier. If passed as null, the classifier
will create a new array.
An array where the scores will be stored,
avoiding unnecessary memory allocations.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
A location to store the output, avoiding unnecessary memory allocations.
The output generated by applying this transformation to the given input.
Views this instance as a multi-class generative classifier.
This instance seen as an .
Views this instance as a multi-class generative classifier.
This instance seen as an .
Base implementation for generative observation sequence taggers. A sequence
tagger can predict the class label of each individual observation in a
input sequence vector.
The data type for the input data. Default is double[].
Predicts a the probability that the sequence vector
has been generated by this log-likelihood tagger.
Predicts a the probability that the sequence vector
has been generated by this log-likelihood tagger.
Predicts a the log-likelihood for each of the observations in
the sequence vector assuming each of the possible states in the
tagger model.
Predicts a the log-likelihood for each of the observations in
the sequence vector assuming each of the possible states in the
tagger model.
Predicts a the probability that the sequence vector
has been generated by this log-likelihood tagger.
Predicts a the probability that the sequence vector
has been generated by this log-likelihood tagger.
Predicts a the probability that the sequence vector
has been generated by this log-likelihood tagger.
Predicts a the probability that the sequence vector
has been generated by this log-likelihood tagger.
Predicts a the probability that the sequence vector
has been generated by this log-likelihood tagger.
Predicts a the probability that the sequence vector
has been generated by this log-likelihood tagger.
Predicts a the probability that the sequence vector
has been generated by this log-likelihood tagger.
Predicts a the probability that the sequence vector
has been generated by this log-likelihood tagger.
Predicts a the probability that the sequence vector
has been generated by this log-likelihood tagger.
Predicts a the probability that the sequence vector
has been generated by this log-likelihood tagger.
Predicts a the probabilities for each of the observations in
the sequence vector assuming each of the possible states in the
tagger model.
Predicts a the probabilities for each of the observations in
the sequence vector assuming each of the possible states in the
tagger model.
Predicts a the log-likelihood for each of the observations in
the sequence vector assuming each of the possible states in the
tagger model.
Predicts a the log-likelihood for each of the observations in
the sequence vector assuming each of the possible states in the
tagger model.
Predicts a the log-likelihood for each of the observations in
the sequence vector assuming each of the possible states in the
tagger model.
Predicts a the log-likelihood for each of the observations in
the sequence vector assuming each of the possible states in the
tagger model.
Predicts a the log-likelihood for each of the observations in
the sequence vector assuming each of the possible states in the
tagger model.
Predicts a the log-likelihood for each of the observations in
the sequence vector assuming each of the possible states in the
tagger model.
Predicts a the probabilities for each of the observations in
the sequence vector assuming each of the possible states in the
tagger model.
Predicts a the probabilities for each of the observations in
the sequence vector assuming each of the possible states in the
tagger model.
Predicts a the log-likelihood for each of the observations in
the sequence vector assuming each of the possible states in the
tagger model.
Predicts a the log-likelihood for each of the observations in
the sequence vector assuming each of the possible states in the
tagger model.
Predicts a the log-likelihood for each of the observations in
the sequence vector assuming each of the possible states in the
tagger model.
Predicts a the log-likelihood for each of the observations in
the sequence vector assuming each of the possible states in the
tagger model.
Computes numerical scores measuring the association between
each of the given vectors and each
possible class.
Computes numerical scores measuring the association between
each of the given vectors and each
possible class.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
The output generated by applying this transformation to the given input.
Common base class for observation sequence taggers.
Computes numerical scores measuring the association between
each of the given vectors and each
possible class.
Computes numerical scores measuring the association between
each of the given vectors and each
possible class.
Computes numerical scores measuring the association between
each of the given vectors and each
possible class.
Computes numerical scores measuring the association between
each of the given vectors and each
possible class.
Computes numerical scores measuring the association between
each of the given vectors and each
possible class.
Computes numerical scores measuring the association between
each of the given vectors and each
possible class.
Computes numerical scores measuring the association between
each of the given vectors and each
possible class.
Computes numerical scores measuring the association between
each of the given vectors and each
possible class.
Base class for multi-class and multi-label classifiers.
The data type for the input data. Default is double[].
Gets the number of classes expected and recognized by the classifier.
The number of classes.
Computes class-label decisions for the given .
The input vectors that should be classified as
any of the possible classes.
The location where to store the class-labels.
A set of class-labels that best describe the
vectors according to this classifier.
Computes class-label decisions for the given .
The input vectors that should be classified as
any of the possible classes.
A set of class-labels that best describe the
vectors according to this classifier.
Computes class-label decisions for the given .
The input vectors that should be classified as
any of the possible classes.
A set of class-labels that best describe the
vectors according to this classifier.
Computes class-label decisions for the given .
The input vectors that should be classified as
any of the possible classes.
The location where to store the class-labels.
A set of class-labels that best describe the
vectors according to this classifier.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
The output generated by applying this transformation to the given input.
Base class for data transformation algorithms.
The type for the output data that enters in the model. Default is double[].
The type for the input data that exits from the model. Default is double[].
Gets the number of inputs accepted by the model.
Gets the number of outputs generated by the model.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
The output generated by applying this transformation to the given input.
Applies the transformation to a set of input vectors,
producing an associated set of output vectors.
The input data to which
the transformation should be applied.
The output generated by applying this
transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
The location to where to store the result of this transformation.
The output generated by applying this transformation to the given input.
Base class for data transformation algorithms.
The type for the output data that enters in the model. Default is double[].
The type for the input data that exits from the model. Default is double[].
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
The location to where to store the result of this transformation.
The output generated by applying this transformation to the given input.
Applies the transformation to a set of input vectors,
producing an associated set of output vectors.
The input data to which
the transformation should be applied.
The output generated by applying this
transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
The location to where to store the result of this transformation.
The output generated by applying this transformation to the given input.
Base class for data transformation algorithms.
The type for the output data that enters in the model. Default is double[].
Base class for data transformation algorithms.
Common interface for unsupervised learning algorithms.
The type for the model being learned.
The type for the output data that originates from the model.
Learns a model that can map the given inputs to the desired outputs.
The model inputs.
A model that has learned how to produce suitable outputs
given the input data .
Minimum (Mean) Distance Classifier.
This is one of the simplest possible pattern recognition classifiers.
This classifier works by comparing a new input vector against the mean
value of the other classes. The class which is closer to this new input
vector is considered the winner, and the vector will be classified as
having the same label as this class.
Base class for Principal Component Analyses (PCA and KPCA).
Obsolete
Obsolete
Obsolete
Obsolete
Obsolete
Obsolete
Initializes a new instance of the class.
Gets the column standard deviations of the source data given at method construction.
Gets the column mean of the source data given at method construction.
Gets or sets the method used by this analysis.
Gets or sets whether calculations will be performed overwriting
data in the original source matrix, using less memory.
Gets or sets whether the transformation result should be whitened
(have unit standard deviation) before it is returned.
Gets or sets the number of outputs (dimensionality of the output vectors)
that should be generated by this model.
Gets the maximum number of outputs (dimensionality of the output vectors)
that can be generated by this model.
Gets or sets the amount of explained variance that should be generated
by this model. This value will alter the
that can be generated by this model.
Provides access to the Singular Values stored during the analysis.
If a covariance method is chosen, then it will contain an empty vector.
The singular values.
Returns the original data supplied to the analysis.
The original data matrix supplied to the analysis.
Gets the resulting projection of the source
data given on the creation of the analysis
into the space spawned by principal components.
The resulting projection in principal component space.
Gets a matrix whose columns contain the principal components. Also known as the Eigenvectors or loadings matrix.
The matrix of principal components.
Gets a matrix whose columns contain the principal components. Also known as the Eigenvectors or loadings matrix.
The matrix of principal components.
Provides access to the Eigenvalues stored during the analysis.
The Eigenvalues.
Gets or sets a cancellation token that can be used
to cancel the algorithm while it is running.
The respective role each component plays in the data set.
The component proportions.
The cumulative distribution of the components proportion role. Also known
as the cumulative energy of the principal components.
The cumulative proportions.
Gets the Principal Components in a object-oriented structure.
The collection of principal components.
Returns the minimal number of principal components
required to represent a given percentile of the data.
The percentile of the data requiring representation.
The minimal number of components required.
Creates additional information about principal components.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
The output generated by applying this transformation to the given input.
Obsolete.
Projects a given matrix into principal component space.
The matrix to be projected.
The number of components to consider.
Projects a given matrix into principal component space.
The matrix to be projected.
The number of components to consider.
Projects a given matrix into principal component space.
The matrix to be projected.
The number of components to consider.
Common interface for information components. Those are
present in multivariate analysis, such as
and .
Gets the index for this component.
Gets the proportion, or amount of information explained by this component.
Gets the cumulative proportion of all discriminants until this component.
Determines the method to be used in a statistical analysis.
By choosing Center, the method will be run on the mean-centered data.
In Principal Component Analysis this means the method will operate
on the Covariance matrix of the given data.
By choosing Standardize, the method will be run on the mean-centered and
standardized data.
In Principal Component Analysis this means the method
will operate on the Correlation matrix of the given data. One should always
choose to standardize when dealing with different units of variables.
Determines the method to be used in a statistical analysis.
By choosing Center, the method will be run on the mean-centered data.
In Principal Component Analysis this means the method will operate
on the Covariance matrix of the given data.
By choosing Standardize, the method will be run on the mean-centered and
standardized data.
In Principal Component Analysis this means the method
will operate on the Correlation matrix of the given data. One should always
choose to standardize when dealing with different units of variables.
By choosing CorrelationMatrix, the method will interpret the given data
as a correlation matrix.
By choosing CovarianceMatrix, the method will interpret the given data
as a correlation matrix.
By choosing KernelMatrix, the method will interpret the given data
as a Kernel (Gram) matrix.
Common interface for statistical analysis.
Computes the analysis using given source data and parameters.
Common interface for descriptive measures, such as
and
.
Gets the variable's index.
Gets the variable's name
Gets the variable's total sum.
Gets the variable's mean.
Gets the variable's standard deviation.
Gets the variable's median.
Gets the variable's outer fences range.
Gets the variable's inner fence range.
Gets the variable's interquartile range.
Gets the variable's mode.
Gets the variable's variance.
Gets the variable's skewness.
Gets the variable's kurtosis.
Gets the variable's standard error of the mean.
Gets the variable's maximum value.
Gets the variable's minimum value.
Gets the variable's length.
Gets the number of distinct values for the variable.
Gets the number of samples for the variable.
Gets the 95% confidence interval around the .
Gets the 95% deviance interval around the .
Gets the variable's observations.
Gets a confidence interval for the
within the given confidence level percentage.
The confidence level. Default is 0.95.
A confidence interval for the estimated value.
Gets a deviance interval for the
within the given confidence level percentage (i.e. uses
the standard deviation rather than the standard error to
compute the range interval for the variable).
The confidence level. Default is 0.95.
A confidence interval for the estimated value.
Common interface for projective statistical analysis.
Projects new data into latent space.
Projects new data into latent space with
given number of dimensions.
Common interface for multivariate regression analysis.
Regression analysis attempt to express many numerical dependent
variables as a combinations of other features or measurements.
Gets the dependent variables' values
for each of the source input points.
Common interface for regression analysis.
Regression analysis attempt to express one numerical dependent variable
as a combinations of other features or measurements.
When the dependent variable is a category label, the class of analysis methods
is known as discriminant analysis.
Gets the dependent variable value
for each of the source input points.
Common interface for discriminant analysis.
Discriminant analysis attempt to express one categorical dependent variable
as a combinations of other features or measurements.
When the dependent variable is a numerical quantity, the class of analysis methods
is known as regression analysis.
Gets the classification labels (the dependent variable)
for each of the source input points.
Exponential contrast function.
According to Hyvärinen, the Exponential contrast function may be
used when the independent components are highly super-Gaussian or
when robustness is very important.
Initializes a new instance of the class.
The exponential alpha constant. Default is 1.
Gets the exponential alpha constant.
Initializes a new instance of the class.
Contrast function.
The vector of observations.
At method's return, this parameter
should contain the evaluation of function over the vector
of observations .
At method's return, this parameter
should contain the evaluation of function derivative over
the vector of observations .
Common interface for contrast functions.
Contrast functions are used as objective functions in
neg-entropy calculations.
Contrast function.
The vector of observations.
At method's return, this parameter
should contain the evaluation of function over the vector
of observations .
At method's return, this parameter
should contain the evaluation of function derivative over
the vector of observations .
Kurtosis contrast function.
According to Hyvärinen, the kurtosis contrast function is
justified on statistical grounds only for estimating sub-Gaussian
independent components when there are no outliers.
Initializes a new instance of the class.
Contrast function.
The vector of observations.
At method's return, this parameter
should contain the evaluation of function over the vector
of observations .
At method's return, this parameter
should contain the evaluation of function derivative over
the vector of observations .
Log-cosh (Hyperbolic Tangent) contrast function.
According to Hyvärinen, the Logcosh contrast function
is a good general-purpose contrast function.
Initializes a new instance of the class.
Initializes a new instance of the class.
The log-cosh alpha constant. Default is 1.
Gets the exponential log-cosh constant.
Contrast function.
The vector of observations.
At method's return, this parameter
should contain the evaluation of function over the vector
of observations .
At method's return, this parameter
should contain the evaluation of function derivative over
the vector of observations .
Descriptive statistics analysis for circular data.
Constructs the Circular Descriptive Analysis.
The source data to perform analysis.
The length of each circular variable (i.e. 24 for hours).
The names for the analyzed variable.
Whether the analysis should conserve memory by doing
operations over the original array.
Constructs the Circular Descriptive Analysis.
The source data to perform analysis.
The length of each circular variable (i.e. 24 for hours).
Whether the analysis should conserve memory by doing
operations over the original array.
Constructs the Circular Descriptive Analysis.
The source data to perform analysis.
The length of each circular variable (i.e. 24 for hours).
Constructs the Circular Descriptive Analysis.
The source data to perform analysis.
The length of each circular variable (i.e. 24 for hours).
Names for the analyzed variables.
Constructs the Circular Descriptive Analysis.
The source data to perform analysis.
The length of each circular variable (i.e. 24 for hours).
Constructs the Circular Descriptive Analysis.
The source data to perform analysis.
The length of each circular variable (i.e. 24 for hours).
Names for the analyzed variables.
Constructs the Circular Descriptive Analysis.
The length of each circular variable (i.e. 24 for hours).
Names for the analyzed variables.
Constructs the Circular Descriptive Analysis.
The length of each circular variable (i.e. 24 for hours).
Computes the analysis using given source data and parameters.
Learns a model that can map the given inputs to the desired outputs.
The model inputs.
A model that has learned how to produce suitable outputs
given the input data .
Gets or sets whether all reported statistics should respect the circular
interval. For example, setting this property to false would allow
the , ,
and properties report minimum and maximum values
outside the variable's allowed circular range. Default is true.
Gets the source matrix from which the analysis was run.
Gets the source matrix from which the analysis was run.
Gets or sets the method to be used when computing quantiles (median and quartiles).
The quantile method.
Gets or sets whether the properties of this class should
be computed only when necessary. If set to true, a copy
of the input data will be maintained inside an instance
of this class, using more memory.
Gets the source matrix from which the analysis was run.
Gets the column names from the variables in the data.
Gets a vector containing the length of
the circular domain for each data column.
Gets a vector containing the Mean of each data column.
Gets a vector containing the Mode of each data column.
Gets a vector containing the Standard Deviation of each data column.
Gets a vector containing the Standard Error of the Mean of each data column.
Gets the 95% confidence intervals for the .
Gets the 95% deviance intervals for the .
A deviance interval uses the standard deviation rather
than the standard error to compute the range interval
for a variable.
Gets a vector containing the Median of each data column.
Gets a vector containing the Variance of each data column.
Gets a vector containing the number of distinct elements for each data column.
Gets an array containing the Ranges of each data column.
Gets an array containing the interquartile range of each data column.
Gets an array containing the inner fences of each data column.
Gets an array containing the outer fences of each data column.
Gets an array containing the sum of each data column. If
the analysis has been computed in place, this will contain
the sum of the transformed angle values instead.
Gets an array containing the sum of cosines for each data column.
Gets an array containing the sum of sines for each data column.
Gets an array containing the circular concentration for each data column.
Gets an array containing the skewness for of each data column.
Gets an array containing the kurtosis for of each data column.
Gets the number of samples (or observations) in the data.
Gets the number of variables (or features) in the data.
Gets a collection of DescriptiveMeasures objects that can be bound to a DataGridView.
Gets a confidence interval for the
within the given confidence level percentage.
The confidence level. Default is 0.95.
The index of the data column whose confidence
interval should be calculated.
A confidence interval for the estimated value.
Gets a deviance interval for the
within the given confidence level percentage (i.e. uses
the standard deviation rather than the standard error to
compute the range interval for the variable).
The confidence level. Default is 0.95.
The index of the data column whose confidence
interval should be calculated.
A confidence interval for the estimated value.
Circular descriptive measures for a variable.
Gets the circular analysis
that originated this measure.
Gets the variable's index.
Gets the variable's name
Gets the variable's total sum.
Gets the variable's mean.
Gets the variable's standard deviation.
Gets the variable's median.
Gets the variable's mode.
Gets the variable's outer fences range.
Gets the variable's inner fence range.
Gets the variable's interquartile range.
Gets the variable's variance.
Gets the variable's maximum value.
Gets the variable's minimum value.
Gets the variable's length.
Gets the number of distinct values for the variable.
Gets the number of samples for the variable.
Gets the sum of cosines for the variable.
Gets the sum of sines for the variable.
Gets the transformed variable's observations.
Gets the variable's standard error of the mean.
Gets the 95% confidence interval around the .
Gets the 95% deviance interval around the .
Gets the variable's observations.
Gets the variable skewness.
Gets the variable kurtosis.
Gets a confidence interval for the
within the given confidence level percentage.
The confidence level. Default is 0.95.
A confidence interval for the estimated value.
Gets a deviance interval for the
within the given confidence level percentage (i.e. uses
the standard deviation rather than the standard error to
compute the range interval for the variable).
The confidence level. Default is 0.95.
A confidence interval for the estimated value.
Collection of descriptive measures.
Gets the key for item.
Distribution fitness analysis.
The distribution analysis class can be used to perform a battery
of distribution fitting tests in order to check from which distribution
a sample is more likely to have come from.
-
Wikipedia contributors. "Multinomial logistic regression." Wikipedia, The Free Encyclopedia, 1st April, 2015.
Available at: https://en.wikipedia.org/wiki/Multinomial_logistic_regression
The first example shows how to reproduce a textbook example using categorical and categorical-with-baseline
variables. Those variables can be transformed/factored to their respective representations using the
class. However, please note that while this example uses features from the
class, the use of this class is not required when learning a
modoel.
-
R. G. Congalton. A Review of Assessing the Accuracy of Classifications
of Remotely Sensed Data. Available on: http://uwf.edu/zhu/evr6930/2.pdf
-
G. Banko. A Review of Assessing the Accuracy of Classifications of Remotely Sensed Data and
of Methods Including Remote Sensing Data in Forest Inventory. Interim report. Available on:
http://www.iiasa.ac.at/Admin/PUB/Documents/IR-98-081.pdf
Gets the Weights matrix.
Creates a new Confusion Matrix.
Creates a new Confusion Matrix.
Creates a new Confusion Matrix.
Gets the row marginals (proportions).
Gets the column marginals (proportions).
Gets the Kappa coefficient of performance.
Gets the standard error of the
coefficient of performance.
Gets the variance of the
coefficient of performance.
Gets the variance of the
under the null hypothesis that the underlying
Kappa value is 0.
Gets the standard error of the
under the null hypothesis that the underlying Kappa
value is 0.
Overall agreement.
Chance agreement.
The chance agreement tells how many samples
were correctly classified by chance alone.
Creates a new Weighted Confusion Matrix with linear weighting.
Creates a new Weighted Confusion Matrix with linear weighting.
Class to perform a Procrustes Analysis
Procrustes analysis is a form of statistical shape analysis used to
analyze the distribution of a set of shapes. It allows to compare shapes (datasets) that have
different rotations, scales and positions. It defines a measure called Procrustes distance that
is an image of how different the shapes are.
References:
-
Wikipedia contributors. "Procrustes analysis" Wikipedia, The Free Encyclopedia, 21 Sept. 2015.
Available at: https://en.wikipedia.org/wiki/Procrustes_analysis
-
Amy Ross. "Procrustes Analysis"
Available at :
// This examples shows how to use the Procrustes Analysis on basic shapes
// We're about to demonstrate that a diamond with four identical edges is also a square !
// Define a square
double[,] square = { { 100, 100 },
{ 300, 100 },
{ 300, 300 },
{ 100, 300 }
};
// Define a diamond with different orientation and scale
double[,] diamond = { { 170, 120 },
{ 220, 170 },
{ 270, 120 },
{ 220, 70 }
};
// Create the Procrustes analysis object
ProcrustesAnalysis pa = new ProcrustesAnalysis(square, diamond);
// Compute the analysis on the square and the diamond
pa.Compute();
// Assert that the diamond is a square
Debug.Assert(pa.ProcrustesDistances[0, 1].IsEqual(0.0, 1E-11));
// Transform the diamond to a square
double[,] diamond_to_a_square = pa.ProcrustedDatasets[1].Transform(pa.ProcrustedDatasets[0]);
Creates a Procrustes Analysis object using the given sample data
Data containing multiple dataset to run the analysis on
Creates an empty Procrustes Analysis object
Source data given to run the analysis
Applies the translation operator to translate the dataset to the zero coordinate
The dataset to translate
The translated dataset
Applies the translation operator to translate the dataset to the given coordinate
Dataset to translate
New center of the dataset
The translated dataset
Applies the scale operator to scale the data to the unitary scale
Dataset to scale
The scaled dataset
Calculates the scale of the given dataset
Dataset to find the scale
The scale of the dataset
Applies the scale operator to scale the data to the given scale
Dataset to scale
Final scale of the output dataset
Scaled dataset
Applies the rotation operator to the given dataset according to the reference dataset
Procrusted dataset to rotate
Reference procrusted dataset
The rotated dataset
Updates the Procrustes Distances according to a set of Procrusted samples
Procrusted samples
Calculate the Procrustes Distance between two sets of data
First data set
Second data set
The Procrustes distance
Procrustes Distances of the computed samples
Procrusted models produced from the computed sample data
Apply Procrustes translation and scale to the given dataset
Procrusted dataset to process and store the results to
The dataset itself
Compute the Procrustes analysis to extract Procrustes distances and models using the constructor parameters
Compute the Procrustes analysis to extract Procrustes distances and models
List of sample data sets to analyze
Procrustes distances of the analyzed samples
Compute the Procrustes analysis to extract Procrustes distances and models by specifying the reference dataset
Index of the reference dataset. If out of bounds of the sample array, the first dataset is used.
List of sample data sets to analyze
Procrustes distances of the analyzed samples
Class to represent an original dataset, its Procrustes form and all necessary data (i.e. rotation, center, scale...)
Original dataset
Procrustes dataset (i.e. original dataset after Procrustes analysis)
Original dataset center
Original dataset scale
Original dataset rotation matrix
Transforms the dataset to match the given reference original dataset
Dataset to match
The transformed dataset matched to the reference
Cox's Proportional Hazards Survival Analysis.
Proportional hazards models are a class of survival models in statistics. Survival models
relate the time that passes before some event occurs to one or more covariates that may be
associated with that quantity. In a proportional hazards model, the unique effect of a unit
increase in a covariate is multiplicative with respect to the hazard rate.
For example, taking a drug may halve one's hazard rate for a stroke occurring, or, changing
the material from which a manufactured component is constructed may double its hazard rate
for failure. Other types of survival models such as accelerated failure time models do not
exhibit proportional hazards. These models could describe a situation such as a drug that
reduces a subject's immediate risk of having a stroke, but where there is no reduction in
the hazard rate after one year for subjects who do not have a stroke in the first year of
analysis.
This class uses the to extract more detailed
information about a given problem, such as confidence intervals, hypothesis tests
and performance measures.
This class can also be bound to standard controls such as the
DataGridView
by setting their DataSource property to the analysis' property.
-
Wikipedia contributors. "Linear regression." Wikipedia, The Free Encyclopedia, 4 Nov. 2012.
Available at: http://en.wikipedia.org/wiki/Linear_regression
The first example shows how to learn a multiple linear regression analysis
from a dataset already given in matricial form (using jagged double[][] arrays).
// Now we can show a summary of analysis
// Accord.Controls.DataGridBox.Show(regression.Coefficients);
// We can also show a summary ANOVA
DataGridBox.Show(regression.Table);
-
R. G. Congalton. A Review of Assessing the Accuracy of Classifications
of Remotely Sensed Data. Available on: http://uwf.edu/zhu/evr6930/2.pdf
-
G. Banko. A Review of Assessing the Accuracy of Classifications of Remotely Sensed Data and
of Methods Including Remote Sensing Data in Forest Inventory. Interim report. Available on:
http://www.iiasa.ac.at/Admin/PUB/Documents/IR-98-081.pdf
The first example shows how a confusion matrix can be constructed from a vector of expected (ground-truth)
values and their associated predictions (as done by a test, procedure or machine learning classifier):
-
R. G. Congalton. A Review of Assessing the Accuracy of Classifications
of Remotely Sensed Data. Available on: http://uwf.edu/zhu/evr6930/2.pdf
-
G. Banko. A Review of Assessing the Accuracy of Classifications of Remotely Sensed Data and
of Methods Including Remote Sensing Data in Forest Inventory. Interim report. Available on:
http://www.iiasa.ac.at/Admin/PUB/Documents/IR-98-081.pdf
The first example shows how a confusion matrix can be constructed from a vector of expected (ground-truth)
values and their associated predictions (as done by a test, procedure or machine learning classifier):
-
http://en.wikipedia.org/wiki/Kendall_tau_rank_correlation_coefficient
-
LEVADA, Alexandre Luis Magalhães. Combinação de modelos de campos aleatórios markovianos
para classificação contextual de imagens multiespectrais [online]. São Carlos : Instituto
de Física de São Carlos, Universidade de São Paulo, 2010. Tese de Doutorado em Física Aplicada.
Disponível em: http://www.teses.usp.br/teses/disponiveis/76/76132/tde-11052010-165642/.
-
MA, Z.; REDMOND, R. L. Tau coefficients for accuracy assessment of
classification of remote sensing data.
Phi coefficient.
The Pearson correlation coefficient (phi) ranges from −1 to +1, where
a value of +1 indicates perfect agreement, a value of -1 indicates perfect
disagreement and a value 0 indicates no agreement or relationship.
References:
http://en.wikipedia.org/wiki/Phi_coefficient,
http://www.psychstat.missouristate.edu/introbook/sbk28m.htm
Gets the Chi-Square statistic for the contingency table.
Tschuprow's T association measure.
Tschuprow's T is a measure of association between two nominal variables, giving
a value between 0 and 1 (inclusive). It is closely related to
Cramér's V, coinciding with it for square contingency tables.
References:
http://en.wikipedia.org/wiki/Tschuprow's_T
Pearson's contingency coefficient C.
Pearson's C measures the degree of association between the two variables. However,
C suffers from the disadvantage that it does not reach a maximum of 1 or the minimum
of -1; the highest it can reach in a 2 x 2 table is .707; the maximum it can reach in
a 4 × 4 table is 0.870. It can reach values closer to 1 in contingency tables with more
categories. It should, therefore, not be used to compare associations among tables with
different numbers of categories. For a improved version of C, see .
References:
http://en.wikipedia.org/wiki/Contingency_table
Sakoda's contingency coefficient V.
Sakoda's V is an adjusted version of Pearson's C
so it reaches a maximum of 1 when there is complete association in a table
of any number of rows and columns.
References:
http://en.wikipedia.org/wiki/Contingency_table
Cramer's V association measure.
Cramér's V varies from 0 (corresponding to no association between the variables)
to 1 (complete association) and can reach 1 only when the two variables are equal
to each other. In practice, a value of 0.1 already provides a good indication that
there is substantive relationship between the two variables.
References:
http://en.wikipedia.org/wiki/Cram%C3%A9r%27s_V,
http://www.acastat.com/Statbook/chisqassoc.htm
Overall agreement.
The overall agreement is the sum of the diagonal elements
of the contingency table divided by the number of samples.
Accuracy. This is the same value as .
The accuracy, or .
Error. This is the same value as 1.0 - .
The average error, or 1.0 - .
Geometric agreement.
The geometric agreement is the geometric mean of the
diagonal elements of the confusion matrix.
Chance agreement.
The chance agreement tells how many samples
were correctly classified by chance alone.
Expected values, or values that could
have been generated just by chance.
Gets binary confusion matrices for each class in the multi-class
classification problem. You can use this property to obtain recall,
precision and other metrics for each of the classes.
Combines several confusion matrices into one single matrix.
The matrices to combine.
Estimates a directly from a classifier, a set of inputs and its expected outputs.
The type of the inputs accepted by the classifier.
The classifier.
The input vectors.
The expected outputs associated with each input vector.
A capturing the performance of the classifier when
trying to predict the outputs from the .
Estimates a directly from a classifier, a set of inputs and its expected outputs.
The type of the inputs accepted by the classifier.
The classifier.
The input vectors.
The expected outputs associated with each input vector.
A capturing the performance of the classifier when
trying to predict the outputs from the .
Descriptive statistics analysis.
Descriptive statistics are used to describe the basic features of the data
in a study. They provide simple summaries about the sample and the measures.
Together with simple graphics analysis, they form the basis of virtually
every quantitative analysis of data.
This class can also be bound to standard controls such as the
DataGridView
by setting their DataSource property to the analysis' property.
References:
-
Wikipedia, The Free Encyclopedia. Descriptive Statistics. Available on:
http://en.wikipedia.org/wiki/Descriptive_statistics
-
Hyvärinen, A (1999). Fast and Robust Fixed-Point Algorithms for Independent Component
Analysis. IEEE Transactions on Neural Networks, 10(3),626-634. Available on:
http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.50.4731
-
E. Bingham and A. Hyvärinen A fast fixed-point algorithm for independent component
analysis of complex-valued signals. Int. J. of Neural Systems, 10(1):1-8, 2000.
-
FastICA: FastICA Algorithms to perform ICA and Projection Pursuit. Available on:
http://cran.r-project.org/web/packages/fastICA/index.html
-
Wikipedia, The Free Encyclopedia. Independent component analysis. Available on:
http://en.wikipedia.org/wiki/Independent_component_analysis
-
Mika et al, Fisher discriminant analysis with kernels (1999). Available on
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.35.9904
-
Heiko Hoffmann, Unsupervised Learning of Visuomotor Associations (Kernel PCA topic).
PhD thesis. 2005. Available on: http://www.heikohoffmann.de/htmlthesis/hoffmann_diss.html
-
James T. Kwok, Ivor W. Tsang. The Pre-Image Problem in Kernel Methods. 2003. Available on:
http://www.hpl.hp.com/conferences/icml2003/papers/345.pdf
The example below shows a typical usage of the analysis. We will be replicating
the exact same example which can be found on the
documentation page. However, while we will be using a kernel,
any other kernel function could have been used.
- The sample size shall exceed the number of variables, and
- Classes may overlap, but their centers shall be distant from each other.
Moreover, LDA requires the following assumptions to be true:
- Independent subjects.
- Normality: the variance-covariance matrix of the
predictors is the same in all groups.
If the latter assumption is violated, it is common to use quadratic discriminant analysis in
the same manner as linear discriminant analysis instead.
This class can also be bound to standard controls such as the
DataGridView
by setting their DataSource property to the analysis' property.
References:
-
R. Gutierrez-Osuna, Linear Discriminant Analysis. Available on:
http://research.cs.tamu.edu/prism/lectures/pr/pr_l10.pdf
-
E. F. Connor. Logistic Regression. Available on:
http://userwww.sfsu.edu/~efc/classes/biol710/logistic/logisticreg.htm
-
C. Shalizi. Logistic Regression and Newton's Method. Lecture notes. Available on:
http://www.stat.cmu.edu/~cshalizi/350/lectures/26/lecture-26.pdf
-
A. Storkey. Learning from Data: Learning Logistic Regressors. Available on:
http://www.inf.ed.ac.uk/teaching/courses/lfd/lectures/logisticlearn-print.pdf
The following example shows to create a Logistic regresion analysis using a full
dataset composed of input vectors and a binary output vector. Each input vector
has an associated label (1 or 0) in the output vector, where 1 represents a positive
label (yes, or true) and 0 represents a negative label (no, or false).
-
Abdi, H. (2010). Partial least square regression, projection on latent structure regression,
PLS-Regression. Wiley Interdisciplinary Reviews: Computational Statistics, 2, 97-106.
Available in: http://www.utdallas.edu/~herve/abdi-wireCS-PLS2010.pdf
-
Abdi, H. (2007). Partial least square regression (PLS regression). In N.J. Salkind (Ed.):
Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage. pp. 740-744.
Resource available online in: http://www.utdallas.edu/~herve/Abdi-PLS-pretty.pdf
-
Martin Anderson, "A comparison of nine PLS1 algorithms". Available on:
http://onlinelibrary.wiley.com/doi/10.1002/cem.1248/pdf
-
Mevik, B-H. Wehrens, R. (2007). The pls Package: Principal Component and Partial Least Squares
Regression in R. Journal of Statistical Software, Volume 18, Issue 2.
Resource available online in: http://www.jstatsoft.org/v18/i02/paper
-
Garson, D. Partial Least Squares Regression (PLS).
http://faculty.chass.ncsu.edu/garson/PA765/pls.htm
-
De Jong, S. (1993). SIMPLS: an alternative approach to partial least squares regression.
Chemometrics and Intelligent Laboratory Systems, 18: 251–263.
http://dx.doi.org/10.1016/0169-7439(93)85002-X
-
Rosipal, Roman and Nicole Kramer. (2006). Overview and Recent Advances in Partial Least
Squares, in Subspace, Latent Structure and Feature Selection Techniques, pp 34–51.
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.85.7735
-
Yi Cao. (2008). Partial Least-Squares and Discriminant Analysis: A tutorial and tool
using PLS for discriminant analysis.
-
Wikipedia contributors. Partial least squares regression. Wikipedia, The Free Encyclopedia;
2009. Available from: http://en.wikipedia.org/wiki/Partial_least_squares_regression.
-
Abdi, H. (2010). Partial least square regression, projection on latent structure regression,
PLS-Regression. Wiley Interdisciplinary Reviews: Computational Statistics, 2, 97-106.
Available in: http://www.utdallas.edu/~herve/abdi-wireCS-PLS2010.pdf
-
Yi Cao. (2008). Partial Least-Squares and Discriminant Analysis: A tutorial and tool
using PLS for discriminant analysis.
Computes PLS parameters using SIMPLS algorithm.
The number of factors to compute.
The mean-centered input values X.
The mean-centered output values Y.
The algorithm implementation is based on the appendix code by Martin Anderson,
with modifications for multiple output variables as given in the sources listed
below.
References:
-
Martin Anderson, "A comparison of nine PLS1 algorithms". Available on:
http://onlinelibrary.wiley.com/doi/10.1002/cem.1248/pdf
-
Abdi, H. (2010). Partial least square regression, projection on latent structure regression,
PLS-Regression. Wiley Interdisciplinary Reviews: Computational Statistics, 2, 97-106.
Available from: http://www.utdallas.edu/~herve/abdi-wireCS-PLS2010.pdf
-
StatSoft, Inc. (2012). Electronic Statistics Textbook: Partial Least Squares (PLS).
Tulsa, OK: StatSoft. Available from: http://www.statsoft.com/textbook/partial-least-squares/#SIMPLS
-
De Jong, S. (1993). SIMPLS: an alternative approach to partial least squares regression.
Chemometrics and Intelligent Laboratory Systems, 18: 251–263.
http://dx.doi.org/10.1016/0169-7439(93)85002-X
Adjusts a data matrix, centering and standardizing its values
using the already computed column's means and standard deviations.
Adjusts a data matrix, centering and standardizing its values
using the already computed column's means and standard deviations.
Returns the index for the column with largest squared sum.
Computes the variable importance in projection (VIP).
A predictor factors matrix in which each row represents
the importance of the variable in a projection considering
the number of factors indicated by the column's index.
References:
-
Il-Gyo Chong, Chi-Hyuck Jun, Performance of some variable selection methods
when multicollinearity is present, Chemometrics and Intelligent Laboratory
Systems, Volume 78, Issues 1-2, 28 July 2005, Pages 103-112, ISSN 0169-7439,
DOI: 10.1016/j.chemolab.2004.12.011.
Represents a Partial Least Squares Factor found in the Partial Least Squares
Analysis, allowing it to be directly bound to controls like the DataGridView.
Creates a partial least squares factor representation.
The analysis to which this component belongs.
The component index.
Gets the Index of this component on the original factor collection.
Returns a reference to the parent analysis object.
Gets the proportion of prediction variables
variance explained by this factor.
Gets the cumulative proportion of dependent variables
variance explained by this factor.
Gets the proportion of dependent variable
variance explained by this factor.
Gets the cumulative proportion of dependent variable
variance explained by this factor.
Gets the input variable's latent vectors for this factor.
Gets the output variable's latent vectors for this factor.
Gets the importance of each variable for the given component.
Gets the proportion, or amount of information explained by this component.
Gets the cumulative proportion of all discriminants until this component.
Represents a Collection of Partial Least Squares Factors found in
the Partial Least Squares Analysis. This class cannot be instantiated.
Represents source variables used in Partial Least Squares Analysis. Can represent either
input variables (predictor variables) or output variables (independent variables or regressors).
Source data used in the analysis. Can be either input data
or output data depending if the variables chosen are predictor
variables or dependent variables, respectively.
Gets the resulting projection (scores) of the source data
into latent space. Can be either from input data or output
data depending if the variables chosen are predictor variables
or dependent variables, respectively.
Gets the column means of the source data. Can be either from
input data or output data, depending if the variables chosen
are predictor variables or dependent variables, respectively.
Gets the column standard deviations of the source data. Can be either
from input data or output data, depending if the variables chosen are
predictor variables or dependent variables, respectively.
Gets the loadings (a.k.a factors or components) for the
variables obtained during the analysis. Can be either from
input data or output data, depending if the variables chosen
are predictor variables or dependent variables, respectively.
Gets the amount of variance explained by each latent factor.
Can be either by input variables' latent factors or output
variables' latent factors, depending if the variables chosen
are predictor variables or dependent variables, respectively.
Gets the cumulative variance explained by each latent factor.
Can be either by input variables' latent factors or output
variables' latent factors, depending if the variables chosen
are predictor variables or dependent variables, respectively.
Projects a given dataset into latent space. Can be either input variable's
latent space or output variable's latent space, depending if the variables
chosen are predictor variables or dependent variables, respectively.
Projects a given dataset into latent space. Can be either input variable's
latent space or output variable's latent space, depending if the variables
chosen are predictor variables or dependent variables, respectively.
Principal component analysis (PCA) is a technique used to reduce
multidimensional data sets to lower dimensions for analysis.
Principal Components Analysis or the Karhunen-Loève expansion is a
classical method for dimensionality reduction or exploratory data
analysis.
Mathematically, PCA is a process that decomposes the covariance matrix of a matrix
into two parts: Eigenvalues and column eigenvectors, whereas Singular Value Decomposition
(SVD) decomposes a matrix per se into three parts: singular values, column eigenvectors,
and row eigenvectors. The relationships between PCA and SVD lie in that the eigenvalues
are the square of the singular values and the column vectors are the same for both.
This class uses SVD on the data set which generally gives better numerical accuracy.
This class can also be bound to standard controls such as the
DataGridView
by setting their DataSource property to the analysis' property.
The example below shows a typical usage of the analysis. However, users
often ask why the framework produces different values than other packages
such as STATA or MATLAB. After the simple introductory example below, we
will be exploring why those results are often different.
-
Wikipedia, The Free Encyclopedia. Receiver Operating Characteristic. Available on:
http://en.wikipedia.org/wiki/Receiver_operating_characteristic
-
Anaesthesist. The magnificent ROC. Available on:
http://www.anaesthetist.com/mnm/stats/roc/Findex.htm
The following example shows how to measure the accuracy
of a binary classifier using a ROC curve.
// This example shows how to measure the accuracy of a
// binary classifier using a ROC curve. For this example,
// we will be creating a Support Vector Machine trained
// on the following training instances:
double[][] inputs =
{
// Those are from class -1
new double[] { 2, 4, 0 },
new double[] { 5, 5, 1 },
new double[] { 4, 5, 0 },
new double[] { 2, 5, 5 },
new double[] { 4, 5, 1 },
new double[] { 4, 5, 0 },
new double[] { 6, 2, 0 },
new double[] { 4, 1, 0 },
// Those are from class +1
new double[] { 1, 4, 5 },
new double[] { 7, 5, 1 },
new double[] { 2, 6, 0 },
new double[] { 7, 4, 7 },
new double[] { 4, 5, 0 },
new double[] { 6, 2, 9 },
new double[] { 4, 1, 6 },
new double[] { 7, 2, 9 },
};
int[] outputs =
{
-1, -1, -1, -1, -1, -1, -1, -1, // fist eight from class -1
+1, +1, +1, +1, +1, +1, +1, +1 // last eight from class +1
};
// Next, we create a linear Support Vector Machine with 4 inputs
SupportVectorMachine machine = new SupportVectorMachine(inputs: 3);
// Create the sequential minimal optimization learning algorithm
var smo = new SequentialMinimalOptimization(machine, inputs, outputs);
// We learn the machine
double error = smo.Run();
// And then extract its predicted labels
double[] predicted = new double[inputs.Length];
for (int i = 0; i < predicted.Length; i++)
predicted[i] = machine.Compute(inputs[i]);
// At this point, the output vector contains the labels which
// should have been assigned by the machine, and the predicted
// vector contains the labels which have been actually assigned.
// Create a new ROC curve to assess the performance of the model
var roc = new ReceiverOperatingCharacteristic(outputs, predicted);
roc.Compute(100); // Compute a ROC curve with 100 cut-off points
// Generate a connected scatter plot for the ROC curve and show it on-screen
ScatterplotBox.Show(roc.GetScatterplot(includeRandom: true), nonBlocking: true)
.SetSymbolSize(0) // do not display data points
.SetLinesVisible(true) // show lines connecting points
.SetScaleTight(true) // tighten the scale to points
.WaitForClose();
The resulting graph is shown below.
Constructs a new Receiver Operating Characteristic model
An array of binary values. Typically represented as 0 and 1, or -1 and 1,
indicating negative and positive cases, respectively. The maximum value
will be treated as the positive case, and the lowest as the negative.
An array of continuous values trying to approximate the measurement array.
Constructs a new Receiver Operating Characteristic model
An array of binary values. Typically represented as 0 and 1, or -1 and 1,
indicating negative and positive cases, respectively. The maximum value
will be treated as the positive case, and the lowest as the negative.
An array of continuous values trying to approximate the measurement array.
Constructs a new Receiver Operating Characteristic model
An array of binary values. Typically represented as 0 and 1, or -1 and 1,
indicating negative and positive cases, respectively. The maximum value
will be treated as the positive case, and the lowest as the negative.
An array of continuous values trying to approximate the measurement array.
Gets the points of the curve.
Gets the number of actual positive cases.
Gets the number of actual negative cases.
Gets the number of cases (observations) being analyzed.
Gets the area under this curve (AUC).
Gets the standard error for the .
Gets the variance of the curve's .
Gets the ground truth values, or the values
which should have been given by the test if
it was perfect.
Gets the actual values given by the test.
Gets the actual test results for subjects
which should have been labeled as positive.
Gets the actual test results for subjects
which should have been labeled as negative.
Gets DeLong's pseudoaccuracies for the positive subjects.
Gets DeLong's pseudoaccuracies for the negative subjects
Computes a n-points ROC curve.
Each point in the ROC curve will have a threshold increase of
1/npoints over the previous point, starting at zero.
The number of points for the curve.
Computes a ROC curve with 1/increment points
The increment over the previous point for each point in the curve.
Computes a ROC curve with 1/increment points
The increment over the previous point for each point in the curve.
True to force the inclusion of the (0,0) point, false otherwise. Default is false.
Computes a ROC curve with the given increment points
Computes a single point of a ROC curve using the given cutoff value.
Generates a representing the ROC curve.
True to include a plot of the random curve (a diagonal line
going from lower left to upper right); false otherwise.
Returns a that represents this curve.
A that represents this curve.
Calculates the area under the ROC curve using the trapezium method.
The area under a ROC curve can never be less than 0.50. If the area is first calculated as
less than 0.50, the definition of abnormal will be reversed from a higher test value to a
lower test value.
Saves the curve to a stream.
The stream to which the curve is to be serialized.
Loads a curve from a stream.
The stream from which the curve is to be deserialized.
The deserialized curve.
Loads a curve from a file.
The path to the file from which the curve is to be deserialized.
The deserialized curve.
Saves the curve to a stream.
The path to the file to which the curve is to be serialized.
Object to hold information about a Receiver Operating Characteristic Curve Point
Constructs a new Receiver Operating Characteristic point.
Gets the cutoff value (discrimination threshold) for this point.
Returns a System.String that represents the current ReceiverOperatingCharacteristicPoint.
Represents a Collection of Receiver Operating Characteristic (ROC) Curve points.
This class cannot be instantiated.
Gets the (1-specificity, sensitivity) values as (x,y) coordinates.
An jagged double array where each element is a double[] vector
with two positions; the first is the value for 1-specificity (x)
and the second the value for sensitivity (y).
Gets an array containing (1-specificity)
values for each point in the curve.
Gets an array containing (sensitivity)
values for each point in the curve.
Returns a that represents this instance.
A that represents this instance.
Backward Stepwise Logistic Regression Analysis.
The Backward Stepwise regression is an exploratory analysis procedure,
where the analysis begins with a full (saturated) model and at each step
variables are eliminated from the model in a iterative fashion.
Significance tests are performed after each removal to track which of
the variables can be discarded safely without implying in degradation.
When no more variables can be removed from the model without causing
a significant loss in the model likelihood, the method can stop.
// Suppose we have the following data about some patients.
// The first variable is continuous and represent patient
// age. The second variable is dichotomic and give whether
// they smoke or not (this is completely fictional data).
double[][] inputs =
{
// Age Smoking
new double[] { 55, 0 }, // 1
new double[] { 28, 0 }, // 2
new double[] { 65, 1 }, // 3
new double[] { 46, 0 }, // 4
new double[] { 86, 1 }, // 5
new double[] { 56, 1 }, // 6
new double[] { 85, 0 }, // 7
new double[] { 33, 0 }, // 8
new double[] { 21, 1 }, // 9
new double[] { 42, 1 }, // 10
new double[] { 33, 0 }, // 11
new double[] { 20, 1 }, // 12
new double[] { 43, 1 }, // 13
new double[] { 31, 1 }, // 14
new double[] { 22, 1 }, // 15
new double[] { 43, 1 }, // 16
new double[] { 46, 0 }, // 17
new double[] { 86, 1 }, // 18
new double[] { 56, 1 }, // 19
new double[] { 55, 0 }, // 20
};
// Additionally, we also have information about whether
// or not they those patients had lung cancer. The array
// below gives 0 for those who did not, and 1 for those
// who did.
double[] output =
{
0, 0, 0, 1, 1, 1, 0, 0, 0, 1,
0, 1, 1, 1, 1, 1, 0, 1, 1, 0
};
// Create a Stepwise Logistic Regression analysis
var regression = new StepwiseLogisticRegressionAnalysis(inputs, output,
new[] { "Age", "Smoking" }, "Cancer");
regression.Compute(); // compute the analysis.
// The full model will be stored in the complete property:
StepwiseLogisticRegressionModel full = regression.Complete;
// The best model will be stored in the current property:
StepwiseLogisticRegressionModel best = regression.Current;
// Let's check the full model results
DataGridBox.Show(full.Coefficients);
// We can see only the Smoking variable is statistically significant.
// This is an indication the Age variable could be discarded from
// the model.
// And check the best inner model result
DataGridBox.Show(best.Coefficients);
// This is the best nested model found. This model only has the
// Smoking variable, which is still significant. Since no other
// variables can be dropped, this is the best final model.
// The variables used in the current best model are
string[] inputVariableNames = best.Inputs; // Smoking
// The best model likelihood ratio p-value is
ChiSquareTest test = best.ChiSquare; // {0.816990081334823}
// so the model is distinguishable from a null model. We can also
// query the other nested models by checking the Nested property:
DataGridBox.Show(regression.Nested);
// Finally, we can also use the analysis to classify a new patient
double y = regression.Current.Regression.Compute(new double[] { 1 });
// For a smoking person, the answer probability is approximately 83%.
Gets or sets a cancellation token that can be used to
stop the learning algorithm while it is running.
Constructs a Stepwise Logistic Regression Analysis.
The input data for the analysis.
The output data for the analysis.
Constructs a Stepwise Logistic Regression Analysis.
The input data for the analysis.
The output data for the analysis.
The names for the input variables.
The name for the output variable.
Gets or sets the maximum number of iterations to be
performed by the regression algorithm. Default is 50.
Gets or sets the difference between two iterations of the regression
algorithm when the algorithm should stop. The difference is calculated
based on the largest absolute parameter change of the regression. Default
is 1e-5.
Source data used in the analysis.
Gets the dependent variable value
for each of the source input points.
Gets the resulting probabilities obtained
by the most likely logistic regression model.
Gets the current best nested model.
Gets the full model.
Gets the collection of nested models obtained after
a step of the backward stepwise procedure.
Gets or sets the name of the input variables.
Gets or sets the name of the output variables.
Gets or sets the significance threshold used to
determine if a nested model is significant or not.
Gets the final set of input variables indices
as selected by the stepwise procedure.
Learns a model that can map the given inputs to the given outputs.
The model inputs.
The desired outputs associated with each inputs.
The weight of importance for each input-output pair (if supported by the learning algorithm).
A model that has learned how to produce given .
Computes the Stepwise Logistic Regression.
Computes one step of the Stepwise Logistic Regression Analysis.
Returns the index of the variable discarded in the step or -1
in case no variable could be discarded.
Fits a logistic regression model to data until convergence.
Stepwise Logistic Regression Nested Model.
Gets information about the regression model
coefficients in a object-oriented structure.
Gets the Stepwise Logistic Regression Analysis
from which this model belongs to.
Gets the regression model.
Gets the subset of the original variables used by the model.
Gets the name of the variables used in
this model combined as a single string.
Gets the Chi-Square Likelihood Ratio test for the model.
Gets the subset of the original variables used by the model.
Gets the Odds Ratio for each coefficient
found during the logistic regression.
Gets the Standard Error for each coefficient
found during the logistic regression.
Gets the Wald Tests for each coefficient.
Gets the value of each coefficient.
Gets the 95% Confidence Intervals (C.I.)
for each coefficient found in the regression.
Gets the Likelihood-Ratio Tests for each coefficient.
Constructs a new Logistic regression model.
Stepwise Logistic Regression Nested Model collection.
This class cannot be instantiated.
Represents a Logistic Regression Coefficient found in the Logistic Regression,
allowing it to be bound to controls like the DataGridView. This class cannot
be instantiated outside the .
Gets the name for the current coefficient.
Gets the Odds ratio for the current coefficient.
Gets the Standard Error for the current coefficient.
Gets the 95% confidence interval (C.I.) for the current coefficient.
Gets the upper limit for the 95% confidence interval.
Gets the lower limit for the 95% confidence interval.
Gets the coefficient value.
Gets the Wald's test performed for this coefficient.
Gets the Likelihood-Ratio test performed for this coefficient.
Represents a collection of Logistic Coefficients found in the
. This class cannot be instantiated.
Common interface for multivariate statistical analysis.
Source data used in the analysis.
Set of statistics functions operating over a circular space.
This class represents collection of common functions used in
statistics. The values are handled as belonging to a distribution
defined over a circle, such as the .
Transforms circular data into angles (normalizes the data to be between -PI and PI).
The samples to be transformed.
The maximum possible sample value (such as 24 for hour data).
Whether to perform the transformation in place.
A double array containing the same data in ,
but normalized between -PI and PI.
Transforms circular data into angles (normalizes the data to be between -PI and PI).
The sample to be transformed.
The maximum possible sample value (such as 24 for hour data).
The normalized to be between -PI and PI.
Transforms angular data back into circular data (reverts the
transformation.
The angle to be reconverted into the original unit.
The maximum possible sample value (such as 24 for hour data).
Whether range values should be wrapped to be contained in the circle. If
set to false, range values could be returned outside the [+pi;-pi] range.
The original before being converted.
Computes the sum of cosines and sines for the given angles.
A double array containing the angles in radians.
The sum of cosines, returned as an out parameter.
The sum of sines, returned as an out parameter.
Computes the Mean direction of the given angles.
A double array containing the angles in radians.
The mean direction of the given angles.
Computes the circular Mean direction of the given circular samples.
The minimum possible value for a sample must be zero and the maximum must
be indicated in the parameter .
A double array containing the circular samples.
The maximum possible value of the samples.
The circular Mean direction of the given samples.
Computes the Mean direction of the given angles.
The number of samples.
The sum of the cosines of the samples.
The sum of the sines of the samples.
The mean direction of the given angles.
Computes the mean resultant vector length (r) of the given angles.
A double array containing the angles in radians.
The mean resultant vector length of the given angles.
Computes the resultant vector length (r) of the given circular samples.
The minimum possible value for a sample must be zero and the maximum must
be indicated in the parameter .
A double array containing the circular samples.
The maximum possible value of the samples.
The mean resultant vector length of the given samples.
Computes the mean resultant vector length (r) of the given angles.
The number of samples.
The sum of the cosines of the samples.
The sum of the sines of the samples.
The mean resultant vector length of the given angles.
Computes the circular variance of the given circular samples.
The minimum possible value for a sample must be zero and the maximum must
be indicated in the parameter .
A double array containing the circular samples.
The maximum possible value of the samples.
The circular variance of the given samples.
Computes the Variance of the given angles.
A double array containing the angles in radians.
The variance of the given angles.
Computes the Variance of the given angles.
The number of samples.
The sum of the cosines of the samples.
The sum of the sines of the samples.
The variance of the angles.
Computes the circular standard deviation of the given circular samples.
The minimum possible value for a sample must be zero and the maximum must
be indicated in the parameter .
A double array containing the circular samples.
The maximum possible value of the samples.
The circular standard deviation of the given samples.
Computes the Standard Deviation of the given angles.
A double array containing the angles in radians.
The standard deviation of the given angles.
Computes the Standard Deviation of the given angles.
The number of samples.
The sum of the cosines of the samples.
The sum of the sines of the samples.
The standard deviation of the angles.
Computes the circular angular deviation of the given circular samples.
The minimum possible value for a sample must be zero and the maximum must
be indicated in the parameter .
A double array containing the circular samples.
The maximum possible value of the samples.
The circular angular deviation of the given samples.
Computes the Angular Deviation of the given angles.
A double array containing the angles in radians.
The angular deviation of the given angles.
Computes the Angular Deviation of the given angles.
The number of samples.
The sum of the cosines of the samples.
The sum of the sines of the samples.
The angular deviation of the angles.
Computes the circular standard error of the given circular samples.
The minimum possible value for a sample must be zero and the maximum must
be indicated in the parameter .
A double array containing the circular samples.
The maximum possible value of the samples.
The confidence level. Default is 0.05.
The circular standard error of the given samples.
Computes the standard error of the given angles.
A double array containing the angles in radians.
The confidence level. Default is 0.05.
The standard error of the given angles.
Computes the standard error of the given angles.
The number of samples.
The sum of the cosines of the samples.
The sum of the sines of the samples.
The confidence level. Default is 0.05.
The standard error of the angles.
Computes the angular distance between two angles.
The first angle.
The second angle.
The distance between the two angles.
Computes the distance between two circular samples.
The first sample.
The second sample.
The maximum possible value of the samples.
The distance between the two angles.
Computes the angular distance between two angles.
The cosine of the first sample.
The sin of the first sample.
The cosine of the second sample.
The sin of the second sample.
The distance between the two angles.
Computes the circular Median of the given circular samples.
The minimum possible value for a sample must be zero and the maximum must
be indicated in the parameter .
A double array containing the circular samples.
The maximum possible value of the samples.
The circular Median of the given samples.
Computes the circular Median direction of the given angles.
A double array containing the angles in radians.
The circular Median of the given angles.
Computes the circular quartiles of the given circular samples.
The minimum possible value for a sample must be zero and the maximum must
be indicated in the parameter .
A double array containing the circular samples.
The maximum possible value of the samples.
The first quartile, as an out parameter.
The third quartile, as an out parameter.
Whether range values should be wrapped to be contained in the circle. If
set to false, range values could be returned outside the [+pi;-pi] range.
The quartile definition that should be used. See for datails.
The median of the given samples.
Computes the circular quartiles of the given circular samples.
The minimum possible value for a sample must be zero and the maximum must
be indicated in the parameter .
A double array containing the circular samples.
The maximum possible value of the samples.
The first quartile, as an out parameter.
The third quartile, as an out parameter.
The median value of the , if already known.
Whether range values should be wrapped to be contained in the circle. If
set to false, range values could be returned outside the [+pi;-pi] range.
The quartile definition that should be used. See for datails.
The median of the given samples.
Computes the circular quartiles of the given circular angles.
A double array containing the angles in radians.
The first quartile, as an out parameter.
The third quartile, as an out parameter.
Whether range values should be wrapped to be contained in the circle. If
set to false, range values could be returned outside the [+pi;-pi] range.
The quartile definition that should be used. See for datails.
The median of the given angles.
Computes the circular quartiles of the given circular samples.
The minimum possible value for a sample must be zero and the maximum must
be indicated in the parameter .
A double array containing the circular samples.
The maximum possible value of the samples.
The sample quartiles, as an out parameter.
Whether range values should be wrapped to be contained in the circle. If
set to false, range values could be returned outside the [+pi;-pi] range.
The quartile definition that should be used. See for datails.
The median of the given samples.
Computes the circular quartiles of the given circular samples.
The minimum possible value for a sample must be zero and the maximum must
be indicated in the parameter .
A double array containing the circular samples.
The maximum possible value of the samples.
The sample quartiles, as an out parameter.
The median value of the , if already known.
Whether range values should be wrapped to be contained in the circle. If
set to false, range values could be returned outside the [+pi;-pi] range.
The quartile definition that should be used. See for datails.
The median of the given samples.
Computes the circular quartiles of the given circular angles.
A double array containing the angles in radians.
The sample quartiles, as an out parameter.
Whether range values should be wrapped to be contained in the circle. If
set to false, range values could be returned outside the [+pi;-pi] range.
The quartile definition that should be used. See for datails.
The median of the given angles.
Computes the circular quartiles of the given circular angles.
A double array containing the angles in radians.
The sample quartiles, as an out parameter.
The angular median, if already known.
Whether range values should be wrapped to be contained in the circle. If
set to false, range values could be returned outside the [+pi;-pi] range.
The quartile definition that should be used. See for datails.
The median of the given angles.
Computes the circular quartiles of the given circular angles.
A double array containing the angles in radians.
The first quartile, as an out parameter.
The third quartile, as an out parameter.
The angular median, if already known.
Whether range values should be wrapped to be contained in the circle. If
set to false, range values could be returned outside the [+pi;-pi] range.
The quartile definition that should be used. See for datails.
The median of the given angles.
Computes the concentration (kappa) of the given angles.
A double array containing the angles in radians.
The concentration (kappa) parameter of the
for the given data.
Computes the concentration (kappa) of the given angles.
A double array containing the angles in radians.
The mean of the angles, if already known.
The concentration (kappa) parameter of the
for the given data.
Computes the Weighted Mean of the given angles.
A double array containing the angles in radians.
An unit vector containing the importance of each angle
in . The sum of this array elements should add up to 1.
The mean of the given angles.
Computes the Weighted Concentration of the given angles.
A double array containing the angles in radians.
An unit vector containing the importance of each angle
in . The sum of this array elements should add up to 1.
The mean of the given angles.
Computes the Weighted Concentration of the given angles.
A double array containing the angles in radians.
An unit vector containing the importance of each angle
in . The sum of this array elements should add up to 1.
The mean of the angles, if already known.
The mean of the given angles.
Computes the maximum likelihood estimate
of kappa given by Best and Fisher (1981).
This method implements the approximation to the Maximum Likelihood
Estimative of the kappa concentration parameter as suggested by Best
and Fisher (1981), cited by Zheng Sun (2006) and Hussin and Mohamed
(2008). Other useful approximations are given by Suvrit Sra (2009).
References:
-
A.G. Hussin and I.B. Mohamed, 2008. Efficient Approximation for the von Mises Concentration Parameter.
Asian Journal of Mathematics & Statistics, 1: 165-169.
-
Suvrit Sra, "A short note on parameter approximation for von Mises-Fisher distributions:
and a fast implementation of $I_s(x)$". (revision of Apr. 2009). Computational Statistics (2011).
Available on: http://www.kyb.mpg.de/publications/attachments/vmfnote_7045%5B0%5D.pdf
-
Zheng Sun. M.Sc. Comparing measures of fit for circular distributions. Master thesis, 2006.
Available on: https://dspace.library.uvic.ca:8443/bitstream/handle/1828/2698/zhengsun_master_thesis.pdf
Computes the circular skewness of the given circular angles.
A double array containing the angles in radians.
The circular skewness for the given .
Computes the circular kurtosis of the given circular angles.
A double array containing the angles in radians.
The circular kurtosis for the given .
Computes the complex circular central
moments of the given circular angles.
Computes the complex circular non-central
moments of the given circular angles.
Common interface for divergence measures (between
probability distributions).
The type of the first distribution to be compared.
The type of the second distribution to be compared.
Common interface for divergence measures (between
probability distributions).
The type of the distributions to be compared.
Contains more than 40 statistical distributions, with support
for most probability distribution measures and estimation methods.
This namespace contains a huge collection of probability distributions, ranging the from the common
and simple Normal (Gaussian) and
Poisson distributions to Inverse-Wishart and
multivariate mixture distributions, including many specialized
univariate distributions used in statistical hypothesis testing.
Some of those distributions include the , ,
, and many others.
For a complete list of all
univariate probability distributions, check the
namespace. For a complete list of all
multivariate distributions, please see the
namespace.
A list of density kernels
such as the Gaussian kernel
and the Epanechnikov kernel
are available in the namespace.
The namespace class diagram is shown below.
The namespace class diagram for univariate distributions is shown below.
The namespace class diagram for multivariate distributions is shown below.
Contains density estimation kernels which can be used in combination
with empirical distributions
and multivariate empirical
distributions.
Common interface for density estimation kernels.
Those kernels are different from kernel
functions. Density estimation kernels are required to obey normalization rules in
order to fulfill integrability and behavioral properties. Moreover, they are defined
over a single input vector, the point estimate of a random variable.
Computes the kernel density function.
The input point.
A density estimate around .
Epanechnikov density kernel.
References:
-
Comaniciu, Dorin, and Peter Meer. "Mean shift: A robust approach toward
feature space analysis." Pattern Analysis and Machine Intelligence, IEEE
Transactions on 24.5 (2002): 603-619. Available at:
http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1000236
-
Dan Styer, Oberlin College Department of Physics and Astronomy; Volume of a d-dimensional
sphere. Last updated 30 August 2007. Available at:
http://www.oberlin.edu/physics/dstyer/StatMech/VolumeDSphere.pdf
-
David W. Scott, Multivariate Density Estimation: Theory, Practice, and
Visualization, Wiley, Aug 31, 1992
The following example shows how to fit a
using Epanechnikov kernels.
-
Comaniciu, Dorin, and Peter Meer. "Mean shift: A robust approach toward
feature space analysis." Pattern Analysis and Machine Intelligence, IEEE
Transactions on 24.5 (2002): 603-619. Available at:
http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1000236
-
Dan Styer, Oberlin College Department of Physics and Astronomy; Volume of a d-dimensional
sphere. Last updated 30 August 2007. Available at:
http://www.oberlin.edu/physics/dstyer/StatMech/VolumeDSphere.pdf
-
David W. Scott, Multivariate Density Estimation: Theory, Practice, and
Visualization, Wiley, Aug 31, 1992
The following example shows how to fit a
using Gaussian kernels:
-
Comaniciu, Dorin, and Peter Meer. "Mean shift: A robust approach toward
feature space analysis." Pattern Analysis and Machine Intelligence, IEEE
Transactions on 24.5 (2002): 603-619. Available at:
http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1000236
-
Dan Styer, Oberlin College Department of Physics and Astronomy; Volume of a d-dimensional
sphere. Last updated 30 August 2007. Available at:
http://www.oberlin.edu/physics/dstyer/StatMech/VolumeDSphere.pdf
-
David W. Scott, Multivariate Density Estimation: Theory, Practice, and
Visualization, Wiley, Aug 31, 1992
The following example demonstrates how to use the Mean Shift algorithm with
a uniform kernel to solve a clustering task:
-
Wikipedia, The Free Encyclopedia. Probability distribution. Available on:
http://en.wikipedia.org/wiki/Probability_distribution
-
Weisstein, Eric W. "Statistical Distribution." From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/StatisticalDistribution.html
Constructs a new MultivariateDistribution class.
The number of rows for matrices modeled by the distribution.
The number of rows for matrices modeled by the distribution.
Gets the number of variables for this distribution.
Gets the number of rows that matrices from this distribution should have.
Gets the number of columns that matrices from this distribution should have.
Gets the mean for this distribution.
A vector containing the mean values for the distribution.
Gets the variance for this distribution.
A vector containing the variance values for the distribution.
Gets the variance-covariance matrix for this distribution.
A matrix containing the covariance values for the distribution.
Gets the mode for this distribution.
A vector containing the mode values for the distribution.
Gets the median for this distribution.
A vector containing the median values for the distribution.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
The probability of x occurring
in the current distribution.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
The probability of x occurring
in the current distribution.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
The probability of x occurring
in the current distribution.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
The probability of x occurring
in the current distribution.
Gets the log-probability density function (pdf)
for this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The logarithm of the probability of x
occurring in the current distribution.
Gets the log-probability density function (pdf)
for this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The logarithm of the probability of x
occurring in the current distribution.
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF.
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
A random observations drawn from this distribution.
Generates a random observation from the current distribution.
A random observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
A random observations drawn from this distribution.
Generates a random observation from the current distribution.
The location where to store the sample.
The random number generator to use as a source of randomness.
Default is to use .
A random observation drawn from this distribution.
Generates a random observation from the current distribution.
The location where to store the sample.
A random observation drawn from this distribution.
Generates a random observation from the current distribution.
The location where to store the sample.
The random number generator to use as a source of randomness.
Default is to use .
A random observation drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
A random vector of observations drawn from this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
The x.
System.Double.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
The x.
System.Double.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Multivariate empirical distribution.
Empirical distributions are based solely on the data. This class
uses the empirical distribution function and the Gaussian kernel
density estimation to provide an univariate continuous distribution
implementation which depends only on sampled data.
References:
-
Wikipedia, The Free Encyclopedia. Empirical Distribution Function. Available on:
http://en.wikipedia.org/wiki/Empirical_distribution_function
-
PlanetMath. Empirical Distribution Function. Available on:
http://planetmath.org/encyclopedia/EmpiricalDistributionFunction.html
-
Wikipedia, The Free Encyclopedia. Kernel Density Estimation. Available on:
http://en.wikipedia.org/wiki/Kernel_density_estimation
-
Bishop, Christopher M.; Pattern Recognition and Machine Learning.
Springer; 1st ed. 2006.
-
Buch-Kromann, T.; Nonparametric Density Estimation (Multidimension), 2007. Available in
http://www.buch-kromann.dk/tine/nonpar/Nonparametric_Density_Estimation_multidim.pdf
-
W. Härdle, M. Müller, S. Sperlich, A. Werwatz; Nonparametric and Semiparametric Models, 2004. Available
in http://sfb649.wiwi.hu-berlin.de/fedc_homepage/xplore/ebooks/html/spm/spmhtmlnode18.html
The first example shows how to fit a
using Gaussian kernels:
-
Wikipedia, The Free Encyclopedia. Inverse Wishart distribution.
Available from: http://en.wikipedia.org/wiki/Inverse-Wishart_distribution
// Create a Inverse Wishart with the parameters
var invWishart = new InverseWishartDistribution(
// Degrees of freedom
degreesOfFreedom: 4,
// Scale parameter
inverseScale: new double[,]
{
{ 1.7, -0.2 },
{ -0.2, 5.3 },
}
);
// Common measures
double[] var = invWishart.Variance; // { -3.4, -10.6 }
double[,] cov = invWishart.Covariance; // see below
double[,] mmean = invWishart.MeanMatrix; // see below
// cov mean
// -5.78 -4.56 1.7 -0.2
// -4.56 -56.18 -0.2 5.3
// (the above matrix representations have been transcribed to text using)
string scov = cov.ToString(DefaultMatrixFormatProvider.InvariantCulture);
string smean = mmean.ToString(DefaultMatrixFormatProvider.InvariantCulture);
// For compatibility reasons, .Mean stores a flattened mean matrix
double[] mean = invWishart.Mean; // { 1.7, -0.2, -0.2, 5.3 }
// Probability density functions
double pdf = invWishart.ProbabilityDensityFunction(new double[,]
{
{ 5.2, 0.2 }, // 0.000029806281690351203
{ 0.2, 4.2 },
});
double lpdf = invWishart.LogProbabilityDensityFunction(new double[,]
{
{ 5.2, 0.2 }, // -10.420791391688828
{ 0.2, 4.2 },
});
Creates a new Inverse Wishart distribution.
The degrees of freedom v.
The inverse scale matrix Ψ (psi).
Gets the mean for this distribution.
A vector containing the mean values for the distribution.
Gets the variance for this distribution.
A vector containing the variance values for the distribution.
Gets the variance-covariance matrix for this distribution.
A matrix containing the covariance values for the distribution.
Not supported.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
For a matrix distribution, such as the Wishart's, this
should be a positive-definite matrix or a matrix written
in flat vector form.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
For a matrix distribution, such as the Wishart's, this
should be a positive-definite matrix or a matrix written
in flat vector form.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Not supported.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
The format.
The format provider.
A that represents this instance.
Uniform distribution inside a n-dimensional ball.
Initializes a new instance of the class.
The number of dimensions in the n-dimensional sphere.
Initializes a new instance of the class.
The sphere's mean.
The sphere's radius.
Gets the sphere radius.
Gets the sphere volume.
Gets the sphere center (mean) vector.
A vector containing the mean values for the distribution.
Gets the variance for this distribution.
A vector containing the variance values for the distribution.
Gets the variance-covariance matrix for this distribution.
A matrix containing the covariance values for the distribution.
Not implemented.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the log-probability density function (pdf)
for this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The logarithm of the probability of x
occurring in the current distribution.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Returns a that represents this instance.
The format.
The format provider.
A that represents this instance.
Uniform distribution inside a n-dimensional ball.
Initializes a new instance of the class.
The number of dimensions in the n-dimensional sphere.
Initializes a new instance of the class.
The sphere's mean.
The sphere's radius.
Gets the sphere radius.
Gets the sphere volume.
Gets the sphere center (mean) vector.
A vector containing the mean values for the distribution.
Gets the variance for this distribution.
A vector containing the variance values for the distribution.
Gets the variance-covariance matrix for this distribution.
A matrix containing the covariance values for the distribution.
Not implemented.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the log-probability density function (pdf)
for this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The logarithm of the probability of x
occurring in the current distribution.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The sphere's mean.
The sphere's radius.
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The sphere's mean.
The sphere's radius.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The number of dimensions in the n-dimensional sphere.
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The number of dimensions in the n-dimensional sphere.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The number of dimensions in the n-dimensional sphere.
The location where to store the samples.
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The number of dimensions in the n-dimensional sphere.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Returns a that represents this instance.
The format.
The format provider.
A that represents this instance.
Von-Mises Fisher distribution.
In directional statistics, the von Mises–Fisher distribution is a probability distribution
on the (p-1)-dimensional sphere in R^p. If p = 2 the distribution reduces to the
von Mises distribution on the circle.
References:
-
Wikipedia, The Free Encyclopedia. Von Mises-Fisher Distribution. Available on:
https://en.wikipedia.org/wiki/Von_Mises%E2%80%93Fisher_distribution
Constructs a Von-Mises Fisher distribution with unit mean.
The number of dimensions in the distribution.
The concentration value κ (kappa).
Constructs a Von-Mises Fisher distribution with unit mean.
The mean direction vector (with unit length).
The concentration value κ (kappa).
Gets the mean for this distribution.
A vector containing the mean values for the distribution.
Not supported.
Not supported.
Not supported.
Gets the probability density function (pdf) for this distribution evaluated at point x.
A single point in the distribution range. For a univariate distribution, this should be
a single double value. For a multivariate distribution, this should be a double array.
The probability of x occurring in the current distribution.
x;The vector should have the same dimension as the distribution.
The Probability Density Function (PDF) describes the probability that a given value x will occur.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
The format.
The format provider.
A that represents this instance.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Wishart Distribution.
The Wishart distribution is a generalization to multiple dimensions of
the Chi-Squared distribution, or, in
the case of non-integer degrees of
freedom, of the Gamma distribution
.
References:
-
Wikipedia, The Free Encyclopedia. Wishart distribution.
Available from: http://en.wikipedia.org/wiki/Wishart_distribution
// Create a Wishart distribution with the parameters:
WishartDistribution wishart = new WishartDistribution(
// Degrees of freedom
degreesOfFreedom: 7,
// Scale parameter
scale: new double[,]
{
{ 4, 1, 1 },
{ 1, 2, 2 }, // (must be symmetric and positive definite)
{ 1, 2, 6 },
}
);
// Common measures
double[] var = wishart.Variance; // { 224, 56, 504 }
double[,] cov = wishart.Covariance; // see below
double[,] meanm = wishart.MeanMatrix; // see below
// 224 63 175 28 7 7
// cov = 63 56 112 mean = 7 14 14
// 175 112 504 7 14 42
// (the above matrix representations have been transcribed to text using)
string scov = cov.ToString(DefaultMatrixFormatProvider.InvariantCulture);
string smean = meanm.ToString(DefaultMatrixFormatProvider.InvariantCulture);
// For compatibility reasons, .Mean stores a flattened mean matrix
double[] mean = wishart.Mean; // { 28, 7, 7, 7, 14, 14, 7, 14, 42 }
// Probability density functions
double pdf = wishart.ProbabilityDensityFunction(new double[,]
{
{ 8, 3, 1 },
{ 3, 7, 1 }, // 0.000000011082455043473361
{ 1, 1, 8 },
});
double lpdf = wishart.LogProbabilityDensityFunction(new double[,]
{
{ 8, 3, 1 },
{ 3, 7, 1 }, // -18.317902605850534
{ 1, 1, 8 },
});
Creates a new Wishart distribution.
The number of rows in the covariance matrices.
The degrees of freedom n.
Creates a new Wishart distribution.
The degrees of freedom n.
The positive-definite matrix scale matrix V.
Gets the degrees of freedom for this Wishart distribution.
Gets the mean for this distribution.
A vector containing the mean values for the distribution.
Gets the variance for this distribution.
A vector containing the variance values for the distribution.
Gets the variance-covariance matrix for this distribution.
A matrix containing the covariance values for the distribution.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
For a matrix distribution, such as the Wishart's, this
should be a positive-definite matrix or a matrix written
in flat vector form.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the log-probability density function (pdf)
for this distribution evaluated at point x.
A single point in the distribution range.
For a matrix distribution, such as the Wishart's, this
should be a positive-definite matrix or a matrix written
in flat vector form.
The logarithm of the probability of x
occurring in the current distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Unsupported.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
The format.
The format provider.
A that represents this instance.
Generates a random vector of observations from the current distribution.
The degrees of freedom n.
The positive-definite matrix scale matrix V.
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The degrees of freedom n.
The positive-definite matrix scale matrix V.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The degrees of freedom n.
The positive-definite matrix scale matrix V.
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The degrees of freedom n.
The positive-definite matrix scale matrix V.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The degrees of freedom n.
The positive-definite matrix scale matrix V.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Common interface for components of
and distributions.
The type of the component distribution.
Gets the component distribution.
Gets the weight associated with this component.
Joint distribution assuming independence between vector components.
In probability and statistics, given at least two random variables X,
Y, ..., that are defined on a probability space, the joint probability
distribution for X, Y, ... is a probability distribution that
gives the probability that each of X, Y, ... falls in any particular range or
discrete set of values specified for that variable. In the case of only two
random variables, this is called a bivariate distribution, but the concept
generalizes to any number of random variables, giving a multivariate distribution.
This class is also available in a generic version, allowing for any
choice of component distribution (.
References:
-
Wikipedia, The Free Encyclopedia. Beta distribution.
Available from: http://en.wikipedia.org/wiki/Joint_probability_distribution
The following example shows how to declare and initialize an Independent Joint
Gaussian Distribution using known means and variances for each component.
// Declare two normal distributions
NormalDistribution pa = new NormalDistribution(4.2, 1); // p(a)
NormalDistribution pb = new NormalDistribution(7.0, 2); // p(b)
// Now, create a joint distribution combining these two:
var joint = new Independent(pa, pb);
// This distribution assumes the distributions of the two components are independent,
// i.e. if we have 2D input vectors on the form {a, b}, then p({a,b}) = p(a) * p(b).
// Lets check a simple example. Consider a 2D input vector x = { 4.2, 7.0 } as
//
double[] x = new double[] { 4.2, 7.0 };
// Those two should be completely equivalent:
double p1 = joint.ProbabilityDensityFunction(x);
double p2 = pa.ProbabilityDensityFunction(x[0]) * pb.ProbabilityDensityFunction(x[1]);
bool equal = p1 == p2; // at this point, equal should be true.
Initializes a new instance of the class.
The components.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Joint distribution assuming independence between vector components.
The type of the underlying distributions.
The type for the observations being modeled by the distribution (i.e. double).
The options for fitting the distribution to the observations.
In probability and statistics, given at least two random variables X,
Y, ..., that are defined on a probability space, the joint probability
distribution for X, Y, ... is a probability distribution that
gives the probability that each of X, Y, ... falls in any particular range or
discrete set of values specified for that variable. In the case of only two
random variables, this is called a bivariate distribution, but the concept
generalizes to any number of random variables, giving a multivariate distribution.
References:
-
Wikipedia, The Free Encyclopedia. Beta distribution.
Available from: http://en.wikipedia.org/wiki/Joint_probability_distribution
The following example shows how to declare and initialize an Independent Joint
Gaussian Distribution using known means and variances for each component.
// Declare two normal distributions
NormalDistribution pa = new NormalDistribution(4.2, 1); // p(a)
NormalDistribution pb = new NormalDistribution(7.0, 2); // p(b)
// Now, create a joint distribution combining these two:
var joint = new Independent<NormalDistribution>(pa, pb);
// This distribution assumes the distributions of the two components are independent,
// i.e. if we have 2D input vectors on the form {a, b}, then p({a,b}) = p(a) * p(b).
// Lets check a simple example. Consider a 2D input vector x = { 4.2, 7.0 } as
//
double[] x = new double[] { 4.2, 7.0 };
// Those two should be completely equivalent:
double p1 = joint.ProbabilityDensityFunction(x);
double p2 = pa.ProbabilityDensityFunction(x[0]) * pb.ProbabilityDensityFunction(x[1]);
bool equal = p1 == p2; // at this point, equal should be true.
The following example shows how to fit a distribution (estimate
its parameters) from a given dataset.
// Let's consider an input dataset of 2D vectors. We would
// like to estimate an Independent<NormalDistribution>
// which best models this data.
double[][] data =
{
// x, y
new double[] { 1, 8 },
new double[] { 2, 6 },
new double[] { 5, 7 },
new double[] { 3, 9 },
};
// We start by declaring some initial guesses for the
// distributions of each random variable (x, and y):
//
var distX = new NormalDistribution(0, 1);
var distY = new NormalDistribution(0, 1);
// Next, we declare our initial guess Independent distribution
var joint = new Independent<NormalDistribution>(distX, distY);
// We can now fit the distribution to our data,
// producing an estimate of its parameters:
//
joint.Fit(data);
// At this point, we have estimated our distribution.
double[] mean = joint.Mean; // should be { 2.75, 7.50 }
double[] var = joint.Variance; // should be { 2.917, 1.667 }
// | 2.917, 0.000 |
double[,] cov = joint.Covariance; // Cov = | |
// | 0.000, 1.667 |
// The covariance matrix is diagonal, as it would be expected
// if is assumed there are no interactions between components.
Initializes a new instance of the class.
The number of independent component distributions.
A function that creates a new distribution for each component index.
Initializes a new instance of the class.
The number of independent component distributions.
A function that creates a new distribution for each component index.
Initializes a new instance of the class.
The number of independent component distributions.
A base component which will be cloned to all dimensions.
Initializes a new instance of the class.
The components.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
For an example on how to fit an independent joint distribution, please
take a look at the examples section for .
Joint distribution assuming independence between vector components.
The type of the underlying distributions.
The type for the observations being modeled by the distribution (i.e. double).
In probability and statistics, given at least two random variables X,
Y, ..., that are defined on a probability space, the joint probability
distribution for X, Y, ... is a probability distribution that
gives the probability that each of X, Y, ... falls in any particular range or
discrete set of values specified for that variable. In the case of only two
random variables, this is called a bivariate distribution, but the concept
generalizes to any number of random variables, giving a multivariate distribution.
References:
-
Wikipedia, The Free Encyclopedia. Beta distribution.
Available from: http://en.wikipedia.org/wiki/Joint_probability_distribution
The following example shows how to declare and initialize an Independent Joint
Gaussian Distribution using known means and variances for each component.
// Declare two normal distributions
NormalDistribution pa = new NormalDistribution(4.2, 1); // p(a)
NormalDistribution pb = new NormalDistribution(7.0, 2); // p(b)
// Now, create a joint distribution combining these two:
var joint = new Independent<NormalDistribution>(pa, pb);
// This distribution assumes the distributions of the two components are independent,
// i.e. if we have 2D input vectors on the form {a, b}, then p({a,b}) = p(a) * p(b).
// Lets check a simple example. Consider a 2D input vector x = { 4.2, 7.0 } as
//
double[] x = new double[] { 4.2, 7.0 };
// Those two should be completely equivalent:
double p1 = joint.ProbabilityDensityFunction(x);
double p2 = pa.ProbabilityDensityFunction(x[0]) * pb.ProbabilityDensityFunction(x[1]);
bool equal = p1 == p2; // at this point, equal should be true.
The following example shows how to fit a distribution (estimate
its parameters) from a given dataset.
// Let's consider an input dataset of 2D vectors. We would
// like to estimate an Independent<NormalDistribution>
// which best models this data.
double[][] data =
{
// x, y
new double[] { 1, 8 },
new double[] { 2, 6 },
new double[] { 5, 7 },
new double[] { 3, 9 },
};
// We start by declaring some initial guesses for the
// distributions of each random variable (x, and y):
//
var distX = new NormalDistribution(0, 1);
var distY = new NormalDistribution(0, 1);
// Next, we declare our initial guess Independent distribution
var joint = new Independent<NormalDistribution>(distX, distY);
// We can now fit the distribution to our data,
// producing an estimate of its parameters:
//
joint.Fit(data);
// At this point, we have estimated our distribution.
double[] mean = joint.Mean; // should be { 2.75, 7.50 }
double[] var = joint.Variance; // should be { 2.917, 1.667 }
// | 2.917, 0.000 |
double[,] cov = joint.Covariance; // Cov = | |
// | 0.000, 1.667 |
// The covariance matrix is diagonal, as it would be expected
// if is assumed there are no interactions between components.
Initializes a new instance of the class.
The number of independent component distributions.
A function that creates a new distribution for each component index.
Initializes a new instance of the class.
The number of independent component distributions.
A function that creates a new distribution for each component index.
Initializes a new instance of the class.
The number of independent component distributions.
A base component which will be cloned to all dimensions.
Initializes a new instance of the class.
The components.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF.
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
For an example on how to fit an independent joint distribution, please
take a look at the examples section for .
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
For an example on how to fit an independent joint distribution, please
take a look at the examples section for .
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
For an example on how to fit an independent joint distribution, please
take a look at the examples section for .
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Generates a random observation from the current distribution.
The location where to store the sample.
A random observation drawn from this distribution.
Generates a random observation from the current distribution.
The location where to store the sample.
The random number generator to use as a source of randomness.
Default is to use .
A random observation drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Abstract class for multivariate discrete probability distributions.
A probability distribution identifies either the probability of each value of an
unidentified random variable (when the variable is discrete), or the probability
of the value falling within a particular interval (when the variable is continuous).
The probability distribution describes the range of possible values that a random
variable can attain and the probability that the value of the random variable is
within any (measurable) subset of that range.
The function describing the probability that a given discrete value will
occur is called the probability function (or probability mass function,
abbreviated PMF), and the function describing the cumulative probability
that a given value or any value smaller than it will occur is called the
distribution function (or cumulative distribution function, abbreviated CDF).
References:
-
Wikipedia, The Free Encyclopedia. Probability distribution. Available on:
http://en.wikipedia.org/wiki/Probability_distribution
-
Weisstein, Eric W. "Statistical Distribution." From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/StatisticalDistribution.html
Constructs a new MultivariateDiscreteDistribution class.
Gets the number of variables for this distribution.
Gets the mean for this distribution.
An array of double-precision values containing
the mean values for this distribution.
Gets the mean for this distribution.
An array of double-precision values containing
the variance values for this distribution.
Gets the variance for this distribution.
An multidimensional array of double-precision values
containing the covariance values for this distribution.
Gets the mode for this distribution.
An array of double-precision values containing
the mode values for this distribution.
Gets the median for this distribution.
An array of double-precision values containing
the median values for this distribution.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Probability Mass Function (PMF) describes the
probability that a given value x will occur.
The probability of x occurring
in the current distribution.
Gets the probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Probability Mass Function (PMF) describes the
probability that a given value x will occur.
The probability of x occurring
in the current distribution.
Gets the log-probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Probability Mass Function (PMF) describes the
probability that a given value x will occur.
The logarithm of the probability of x
occurring in the current distribution.
Gets the log-probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Probability Mass Function (PMF) describes the
probability that a given value x will occur.
The logarithm of the probability of x
occurring in the current distribution.
Not supported.
Not supported.
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF.
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF.
Gets the marginal distribution of a given variable.
The variable index.
Gets the marginal distribution of a given variable evaluated at a given value.
The variable index.
The variable value.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
A random observations drawn from this distribution.
Generates a random observation from the current distribution.
A random observations drawn from this distribution.
Generates a random observation from the current distribution.
A random observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
A random observations drawn from this distribution.
Generates a random observation from the current distribution.
A random observations drawn from this distribution.
Generates a random observation from the current distribution.
A random observations drawn from this distribution.
Joint distribution assuming independence between vector components.
The type of the underlying distributions.
In probability and statistics, given at least two random variables X,
Y, ..., that are defined on a probability space, the joint probability
distribution for X, Y, ... is a probability distribution that
gives the probability that each of X, Y, ... falls in any particular range or
discrete set of values specified for that variable. In the case of only two
random variables, this is called a bivariate distribution, but the concept
generalizes to any number of random variables, giving a multivariate distribution.
References:
-
Wikipedia, The Free Encyclopedia. Beta distribution.
Available from: http://en.wikipedia.org/wiki/Joint_probability_distribution
The following example shows how to declare and initialize an Independent Joint
Gaussian Distribution using known means and variances for each component.
// Declare two normal distributions
NormalDistribution pa = new NormalDistribution(4.2, 1); // p(a)
NormalDistribution pb = new NormalDistribution(7.0, 2); // p(b)
// Now, create a joint distribution combining these two:
var joint = new Independent<NormalDistribution>(pa, pb);
// This distribution assumes the distributions of the two components are independent,
// i.e. if we have 2D input vectors on the form {a, b}, then p({a,b}) = p(a) * p(b).
// Lets check a simple example. Consider a 2D input vector x = { 4.2, 7.0 } as
//
double[] x = new double[] { 4.2, 7.0 };
// Those two should be completely equivalent:
double p1 = joint.ProbabilityDensityFunction(x);
double p2 = pa.ProbabilityDensityFunction(x[0]) * pb.ProbabilityDensityFunction(x[1]);
bool equal = p1 == p2; // at this point, equal should be true.
The following example shows how to fit a distribution (estimate
its parameters) from a given dataset.
// Let's consider an input dataset of 2D vectors. We would
// like to estimate an Independent<NormalDistribution>
// which best models this data.
double[][] data =
{
// x, y
new double[] { 1, 8 },
new double[] { 2, 6 },
new double[] { 5, 7 },
new double[] { 3, 9 },
};
// We start by declaring some initial guesses for the
// distributions of each random variable (x, and y):
//
var distX = new NormalDistribution(0, 1);
var distY = new NormalDistribution(0, 1);
// Next, we declare our initial guess Independent distribution
var joint = new Independent<NormalDistribution>(distX, distY);
// We can now fit the distribution to our data,
// producing an estimate of its parameters:
//
joint.Fit(data);
// At this point, we have estimated our distribution.
double[] mean = joint.Mean; // should be { 2.75, 7.50 }
double[] var = joint.Variance; // should be { 2.917, 1.667 }
// | 2.917, 0.000 |
double[,] cov = joint.Covariance; // Cov = | |
// | 0.000, 1.667 |
// The covariance matrix is diagonal, as it would be expected
// if is assumed there are no interactions between components.
Initializes a new instance of the class.
The number of independent component distributions.
Initializes a new instance of the class.
The number of independent component distributions.
A function that creates a new distribution for each component index.
Initializes a new instance of the class.
The number of independent component distributions.
A function that creates a new distribution for each component index.
Initializes a new instance of the class.
The number of independent component distributions.
A base component which will be cloned to all dimensions.
Initializes a new instance of the class.
The components.
Gets or sets the components of this joint distribution.
Gets the components of this joint distribution.
Gets the mean for this distribution.
A vector containing the mean values for the distribution.
Gets the variance for this distribution.
A vector containing the variance values for the distribution.
Gets the variance-covariance matrix for this distribution.
A matrix containing the covariance values for the distribution.
For an independent distribution, this matrix will always be diagonal.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the log-probability density function (pdf)
for this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The logarithm of the probability of x
occurring in the current distribution.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
For an example on how to fit an independent joint distribution, please
take a look at the examples section for .
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
For an example on how to fit an independent joint distribution, please
take a look at the examples section for .
Resets cached values (should be called after re-estimation).
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Represents one component distribution in a
mixture distribution.
The distribution type.
Gets the weight associated with this component.
Gets the component distribution.
Initializes a new instance of the struct.
The mixture distribution.
The component index.
Mixture of multivariate probability distributions.
A mixture density is a probability density function which is expressed
as a convex combination (i.e. a weighted sum, with non-negative weights
that sum to 1) of other probability density functions. The individual
density functions that are combined to make the mixture density are
called the mixture components, and the weights associated with each
component are called the mixture weights.
References:
-
Wikipedia, The Free Encyclopedia. Mixture density. Available on:
http://en.wikipedia.org/wiki/Mixture_density
The type of the multivariate component distributions.
// Randomly initialize some mixture components
MultivariateNormalDistribution[] components = new MultivariateNormalDistribution[2];
components[0] = new MultivariateNormalDistribution(new double[] { 2 }, new double[,] { { 1 } });
components[1] = new MultivariateNormalDistribution(new double[] { 5 }, new double[,] { { 1 } });
// Create an initial mixture
var mixture = new MultivariateMixture<MultivariateNormalDistribution>(components);
// Now, suppose we have a weighted data
// set. Those will be the input points:
double[][] points = new double[] { 0, 3, 1, 7, 3, 5, 1, 2, -1, 2, 7, 6, 8, 6 } // (14 points)
.ToArray();
// And those are their respective unnormalized weights:
double[] weights = { 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 3, 1, 1 }; // (14 weights)
// Let's normalize the weights so they sum up to one:
weights = weights.Divide(weights.Sum());
// Now we can fit our model to the data:
mixture.Fit(points, weights); // done!
// Our model will be:
double mean1 = mixture.Components[0].Mean[0]; // 1.41126
double mean2 = mixture.Components[1].Mean[0]; // 6.53301
// With mixture coefficients
double pi1 = mixture.Coefficients[0]; // 0.51408489193241225
double pi2 = mixture.Coefficients[1]; // 0.48591510806758775
// If we need the GaussianMixtureModel functionality, we can
// use the estimated mixture to initialize a new model:
GaussianMixtureModel gmm = new GaussianMixtureModel(mixture);
mean1 = gmm.Gaussians[0].Mean[0]; // 1.41126 (same)
mean2 = gmm.Gaussians[1].Mean[0]; // 6.53301 (same)
p1 = gmm.Gaussians[0].Proportion; // 0.51408 (same)
p2 = gmm.Gaussians[1].Proportion; // 0.48591 (same)
Initializes a new instance of the class.
The mixture distribution components.
Initializes a new instance of the class.
The mixture weight coefficients.
The mixture distribution components.
Gets the mixture components.
Gets the weight coefficients.
Gets the probability density function (pdf) for one of
the component distributions evaluated at point x.
The index of the desired component distribution.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the log-probability density function (pdf) for one
of the component distributions evaluated at point x.
The index of the desired component distribution.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
The logarithm of the probability of x
occurring in the current distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the cumulative distribution function (cdf) for one
of the component distributions evaluated at point x.
The component distribution's index.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Computes the log-likelihood of the distribution
for a given set of observations.
Computes the log-likelihood of the distribution
for a given set of observations.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Gets the mean for this distribution.
Gets the variance-covariance matrix for this distribution.
Gets the variance vector for this distribution.
Estimates a new mixture model from a given set of observations.
A set of observations.
The initial components of the mixture model.
Returns a new Mixture fitted to the given observations.
Estimates a new mixture model from a given set of observations.
A set of observations.
The initial components of the mixture model.
The initial mixture coefficients.
Returns a new Mixture fitted to the given observations.
Estimates a new mixture model from a given set of observations.
A set of observations.
The initial components of the mixture model.
The initial mixture coefficients.
The convergence threshold for the Expectation-Maximization estimation.
Returns a new Mixture fitted to the given observations.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Returns a that represents this instance.
A that represents this instance.
Multinomial probability distribution.
The multinomial distribution is a generalization of the binomial
distribution. The binomial distribution is the probability distribution
of the number of "successes" in n independent
Bernoulli
trials, with the same probability of "success" on each trial.
In a multinomial distribution, the analog of the
Bernoulli distribution is the
categorical distribution,
where each trial results in exactly one of some fixed finite number
k of possible outcomes, with probabilities p1, ..., pk
and there are n independent trials.
References:
-
Wikipedia, The Free Encyclopedia. Multinomial distribution. Available on:
http://en.wikipedia.org/wiki/Multinomial_distribution
// distribution parameters
int numberOfTrials = 5;
double[] probabilities = { 0.25, 0.75 };
// Create a new Multinomial distribution with 5 trials for 2 symbols
var dist = new MultinomialDistribution(numberOfTrials, probabilities);
int dimensions = dist.Dimension; // 2
double[] mean = dist.Mean; // { 1.25, 3.75 }
double[] median = dist.Median; // { 1.25, 3.75 }
double[] var = dist.Variance; // { -0.9375, -0.9375 }
double pdf1 = dist.ProbabilityMassFunction(new[] { 2, 3 }); // 0.26367187499999994
double pdf2 = dist.ProbabilityMassFunction(new[] { 1, 4 }); // 0.3955078125
double pdf3 = dist.ProbabilityMassFunction(new[] { 5, 0 }); // 0.0009765625
double lpdf = dist.LogProbabilityMassFunction(new[] { 1, 4 }); // -0.9275847384929139
// output is "Multinomial(x; n = 5, p = { 0.25, 0.75 })"
string str = dist.ToString(CultureInfo.InvariantCulture);
Initializes a new instance of the class.
The total number of trials N.
A vector containing the probabilities of seeing each of possible outcomes.
Gets the event probabilities associated with the trials.
Gets the number of Bernoulli trials N.
Gets the mean for this distribution.
Gets the variance vector for this distribution.
Gets the variance-covariance matrix for this distribution.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Not supported.
Gets the probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
The Probability Mass Function (PMF) describes the
probability that a given value x will occur.
Gets the log-probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Mass Function (PMF) describes the
probability that a given value x will occur.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Dirichlet distribution.
The Dirichlet distribution, often denoted Dir(α), is a family of continuous
multivariate probability distributions parameterized by a vector α of positive
real numbers. It is the multivariate generalization of the beta distribution.
Dirichlet distributions are very often used as prior distributions in Bayesian
statistics, and in fact the Dirichlet distribution is the conjugate prior of the
categorical distribution and multinomial distribution. That is, its probability
density function returns the belief that the probabilities of K rival events are
xi given that each event has been observed αi−1 times.
References:
-
Wikipedia, The Free Encyclopedia. Dirichlet distribution.
Available from: http://en.wikipedia.org/wiki/Dirichlet_distribution
// Create a Dirichlet with the following concentrations
var dirich = new DirichletDistribution(0.42, 0.57, 1.2);
// Common measures
double[] mean = dirich.Mean; // { 0.19, 0.26, 0.54 }
double[] median = dirich.Median; // { 0.19, 0.26, 0.54 }
double[] var = dirich.Variance; // { 0.048, 0.060, 0.077 }
double[,] cov = dirich.Covariance; // see below
// 0.0115297440926238 0.0156475098399895 0.0329421259789253
// cov = 0.0156475098399895 0.0212359062114143 0.0447071709713986
// 0.0329421259789253 0.0447071709713986 0.0941203599397865
// (the above matrix representation has been transcribed to text using)
string str = cov.ToString(DefaultMatrixFormatProvider.InvariantCulture);
// Probability mass functions
double pdf1 = dirich.ProbabilityDensityFunction(new double[] { 2, 5 }); // 0.12121671541846207
double pdf2 = dirich.ProbabilityDensityFunction(new double[] { 4, 2 }); // 0.12024840322466089
double pdf3 = dirich.ProbabilityDensityFunction(new double[] { 3, 7 }); // 0.082907634905068528
double lpdf = dirich.LogProbabilityDensityFunction(new double[] { 3, 7 }); // -2.4900281233124044
Creates a new symmetric Dirichlet distribution.
The number k of categories.
The common concentration parameter α (alpha).
Creates a new Dirichlet distribution.
The concentration parameters αi (alpha_i).
Gets the mean for this distribution.
A vector containing the mean values for the distribution.
Gets the variance for this distribution.
A vector containing the variance values for the distribution.
Gets the variance-covariance matrix for this distribution.
A matrix containing the covariance values for the distribution.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Not supported.
Gets the log-probability density function (pdf)
for this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The logarithm of the probability of x
occurring in the current distribution.
Not supported.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
The format.
The format provider.
A that represents this instance.
Hidden Markov Model probability distribution.
Initializes a new instance of the class.
The model.
Gets the mean for this distribution.
An array of double-precision values containing
the mean values for this distribution.
Gets the mean for this distribution.
An array of double-precision values containing
the variance values for this distribution.
Gets the variance for this distribution.
An multidimensional array of double-precision values
containing the covariance values for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
The Probability Mass Function (PMF) describes the
probability that a given value x will occur.
Gets the log-probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Mass Function (PMF) describes the
probability that a given value x will occur.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
The format.
The format provider.
A that represents this instance.
Abstract class for Multivariate Probability Distributions.
A probability distribution identifies either the probability of each value of an
unidentified random variable (when the variable is discrete), or the probability
of the value falling within a particular interval (when the variable is continuous).
The probability distribution describes the range of possible values that a random
variable can attain and the probability that the value of the random variable is
within any (measurable) subset of that range.
The function describing the probability that a given value will occur is called
the probability function (or probability density function, abbreviated PDF), and
the function describing the cumulative probability that a given value or any value
smaller than it will occur is called the distribution function (or cumulative
distribution function, abbreviated CDF).
References:
-
Wikipedia, The Free Encyclopedia. Probability distribution. Available on:
http://en.wikipedia.org/wiki/Probability_distribution
-
Weisstein, Eric W. "Statistical Distribution." From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/StatisticalDistribution.html
Constructs a new MultivariateDistribution class.
The number of dimensions in the distribution.
Gets the number of variables for this distribution.
Gets the mean for this distribution.
A vector containing the mean values for the distribution.
Gets the variance for this distribution.
A vector containing the variance values for the distribution.
Gets the variance-covariance matrix for this distribution.
A matrix containing the covariance values for the distribution.
Gets the mode for this distribution.
A vector containing the mode values for the distribution.
Gets the median for this distribution.
A vector containing the median values for the distribution.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
The probability of x occurring
in the current distribution.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
The probability of x occurring
in the current distribution.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
The probability of x occurring
in the current distribution.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
The probability of x occurring
in the current distribution.
Gets the log-probability density function (pdf)
for this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The logarithm of the probability of x
occurring in the current distribution.
Gets the log-probability density function (pdf)
for this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The logarithm of the probability of x
occurring in the current distribution.
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF.
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
A random observations drawn from this distribution.
Generates a random observation from the current distribution.
A random observations drawn from this distribution.
Generates a random observation from the current distribution.
The location where to store the sample.
A random observation drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the observations.
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
A random observations drawn from this distribution.
Generates a random observation from the current distribution.
A random observations drawn from this distribution.
Generates a random observation from the current distribution.
The location where to store the sample.
The random number generator to use as a source of randomness.
Default is to use .
A random observation drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the observations.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Joint distribution of multiple discrete univariate distributions.
This class builds a (potentially huge) lookup table for discrete
symbol distributions. For example, given a discrete variable A
which may take symbols a, b, c; and a discrete variable B which
may assume values x, y, z, this class will build the probability
table:
x y z
a p(a,x) p(a,y) p(a,z)
b p(b,x) p(b,y) p(b,z)
c p(c,x) p(c,y) p(c,z)
Thus comprising the probabilities for all possible simple combination. This
distribution is a generalization of the
for multivariate discrete observations.
The following example should demonstrate how to estimate a joint
distribution of two discrete variables. The first variable can
take up to three distinct values, whereas the second can assume
up to five.
// Lets create a joint distribution for two discrete variables:
// the first of which can assume 3 distinct symbol values: 0, 1, 2
// the second which can assume 5 distinct symbol values: 0, 1, 2, 3, 4
int[] symbols = { 3, 5 }; // specify the symbol counts
// Create the joint distribution for the above variables
JointDistribution joint = new JointDistribution(symbols);
// Now, suppose we would like to fit the distribution (estimate
// its parameters) from the following multivariate observations:
//
double[][] observations =
{
new double[] { 0, 0 },
new double[] { 1, 1 },
new double[] { 2, 1 },
new double[] { 0, 0 },
};
// Estimate parameters
joint.Fit(observations);
// At this point, we can query the distribution for observations:
double p1 = joint.ProbabilityMassFunction(new[] { 0, 0 }); // should be 0.50
double p2 = joint.ProbabilityMassFunction(new[] { 1, 1 }); // should be 0.25
double p3 = joint.ProbabilityMassFunction(new[] { 2, 1 }); // should be 0.25
// As it can be seem, indeed {0,0} appeared twice at the data,
// and {1,1} and {2,1 appeared one fourth of the data each.
Gets the frequency of observation of each discrete variable.
Gets the integer value where the
discrete distribution starts.
Gets the integer value where the
discrete distribution ends.
Gets the number of symbols in the distribution.
Gets the number of symbols for each discrete variable.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Constructs a new joint discrete distribution.
Constructs a new joint discrete distribution.
Constructs a new joint discrete distribution.
Constructs a new joint discrete distribution.
Constructs a new joint discrete distribution.
Gets or sets the probability value attached to the given index.
Constructs a new multidimensional uniform discrete distribution.
The number of dimensions in the joint distribution.
The integer value where the distribution starts, also
known as a. Default value is 0.
The integer value where the distribution ends, also
known as b.
Gets the probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Probability Mass Function (PMF) describes the
probability that a given value x will occur.
The probability of x occurring
in the current distribution.
Gets the log-probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Mass Function (PMF) describes the
probability that a given value x will occur.
Gets the mean for this distribution.
An array of double-precision values containing
the mean values for this distribution.
Gets the mean for this distribution.
An array of double-precision values containing
the variance values for this distribution.
Gets the variance for this distribution.
An multidimensional array of double-precision values
containing the covariance values for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
This method does not accept fitting options.
weights;The weight vector should have the same size as the observations
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Estimates a new from a given set of observations.
Please see .
Multivariate Normal (Gaussian) distribution.
The Gaussian is the most widely used distribution for continuous
variables. In the case of many variables, it is governed by two
parameters, the mean vector and the variance-covariance matrix.
When a covariance matrix given to the class constructor is not positive
definite, the distribution is degenerate and this may be an indication
indication that it may be entirely contained in a r-dimensional subspace.
Applying a rotation to an orthogonal basis to recover a non-degenerate
r-dimensional distribution may help in this case.
References:
-
Ai Access. Glossary of Data Modeling. Positive definite matrix. Available on:
http://www.aiaccess.net/English/Glossaries/GlosMod/e_gm_positive_definite_matrix.htm
The following example shows how to create a Multivariate Gaussian
distribution with known parameters mean vector and covariance matrix
// Create a multivariate Gaussian distribution
var dist = new MultivariateNormalDistribution(
// mean vector mu
mean: new double[]
{
4, 2
},
// covariance matrix sigma
covariance: new double[,]
{
{ 0.3, 0.1 },
{ 0.1, 0.7 }
}
);
// Common measures
double[] mean = dist.Mean; // { 4, 2 }
double[] median = dist.Median; // { 4, 2 }
double[] var = dist.Variance; // { 0.3, 0.7 } (diagonal from cov)
double[,] cov = dist.Covariance; // { { 0.3, 0.1 }, { 0.1, 0.7 } }
// Probability mass functions
double pdf1 = dist.ProbabilityDensityFunction(new double[] { 2, 5 }); // 0.000000018917884164743237
double pdf2 = dist.ProbabilityDensityFunction(new double[] { 4, 2 }); // 0.35588127170858852
double pdf3 = dist.ProbabilityDensityFunction(new double[] { 3, 7 }); // 0.000000000036520107734505265
double lpdf = dist.LogProbabilityDensityFunction(new double[] { 3, 7 }); // -24.033158110192296
// Cumulative distribution function (for up to two dimensions)
double cdf = dist.DistributionFunction(new double[] { 3, 5 }); // 0.033944035782101548
// Generate samples from this distribution:
double[][] sample = dist.Generate(1000000);
The following example demonstrates how to fit a multivariate Gaussian to
a set of observations. Since those observations would lead to numerical
difficulties, the example also demonstrates how to specify a regularization
constant to avoid getting a .
double[][] observations =
{
new double[] { 1, 2 },
new double[] { 1, 2 },
new double[] { 1, 2 },
new double[] { 1, 2 }
};
// Create a multivariate Gaussian for 2 dimensions
var normal = new MultivariateNormalDistribution(2);
// Specify a regularization constant in the fitting options
NormalOptions options = new NormalOptions() { Regularization = double.Epsilon };
// Fit the distribution to the data
normal.Fit(observations, options);
// Check distribution parameters
double[] mean = normal.Mean; // { 1, 2 }
double[] var = normal.Variance; // { 4.9E-324, 4.9E-324 } (almost machine zero)
The next example shows how to estimate a Gaussian distribution from data
available inside a Microsoft Excel spreadsheet using the ExcelReader class.
// Create a new Excel reader to read data from a spreadsheet
ExcelReader reader = new ExcelReader(@"test.xls", hasHeaders: false);
// Extract the "Data" worksheet from the xls
DataTable table = reader.GetWorksheet("Data");
// Convert the data table to a jagged matrix
double[][] observations = table.ToArray();
// Estimate a new Multivariate Normal Distribution from the observations
var dist = MultivariateNormalDistribution.Estimate(observations, new NormalOptions()
{
Regularization = 1e-10 // this value will be added to the diagonal until it becomes positive-definite
});
Constructs a multivariate Gaussian distribution
with zero mean vector and identity covariance matrix.
The number of dimensions in the distribution.
Constructs a multivariate Gaussian distribution
with given mean vector and covariance matrix.
The mean vector μ (mu) for the distribution.
The covariance matrix Σ (sigma) for the distribution.
Constructs a multivariate Gaussian distribution
with given mean vector and covariance matrix.
The mean vector μ (mu) for the distribution.
Constructs a multivariate Gaussian distribution
with given mean vector and covariance matrix.
The mean vector μ (mu) for the distribution.
The covariance matrix Σ (sigma) for the distribution.
Gets the Mean vector μ (mu) for
the Gaussian distribution.
Gets the Variance vector diag(Σ), the diagonal of
the sigma matrix, for the Gaussian distribution.
Gets the variance-covariance matrix
Σ (sigma) for the Gaussian distribution.
Computes the cumulative distribution function for distributions
up to two dimensions. For more than two dimensions, this method
is not supported.
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the log-probability density function (pdf)
for this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The logarithm of the probability of x
occurring in the current distribution.
Gets the Mahalanobis distance between a sample and this distribution.
A point in the distribution space.
The Mahalanobis distance between the point and this distribution.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Please see .
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Please see .
Estimates a new Normal distribution from a given set of observations.
Estimates a new Normal distribution from a given set of observations.
Please see .
Estimates a new Normal distribution from a given set of observations.
Please see .
Estimates a new Normal distribution from a given set of observations.
Please see .
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Converts this multivariate
normal distribution into a joint distribution
of independent normal distributions.
A independent joint distribution of
normal distributions.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Creates a new univariate Normal distribution.
The mean value for the distribution.
The standard deviation for the distribution.
A object that
actually represents a .
Creates a new bivariate Normal distribution.
The mean value for the first variate in the distribution.
The mean value for the second variate in the distribution.
The standard deviation for the first variate.
The standard deviation for the second variate.
The correlation coefficient between the two distributions.
A bi-dimensional .
Returns a that represents this instance.
The format.
The format provider.
A that represents this instance.
Generates a random vector of observations from a distribution with the given parameters.
The number of samples to generate.
The mean vector μ (mu) for the distribution.
The covariance matrix Σ (sigma) for the distribution.
A random vector of observations drawn from this distribution.
Common interface for multivariate probability distributions.
This interface is implemented by both multivariate
Discrete Distributions and Continuous
Distributions.
For Univariate distributions, see .
Gets the number of variables for the distribution.
Gets the Mean vector for the distribution.
An array of double-precision values containing
the mean values for this distribution.
Gets the Median vector for the distribution.
An array of double-precision values containing
the median values for this distribution.
Gets the Mode vector for the distribution.
An array of double-precision values containing
the mode values for this distribution.
Gets the Variance vector for the distribution.
An array of double-precision values containing
the variance values for this distribution.
Gets the Variance-Covariance matrix for the distribution.
An multidimensional array of double-precision values
containing the covariance values for this distribution.
Common interface for multivariate probability distributions.
This interface is implemented by both multivariate
Discrete Distributions and Continuous
Distributions. However, unlike , this interface
has a generic parameter that allows to define the type of the distribution values (i.e.
).
For Univariate distributions, see .
Gets the number of variables for the distribution.
Gets the Mean vector for the distribution.
An array of double-precision values containing
the mean values for this distribution.
Gets the Median vector for the distribution.
An array of double-precision values containing
the median values for this distribution.
Gets the Mode vector for the distribution.
An array of double-precision values containing
the mode values for this distribution.
Gets the Variance vector for the distribution.
An array of double-precision values containing
the variance values for this distribution.
Gets the Variance-Covariance matrix for the distribution.
An multidimensional array of double-precision values
containing the covariance values for this distribution.
Metropolis-Hasting sampling algorithm.
References:
-
Wikipedia, The Free Encyclopedia. Metropolis-Hastings algorithm.
Available on: https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm
-
Darren Wilkinson, Metropolis Hastings MCMC when the proposal and target have differing support.
Available on: https://darrenjw.wordpress.com/2012/06/04/metropolis-hastings-mcmc-when-the-proposal-and-target-have-differing-support/
Gets the target distribution whose samples must be generated.
Initializes a new instance of the class.
The target distribution whose samples should be generated.
The proposal distribution that is used to generate new parameter samples to be explored.
Metropolis-Hasting sampling algorithm.
References:
-
Wikipedia, The Free Encyclopedia. Metropolis-Hastings algorithm.
Available on: https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm
-
Darren Wilkinson, Metropolis Hastings MCMC when the proposal and target have differing support.
Available on: https://darrenjw.wordpress.com/2012/06/04/metropolis-hastings-mcmc-when-the-proposal-and-target-have-differing-support/
Gets or sets the move proposal distribution.
Initializes a new instance of the algorithm.
The number of dimensions in each observation.
A function specifying the log probability density of the distribution to be sampled.
The proposal distribution that is used to generate new parameter samples to be explored.
Initializes a new instance of the class.
Initializes the algorithm.
Metropolis-Hasting sampling algorithm.
References:
-
Wikipedia, The Free Encyclopedia. Metropolis-Hastings algorithm.
Available on: https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm
-
Darren Wilkinson, Metropolis Hastings MCMC when the proposal and target have differing support.
Available on: https://darrenjw.wordpress.com/2012/06/04/metropolis-hastings-mcmc-when-the-proposal-and-target-have-differing-support/
Gets the last successfully generated observation.
Gets or sets a factory method to create random number generators used in this instance.
Gets the log-probability of the last successfully
generated sample.
Gets the log-probability density function of the target distribution.
Gets or sets the move proposal distribution.
Gets the acceptance rate for the proposals generated
by the proposal distribution.
Gets the number of dimensions in each observation.
Gets or sets how many initial samples will get discarded as part
of the initial thermalization (warm-up, initialization) process.
Initializes a new instance of the algorithm.
The number of dimensions in each observation.
A function specifying the log probability density of the distribution to be sampled.
The proposal distribution that is used to generate new parameter samples to be explored.
Initializes a new instance of the class.
Initializes the algorithm.
Attempts to generate a new observation from the target
distribution, storing its value in the
property.
A new observation, if the method has succeed; otherwise, null.
True if the sample was successfully generated; otherwise, returns false.
Attempts to generate a new observation from the target
distribution, storing its value in the
property.
True if the sample was successfully generated; otherwise, false.
Thermalizes the sample generation process, generating up to
samples and discarding them. This step
is done automatically upon the first call to any of the
functions.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
A random observation drawn from this distribution.
Metropolis-Hasting sampling algorithm.
References:
-
Wikipedia, The Free Encyclopedia. Metropolis-Hastings algorithm.
Available on: https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm
-
Darren Wilkinson, Metropolis Hastings MCMC when the proposal and target have differing support.
Available on: https://darrenjw.wordpress.com/2012/06/04/metropolis-hastings-mcmc-when-the-proposal-and-target-have-differing-support/
Initializes a new instance of the algorithm.
The number of dimensions in each observation.
A function specifying the log probability density of the distribution to be sampled.
The proposal distribution that is used to generate new parameter samples to be explored.
Initializes a new instance of the algorithm.
The number of dimensions in each observation.
A function specifying the log probability density of the distribution to be sampled.
Creates a new sampler using independent Normal distributions
as the parameter proposal generation priors.
The number of dimensions in each observation.
A function specifying the log probability density of the distribution to be sampled.
A sampling algorithm that can generate samples from the target distribution.
Creates a new sampler using independent Normal distributions
as the parameter proposal generation priors.
The number of dimensions in each observation.
The target distribution whose samples should be generated.
A sampling algorithm that can generate samples from the target distribution.
Creates a new sampler using symmetric geometric distributions
as the parameter proposal generation priors.
The number of dimensions in each observation.
A function specifying the log probability density of the distribution to be sampled.
A sampling algorithm that can generate samples from the target distribution.
Creates a new sampler using symmetric geometric distributions
as the parameter proposal generation priors.
The number of dimensions in each observation.
The target distribution whose samples should be generated.
A sampling algorithm that can generate samples from the target distribution.
Contains univariate distributions such as Normal,
Cauchy,
Hypergeometric, Poisson,
Bernoulli, and specialized distributions such
as the Kolmogorov-Smirnov,
Nakagami,
Weibull, and Von-Mises distributions.
The namespace class diagram is shown below.
Beta Distribution (of the first kind).
The beta distribution is a family of continuous probability distributions
defined on the interval (0, 1) parameterized by two positive shape parameters,
typically denoted by α and β. The beta distribution can be suited to the
statistical modeling of proportions in applications where values of proportions
equal to 0 or 1 do not occur. One theoretical case where the beta distribution
arises is as the distribution of the ratio formed by one random variable having
a Gamma distribution divided by the sum of it and another independent random
variable also having a Gamma distribution with the same scale parameter (but
possibly different shape parameter).
References:
-
Wikipedia, The Free Encyclopedia. Beta distribution.
Available from: http://en.wikipedia.org/wiki/Beta_distribution
Note: More advanced examples, including distribution estimation and random number
generation are also available at the
page.
The following example shows how to instantiate and use a Beta
distribution given its alpha and beta parameters:
double alpha = 0.42;
double beta = 1.57;
// Create a new Beta distribution with α = 0.42 and β = 1.57
BetaDistribution distribution = new BetaDistribution(alpha, beta);
// Common measures
double mean = distribution.Mean; // 0.21105527638190955
double median = distribution.Median; // 0.11577711097114812
double var = distribution.Variance; // 0.055689279830523512
// Cumulative distribution functions
double cdf = distribution.DistributionFunction(x: 0.27); // 0.69358638272337991
double ccdf = distribution.ComplementaryDistributionFunction(x: 0.27); // 0.30641361727662009
double icdf = distribution.InverseDistributionFunction(p: cdf); // 0.26999999068687469
// Probability density functions
double pdf = distribution.ProbabilityDensityFunction(x: 0.27); // 0.94644031936694828
double lpdf = distribution.LogProbabilityDensityFunction(x: 0.27); // -0.055047364344046057
// Hazard (failure rate) functions
double hf = distribution.HazardFunction(x: 0.27); // 3.0887671630877072
double chf = distribution.CumulativeHazardFunction(x: 0.27); // 1.1828193992944409
// String representation
string str = distribution.ToString(); // B(x; α = 0.42, β = 1.57)
The following example shows to create a Beta distribution
given a discrete number of trials and the number of successes
within those trials. It also shows how to compute the 2.5 and
97.5 percentiles of the distribution:
int trials = 100;
int successes = 78;
BetaDistribution distribution = new BetaDistribution(successes, trials);
double mean = distribution.Mean; // 0.77450980392156865
double median = distribution.Median; // 0.77630912598534851
double p025 = distribution.InverseDistributionFunction(p: 0.025); // 0.68899653915764347
double p975 = distribution.InverseDistributionFunction(p: 0.975); // 0.84983461640764513
The next example shows how to generate 1000 new samples from a Beta distribution:
// Using the distribution's parameters
double[] samples = GeneralizedBetaDistribution
.Random(alpha: 2, beta: 3, min: 0, max: 1, samples: 1000);
// Using an existing distribution
var b = new GeneralizedBetaDistribution(alpha: 1, beta: 2);
double[] new_samples = b.Generate(1000);
And finally, how to estimate the parameters of a Beta distribution from
a set of observations, using either the Method-of-moments or the Maximum
Likelihood Estimate.
// Draw 100000 observations from a Beta with α = 2, β = 3:
double[] samples = GeneralizedBetaDistribution
.Random(alpha: 2, beta: 3, samples: 100000);
// Estimate a distribution from the data
var B = BetaDistribution.Estimate(samples);
// Explicitly using Method-of-moments estimation
var mm = BetaDistribution.Estimate(samples,
new BetaOptions { Method = BetaEstimationMethod.Moments });
// Explicitly using Maximum Likelihood estimation
var mle = BetaDistribution.Estimate(samples,
new BetaOptions { Method = BetaEstimationMethod.MaximumLikelihood });
Creates a new Beta distribution.
Creates a new Beta distribution.
The number of success r. Default is 0.
The number of trials n. Default is 1.
Creates a new Beta distribution.
The shape parameter α (alpha).
The shape parameter β (beta).
Gets the shape parameter α (alpha)
Gets the shape parameter β (beta).
Gets the number of successes r.
Gets the number of trials n.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the mean for this distribution.
The distribution's mean value.
The Beta's mean is computed as μ = a / (a + b).
Gets the variance for this distribution.
The distribution's variance.
The Beta's variance is computed as σ² = (a * b) / ((a + b)² * (a + b + 1)).
Gets the entropy for this distribution.
The distribution's entropy.
Gets the mode for this distribution.
The beta distribution's mode is given
by (a - 1) / (a + b - 2).
The distribution's mode value.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
The Beta's CDF is computed using the Incomplete
(regularized) Beta function I_x(a,b) as CDF(x) = I_x(a,b)
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
A probability value between 0 and 1.
A sample which could original the given probability
value when applied in the .
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
The Beta's PDF is computed as pdf(x) = c * x^(a - 1) * (1 - x)^(b - 1)
where constant c is c = 1.0 / Beta.Function(a, b)
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting,
such as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting,
such as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Computes the Gradient of the Log-Likelihood function for estimating Beta distributions.
The observed values.
The current alpha value.
The current beta value.
A bi-dimensional value containing the gradient w.r.t to alpha in its
first position, and the gradient w.r.t to be in its second position.
Computes the Gradient of the Log-Likelihood function for estimating Beta distributions.
The sum of log(y), where y refers to all observed values.
The sum of log(1 - y), where y refers to all observed values.
The total number of observed values.
The current alpha value.
The current beta value.
A bi-dimensional vector to store the gradient.
A bi-dimensional vector containing the gradient w.r.t to alpha in its
first position, and the gradient w.r.t to be in its second position.
Computes the Log-Likelihood function for estimating Beta distributions.
The observed values.
The current alpha value.
The current beta value.
The log-likelihood value for the given observations and given Beta parameters.
Computes the Log-Likelihood function for estimating Beta distributions.
The sum of log(y), where y refers to all observed values.
The sum of log(1 - y), where y refers to all observed values.
The total number of observed values.
The current alpha value.
The current beta value.
The log-likelihood value for the given observations and given Beta parameters.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
The random number generator to use as a source of randomness.
Default is to use .
A random observations drawn from this distribution.
Generates a random vector of observations from the
Beta distribution with the given parameters.
The shape parameter α (alpha).
The shape parameter β (beta).
The number of samples to generate.
An array of double values sampled from the specified Beta distribution.
Generates a random vector of observations from the
Beta distribution with the given parameters.
The shape parameter α (alpha).
The shape parameter β (beta).
The number of samples to generate.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified Beta distribution.
Generates a random vector of observations from the
Beta distribution with the given parameters.
The shape parameter α (alpha).
The shape parameter β (beta).
The number of samples to generate.
The location where to store the samples.
An array of double values sampled from the specified Beta distribution.
Generates a random vector of observations from the
Beta distribution with the given parameters.
The shape parameter α (alpha).
The shape parameter β (beta).
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified Beta distribution.
Generates a random observation from the
Beta distribution with the given parameters.
The shape parameter α (alpha).
The shape parameter β (beta).
A random double value sampled from the specified Beta distribution.
Generates a random observation from the
Beta distribution with the given parameters.
The shape parameter α (alpha).
The shape parameter β (beta).
The random number generator to use as a source of randomness.
Default is to use .
A random double value sampled from the specified Beta distribution.
Estimates a new Beta distribution from a set of observations.
Estimates a new Beta distribution from a set of weighted observations.
Estimates a new Beta distribution from a set of weighted observations.
Estimates a new Beta distribution from a set of observations.
Beta prime distribution.
In probability theory and statistics, the beta prime distribution (also known as inverted
beta distribution or beta distribution of the second kind) is an absolutely continuous
probability distribution defined for x > 0 with two parameters α and β, having the
probability density function:
x^(α-1) (1+x)^(-α-β)
f(x) = --------------------
B(α,β)
where B is the Beta function. While the related beta distribution is
the conjugate prior distribution of the parameter of a Bernoulli
distribution expressed as a probability, the beta prime distribution is the conjugate prior
distribution of the parameter of a Bernoulli distribution expressed in odds. The distribution is
a Pearson type VI distribution.
References:
-
Wikipedia, The Free Encyclopedia. Beta Prime distribution. Available on:
http://en.wikipedia.org/wiki/Beta_prime_distribution
The following example shows how to create and test the main characteristics
of an Beta prime distribution given its two non-negative shape parameters:
// Create a new Beta-Prime distribution with shape (4,2)
var betaPrime = new BetaPrimeDistribution(alpha: 4, beta: 2);
double mean = betaPrime.Mean; // 4.0
double median = betaPrime.Median; // 2.1866398762435981
double mode = betaPrime.Mode; // 1.0
double var = betaPrime.Variance; // +inf
double cdf = betaPrime.DistributionFunction(x: 0.4); // 0.02570357589099781
double pdf = betaPrime.ProbabilityDensityFunction(x: 0.4); // 0.16999719504628183
double lpdf = betaPrime.LogProbabilityDensityFunction(x: 0.4); // -1.7719733417957513
double ccdf = betaPrime.ComplementaryDistributionFunction(x: 0.4); // 0.97429642410900219
double icdf = betaPrime.InverseDistributionFunction(p: cdf); // 0.39999982363709291
double hf = betaPrime.HazardFunction(x: 0.4); // 0.17448200654307533
double chf = betaPrime.CumulativeHazardFunction(x: 0.4); // 0.026039684773113869
string str = betaPrime.ToString(CultureInfo.InvariantCulture); // BetaPrime(x; α = 4, β = 2)
Constructs a new Beta-Prime distribution with the given
two non-negative shape parameters a and b.
The distribution's non-negative shape parameter a.
The distribution's non-negative shape parameter b.
Gets the distribution's non-negative shape parameter a.
Gets the distribution's non-negative shape parameter b.
Gets the mean for this distribution.
The distribution's mean value.
Gets the mode for this distribution.
The distribution's mode value.
Gets the variance for this distribution.
The distribution's variance.
Not supported.
Gets the support interval for this distribution, which
for the Beta- Prime distribution ranges from 0 to all
positive numbers.
A containing
the support interval for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Cauchy-Lorentz distribution.
The Cauchy distribution, named after Augustin Cauchy, is a continuous probability
distribution. It is also known, especially among physicists, as the Lorentz
distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian)
function, or Breit–Wigner distribution. The simplest Cauchy distribution is called
the standard Cauchy distribution. It has the distribution of a random variable that
is the ratio of two independent standard normal random variables.
References:
-
Wikipedia, The Free Encyclopedia. Cauchy distribution.
Available from: http://en.wikipedia.org/wiki/Cauchy_distribution
The following example demonstrates how to instantiate a Cauchy distribution
with a given location parameter x0 and scale parameter γ (gamma), calculating
its main properties and characteristics:
double location = 0.42;
double scale = 1.57;
// Create a new Cauchy distribution with x0 = 0.42 and γ = 1.57
CauchyDistribution cauchy = new CauchyDistribution(location, scale);
// Common measures
double mean = cauchy.Mean; // NaN - Cauchy's mean is undefined.
double var = cauchy.Variance; // NaN - Cauchy's variance is undefined.
double median = cauchy.Median; // 0.42
// Cumulative distribution functions
double cdf = cauchy.DistributionFunction(x: 0.27); // 0.46968025841608563
double ccdf = cauchy.ComplementaryDistributionFunction(x: 0.27); // 0.53031974158391437
double icdf = cauchy.InverseDistributionFunction(p: 0.69358638272337991); // 1.5130304686978195
// Probability density functions
double pdf = cauchy.ProbabilityDensityFunction(x: 0.27); // 0.2009112009763413
double lpdf = cauchy.LogProbabilityDensityFunction(x: 0.27); // -1.6048922547266871
// Hazard (failure rate) functions
double hf = cauchy.HazardFunction(x: 0.27); // 0.3788491832800277
double chf = cauchy.CumulativeHazardFunction(x: 0.27); // 0.63427516833243092
// String representation
string str = cauchy.ToString(CultureInfo.InvariantCulture); // "Cauchy(x; x0 = 0.42, γ = 1.57)
The following example shows how to fit a Cauchy distribution (estimate its
location and shape parameters) given a set of observation values.
// Create an initial distribution
CauchyDistribution cauchy = new CauchyDistribution();
// Consider a vector of univariate observations
double[] observations = { 0.25, 0.12, 0.72, 0.21, 0.62, 0.12, 0.62, 0.12 };
// Fit to the observations
cauchy.Fit(observations);
// Check estimated values
double location = cauchy.Location; // 0.18383
double gamma = cauchy.Scale; // -0.10530
It is also possible to estimate only some of the Cauchy parameters at
a time. For this, you can specify a object
and pass it alongside the observations:
// Create options to estimate location only
CauchyOptions options = new CauchyOptions()
{
EstimateLocation = true,
EstimateScale = false
};
// Create an initial distribution with a pre-defined scale
CauchyDistribution cauchy = new CauchyDistribution(location: 0, scale: 4.2);
// Fit to the observations
cauchy.Fit(observations, options);
// Check estimated values
double location = cauchy.Location; // 0.3471218110202
double gamma = cauchy.Scale; // 4.2 (unchanged)
Constructs a Cauchy-Lorentz distribution
with location parameter 0 and scale 1.
Constructs a Cauchy-Lorentz distribution
with given location and scale parameters.
The location parameter x0.
The scale parameter gamma (γ).
Gets the distribution's
location parameter x0.
Gets the distribution's
scale parameter gamma.
Gets the median for this distribution.
The distribution's median value.
The Cauchy's median is the location parameter x0.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the mode for this distribution.
The distribution's mode value.
The Cauchy's median is the location parameter x0.
Cauchy's mean is undefined.
Undefined.
Cauchy's variance is undefined.
Undefined.
Gets the entropy for this distribution.
The distribution's entropy.
The Cauchy's entropy is defined as log(scale) + log(4*π).
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
The Cauchy's CDF is defined as CDF(x) = 1/π * atan2(x-location, scale) + 0.5.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
The Cauchy's PDF is defined as PDF(x) = c / (1.0 + ((x-location)/scale)²)
where the constant c is given by c = 1.0 / (π * scale);
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
See .
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
See .
Gets the Standard Cauchy Distribution,
with zero location and unitary shape.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
The random number generator to use as a source of randomness.
Default is to use .
A random observations drawn from this distribution.
Generates a random observation from the
Cauchy distribution with the given parameters.
The location parameter x0.
The scale parameter gamma.
A random double value sampled from the specified Cauchy distribution.
Generates a random vector of observations from the
Cauchy distribution with the given parameters.
The location parameter x0.
The scale parameter gamma.
The number of samples to generate.
An array of double values sampled from the specified Cauchy distribution.
Generates a random vector of observations from the
Cauchy distribution with the given parameters.
The location parameter x0.
The scale parameter gamma.
The number of samples to generate.
The location where to store the samples.
An array of double values sampled from the specified Cauchy distribution.
Generates a random observation from the
Cauchy distribution with the given parameters.
The location parameter x0.
The scale parameter gamma.
The random number generator to use as a source of randomness.
Default is to use .
A random double value sampled from the specified Cauchy distribution.
Generates a random vector of observations from the
Cauchy distribution with the given parameters.
The location parameter x0.
The scale parameter gamma.
The number of samples to generate.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified Cauchy distribution.
Generates a random vector of observations from the
Cauchy distribution with the given parameters.
The location parameter x0.
The scale parameter gamma.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified Cauchy distribution.
Returns a that represents this instance.
A that represents this instance.
Birnbaum-Saunders (Fatigue Life) distribution.
The Birnbaum–Saunders distribution, also known as the fatigue life distribution,
is a probability distribution used extensively in reliability applications to model
failure times. There are several alternative formulations of this distribution in
the literature. It is named after Z. W. Birnbaum and S. C. Saunders.
References:
-
Wikipedia, The Free Encyclopedia. Birnbaum–Saunders distribution.
Available from: http://en.wikipedia.org/wiki/Birnbaum%E2%80%93Saunders_distribution
-
NIST/SEMATECH e-Handbook of Statistical Methods, Birnbaum-Saunders (Fatigue Life) Distribution
Available from: http://www.itl.nist.gov/div898/handbook/eda/section3/eda366a.htm
This example shows how to create a Birnbaum-Saunders distribution
and compute some of its properties.
// Creates a new Birnbaum-Saunders distribution
var bs = new BirnbaumSaundersDistribution(shape: 0.42);
double mean = bs.Mean; // 1.0882000000000001
double median = bs.Median; // 1.0
double var = bs.Variance; // 0.21529619999999997
double cdf = bs.DistributionFunction(x: 1.4); // 0.78956384911580346
double pdf = bs.ProbabilityDensityFunction(x: 1.4); // 1.3618433601225426
double lpdf = bs.LogProbabilityDensityFunction(x: 1.4); // 0.30883919386130815
double ccdf = bs.ComplementaryDistributionFunction(x: 1.4); // 0.21043615088419654
double icdf = bs.InverseDistributionFunction(p: cdf); // 2.0618330099769064
double hf = bs.HazardFunction(x: 1.4); // 6.4715276077824093
double chf = bs.CumulativeHazardFunction(x: 1.4); // 1.5585729930861034
string str = bs.ToString(CultureInfo.InvariantCulture); // BirnbaumSaunders(x; μ = 0, β = 1, γ = 0.42)
Constructs a Birnbaum-Saunders distribution
with location parameter 0, scale 1, and shape 1.
Constructs a Birnbaum-Saunders distribution
with location parameter 0, scale 1, and the
given shape.
The shape parameter gamma (γ). Default is 1.
Constructs a Birnbaum-Saunders distribution
with given location, shape and scale parameters.
The location parameter μ. Default is 0.
The scale parameter beta (β). Default is 1.
The shape parameter gamma (γ). Default is 1.
Gets the distribution's location parameter μ.
Gets the distribution's scale parameter β.
Gets the distribution's shape parameter γ.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the mean for this distribution.
The Birnbaum Saunders mean is defined as
1 + 0.5γ².
The distribution's mean value.
Gets the variance for this distribution.
The Birnbaum Saunders variance is defined as
γ² (1 + (5/4)γ²).
The distribution's mean value.
This method is not supported.
This method is not supported.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
A probability value between 0 and 1.
A sample which could original the given probability
value when applied in the .
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Distribution types supported by the Anderson-Darling distribution.
The statistic should reflect p-values for
a Anderson-Darling comparison against an
Uniform distribution.
The statistic should reflect p-values for
a Anderson-Darling comparison against a
Normal distribution.
Anderson-Darling (A²) distribution.
// Create a new Anderson Darling distribution (A²) for comparing against a Gaussian
var a2 = new AndersonDarlingDistribution(AndersonDarlingDistributionType.Normal, 30);
double median = a2.Median; // 0.33089957635450062
double chf = a2.CumulativeHazardFunction(x: 0.27); // 0.42618068373640966
double cdf = a2.DistributionFunction(x: 0.27); // 0.34700165471995292
double ccdf = a2.ComplementaryDistributionFunction(x: 0.27); // 0.65299834528004708
double icdf = a2.InverseDistributionFunction(p: cdf); // 0.27000000012207787
string str = a2.ToString(CultureInfo.InvariantCulture); // "A²(x; n = 30)"
Gets the type of the distribution that the
Anderson-Darling is being performed against.
Gets the number of samples distribution parameter.
Creates a new Anderson-Darling distribution.
The type of the compared distribution.
The number of samples.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Not supported.
Not supported.
Not supported.
Not supported.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
See .
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
A single point in the distribution range.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF.
Not supported.
Not supported.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Grubb's statistic distribution.
Gets the number of samples for the distribution.
Not supported.
Not supported.
Not supported.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Initializes a new instance of the class.
The number of samples.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
System.Double.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
A probability value between 0 and 1.
A sample which could original the given probability
value when applied in the .
The Inverse Cumulative Distribution Function (ICDF) specifies, for
a given probability, the value which the random variable will be at,
or below, with that probability.
Not supported.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
The format.
The format provider.
A that represents this instance.
Kumaraswamy distribution.
In probability and statistics, the Kumaraswamy's double bounded distribution is a
family of continuous probability distributions defined on the interval [0,1] differing
in the values of their two non-negative shape parameters, a and b.
It is similar to the Beta distribution, but much simpler to use especially in simulation
studies due to the simple closed form of both its probability density function and
cumulative distribution function. This distribution was originally proposed by Poondi
Kumaraswamy for variables that are lower and upper bounded.
A good example of the use of the Kumaraswamy distribution is the storage volume of a
reservoir of capacity zmax whose upper bound is zmax and lower
bound is 0 (Fletcher and Ponnambalam, 1996).
References:
-
Wikipedia, The Free Encyclopedia. Kumaraswamy distribution. Available on:
http://en.wikipedia.org/wiki/Kumaraswamy_distribution
The following example shows how to create and test the main characteristics
of an Kumaraswamy distribution given its two non-negative shape parameters:
// Create a new Kumaraswamy distribution with shape (4,2)
var kumaraswamy = new KumaraswamyDistribution(a: 4, b: 2);
double mean = kumaraswamy.Mean; // 0.71111111111111114
double median = kumaraswamy.Median; // 0.73566031573423674
double mode = kumaraswamy.Mode; // 0.80910671157022118
double var = kumaraswamy.Variance; // 0.027654320987654302
double cdf = kumaraswamy.DistributionFunction(x: 0.4); // 0.050544639999999919
double pdf = kumaraswamy.ProbabilityDensityFunction(x: 0.4); // 0.49889280000000014
double lpdf = kumaraswamy.LogProbabilityDensityFunction(x: 0.4); // -0.69536403596913343
double ccdf = kumaraswamy.ComplementaryDistributionFunction(x: 0.4); // 0.94945536000000008
double icdf = kumaraswamy.InverseDistributionFunction(p: cdf); // 0.40000011480618253
double hf = kumaraswamy.HazardFunction(x: 0.4); // 0.52545155993431869
double chf = kumaraswamy.CumulativeHazardFunction(x: 0.4); // 0.051866764053008864
string str = kumaraswamy.ToString(CultureInfo.InvariantCulture); // Kumaraswamy(x; a = 4, b = 2)
Constructs a new Kumaraswamy's double bounded distribution with
the given two non-negative shape parameters a and b.
The distribution's non-negative shape parameter a.
The distribution's non-negative shape parameter b.
Gets the distribution's non-negative shape parameter a.
Gets the distribution's non-negative shape parameter b.
Gets the mean for this distribution.
The distribution's mean value.
Gets the variance for this distribution.
The distribution's variance.
Gets the median for this distribution.
The distribution's median value.
Gets the mode for this distribution.
The distribution's mode value.
Not supported.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Rademacher distribution.
Gets the mean for this distribution.
The distribution's mean value.
Gets the variance for this distribution.
The distribution's variance.
Gets the median for this distribution.
The distribution's median value.
Gets the entropy for this distribution.
The distribution's entropy.
Returns NaN.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets P(X ≤ k), the cumulative distribution function
(cdf) for this distribution evaluated at point k.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of k occurring
in the current distribution.
The Probability Mass Function (PMF) describes the
probability that a given value x will occur.
Returns a that represents this instance.
The format.
The format provider.
A that represents this instance.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Shapiro-Wilk distribution.
The Shapiro-Wilk distribution models the distribution of
Shapiro-Wilk's test statistic.
References:
-
Royston, P. "Algorithm AS 181: The W test for Normality", Applied Statistics (1982),
Vol. 31, pp. 176–180.
-
Royston, P. "Remark AS R94", Applied Statistics (1995), Vol. 44, No. 4, pp. 547-551.
Available at http://lib.stat.cmu.edu/apstat/R94
-
Royston, P. "Approximating the Shapiro-Wilk W-test for non-normality",
Statistics and Computing (1992), Vol. 2, pp. 117-119.
-
Royston, P. "An Extension of Shapiro and Wilk's W Test for Normality to Large
Samples", Journal of the Royal Statistical Society Series C (1982a), Vol. 31,
No. 2, pp. 115-124.
// Create a new Shapiro-Wilk's W for 5 samples
var sw = new ShapiroWilkDistribution(samples: 5);
double mean = sw.Mean; // 0.81248567196628929
double median = sw.Median; // 0.81248567196628929
double mode = sw.Mode; // 0.81248567196628929
double cdf = sw.DistributionFunction(x: 0.84); // 0.83507812080728383
double pdf = sw.ProbabilityDensityFunction(x: 0.84); // 0.82021062372326459
double lpdf = sw.LogProbabilityDensityFunction(x: 0.84); // -0.1981941135071546
double ccdf = sw.ComplementaryDistributionFunction(x: 0.84); // 0.16492187919271617
double icdf = sw.InverseDistributionFunction(p: cdf); // 0.84000000194587177
double hf = sw.HazardFunction(x: 0.84); // 4.9733281462602292
double chf = sw.CumulativeHazardFunction(x: 0.84); // 1.8022833766369502
string str = sw.ToString(CultureInfo.InvariantCulture); // W(x; n = 12)
Gets the number of samples distribution parameter.
Creates a new Shapiro-Wilk distribution.
The number of samples.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the mean for this distribution.
The distribution's mean value.
Gets the mode for this distribution.
The distribution's mode value.
Not supported.
Gets the median for this distribution.
The distribution's median value.
Not supported.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
See .
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
A single point in the distribution range.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Not supported.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Log-Logistic distribution.
In probability and statistics, the log-logistic distribution (known as the Fisk
distribution in economics) is a continuous probability distribution for a non-negative
random variable. It is used in survival analysis as a parametric model for events
whose rate increases initially and decreases later, for example mortality rate from
cancer following diagnosis or treatment. It has also been used in hydrology to model
stream flow and precipitation, and in economics as a simple model of the distribution
of wealth or income.
The log-logistic distribution is the probability distribution of a random variable
whose logarithm has a logistic distribution. It is similar in shape to the log-normal
distribution but has heavier tails. Its cumulative distribution function can be written
in closed form, unlike that of the log-normal.
References:
-
Wikipedia, The Free Encyclopedia. Log-logistic distribution. Available on:
http://en.wikipedia.org/wiki/Log-logistic_distribution
This examples shows how to create a Log-Logistic distribution
and compute some of its properties and characteristics.
// Create a LLD2 distribution with scale = 0.42, shape = 2.2
var log = new LogLogisticDistribution(scale: 0.42, shape: 2.2);
double mean = log.Mean; // 0.60592605102976937
double median = log.Median; // 0.42
double mode = log.Mode; // 0.26892249963239817
double var = log.Variance; // 1.4357858982592435
double cdf = log.DistributionFunction(x: 1.4); // 0.93393329906725353
double pdf = log.ProbabilityDensityFunction(x: 1.4); // 0.096960115938100763
double lpdf = log.LogProbabilityDensityFunction(x: 1.4); // -2.3334555609306102
double ccdf = log.ComplementaryDistributionFunction(x: 1.4); // 0.066066700932746525
double icdf = log.InverseDistributionFunction(p: cdf); // 1.4000000000000006
double hf = log.HazardFunction(x: 1.4); // 1.4676094699628273
double chf = log.CumulativeHazardFunction(x: 1.4); // 2.7170904270953637
string str = log.ToString(CultureInfo.InvariantCulture); // LogLogistic(x; α = 0.42, β = 2.2)
Constructs a Log-Logistic distribution with unit scale and unit shape.
Constructs a Log-Logistic distribution
with the given scale and unit shape.
The distribution's scale value α (alpha). Default is 1.
Constructs a Log-Logistic distribution
with the given scale and shape parameters.
The distribution's scale value α (alpha). Default is 1.
The distribution's shape value β (beta). Default is 1.
Gets the distribution's scale value (α).
Gets the distribution's shape value (β).
Gets the mean for this distribution.
The distribution's mean value.
Gets the median for this distribution.
The distribution's median value.
Gets the variance for this distribution.
The distribution's variance.
Gets the mode for this distribution.
The distribution's mode value.
Not supported.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
A probability value between 0 and 1.
A sample which could original the given probability
value when applied in the .
Gets the first derivative of the
inverse distribution function (icdf) for this distribution evaluated
at probability p.
A probability value between 0 and 1.
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
A single point in the distribution range.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF.
Gets the hazard function, also known as the failure rate or
the conditional failure density function for this distribution
evaluated at point x.
A single point in the distribution range.
The conditional failure density function h(x)
evaluated at x in the current distribution.
The hazard function is the ratio of the probability
density function f(x) to the survival function, S(x).
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Creates a new using
the location-shape parametrization. In this parametrization,
is taken as 1 / .
The location parameter μ (mu) [taken as μ = α].
The distribution's shape value σ (sigma) [taken as σ = β].
A with α = μ and β = 1/σ.
Inverse chi-Square (χ²) probability distribution
In probability and statistics, the inverse-chi-squared distribution (or
inverted-chi-square distribution) is a continuous probability distribution
of a positive-valued random variable. It is closely related to the
chi-squared distribution and its
specific importance is that it arises in the application of Bayesian
inference to the normal distribution, where it can be used as the
prior and posterior distribution for an unknown variance.
The inverse-chi-squared distribution (or inverted-chi-square distribution) is
the probability distribution of a random variable whose multiplicative inverse
(reciprocal) has a chi-squared distribution.
It is also often defined as the distribution of a random variable whose reciprocal
divided by its degrees of freedom is a chi-squared distribution. That is, if X has
the chi-squared distribution with v degrees of freedom, then according to
the first definition, 1/X has the inverse-chi-squared distribution with v
degrees of freedom; while according to the second definition, vX has the
inverse-chi-squared distribution with v degrees of freedom. Only the first
definition is covered by this class.
References:
-
Wikipedia, The Free Encyclopedia. Inverse-chi-square distribution. Available on:
http://en.wikipedia.org/wiki/Inverse-chi-squared_distribution
The following example demonstrates how to create a new inverse
χ² distribution with the given degrees of freedom.
// Create a new inverse χ² distribution with 7 d.f.
var invchisq = new InverseChiSquareDistribution(degreesOfFreedom: 7);
double mean = invchisq.Mean; // 0.2
double median = invchisq.Median; // 6.345811068141737
double var = invchisq.Variance; // 75
double cdf = invchisq.DistributionFunction(x: 6.27); // 0.50860033566176044
double pdf = invchisq.ProbabilityDensityFunction(x: 6.27); // 0.0000063457380298844403
double lpdf = invchisq.LogProbabilityDensityFunction(x: 6.27); // -11.967727146795536
double ccdf = invchisq.ComplementaryDistributionFunction(x: 6.27); // 0.49139966433823956
double icdf = invchisq.InverseDistributionFunction(p: cdf); // 6.2699998329362963
double hf = invchisq.HazardFunction(x: 6.27); // 0.000012913598625327002
double chf = invchisq.CumulativeHazardFunction(x: 6.27); // 0.71049750196765715
string str = invchisq.ToString(); // "Inv-χ²(x; df = 7)"
Constructs a new Inverse Chi-Square distribution
with the given degrees of freedom.
The degrees of freedom for the distribution.
Gets the Degrees of Freedom for this distribution.
Gets the probability density function (pdf) for
the χ² distribution evaluated at point x.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
The probability of x occurring
in the current distribution.
Gets the cumulative distribution function (cdf) for
the χ² distribution evaluated at point x.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the mean for this distribution.
Gets the variance for this distribution.
Gets the mode for this distribution.
The distribution's mode value.
Gets the entropy for this distribution.
This method is not supported.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Hyperbolic Secant distribution.
In probability theory and statistics, the hyperbolic secant distribution is
a continuous probability distribution whose probability density function and
characteristic function are proportional to the hyperbolic secant function.
The hyperbolic secant function is equivalent to the inverse hyperbolic cosine,
and thus this distribution is also called the inverse-cosh distribution.
References:
-
Wikipedia, The Free Encyclopedia. Hyperbolic secant distribution. Available on:
http://en.wikipedia.org/wiki/Sech_distribution
This examples shows how to create a Sech distribution,
compute some of its properties and generate a number of
random samples from it.
// Create a new hyperbolic secant distribution
var sech = new HyperbolicSecantDistribution();
double mean = sech.Mean; // 0.0
double median = sech.Median; // 0.0
double mode = sech.Mode; // 0.0
double var = sech.Variance; // 1.0
double cdf = sech.DistributionFunction(x: 1.4); // 0.92968538268895873
double pdf = sech.ProbabilityDensityFunction(x: 1.4); // 0.10955386512899701
double lpdf = sech.LogProbabilityDensityFunction(x: 1.4); // -2.2113389316917877
double ccdf = sech.ComplementaryDistributionFunction(x: 1.4); // 0.070314617311041272
double icdf = sech.InverseDistributionFunction(p: cdf); // 1.40
double hf = sech.HazardFunction(x: 1.4); // 1.5580524977385339
string str = sech.ToString(); // Sech(x)
Constructs a Hyperbolic Secant (Sech) distribution.
Gets the mean for this distribution (always zero).
The distribution's mean value.
Gets the median for this distribution (always zero).
The distribution's median value.
Gets the variance for this distribution (always one).
The distribution's variance.
Gets the Standard Deviation (the square root of
the variance) for the current distribution.
The distribution's standard deviation.
Gets the mode for this distribution (always zero).
The distribution's mode value.
Gets the support interval for this distribution (-inf, +inf).
A containing
the support interval for this distribution.
Gets the entropy for this distribution.
The distribution's entropy.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
The format.
The format provider.
A that represents this instance.
Logistic distribution.
In probability theory and statistics, the logistic distribution is a continuous
probability distribution. Its cumulative distribution function is the logistic
function, which appears in logistic regression and feedforward neural networks.
It resembles the normal distribution in shape but has heavier tails (higher
kurtosis). The Tukey lambda distribution
can be considered a generalization of the logistic distribution since it adds a
shape parameter, λ (the Tukey distribution becomes logistic when λ is zero).
References:
-
Wikipedia, The Free Encyclopedia. Logistic distribution. Available on:
http://en.wikipedia.org/wiki/Logistic_distribution
This examples shows how to create a Logistic distribution,
compute some of its properties and generate a number of
random samples from it.
// Create a logistic distribution with μ = 0.42 and scale = 3
var log = new LogisticDistribution(location: 0.42, scale: 1.2);
double mean = log.Mean; // 0.42
double median = log.Median; // 0.42
double mode = log.Mode; // 0.42
double var = log.Variance; // 4.737410112522892
double cdf = log.DistributionFunction(x: 1.4); // 0.693528308197921
double pdf = log.ProbabilityDensityFunction(x: 1.4); // 0.17712232827170876
double lpdf = log.LogProbabilityDensityFunction(x: 1.4); // -1.7309146649427332
double ccdf = log.ComplementaryDistributionFunction(x: 1.4); // 0.306471691802079
double icdf = log.InverseDistributionFunction(p: cdf); // 1.3999999999999997
double hf = log.HazardFunction(x: 1.4); // 0.57794025683160088
double chf = log.CumulativeHazardFunction(x: 1.4); // 1.1826298874077226
string str = log.ToString(CultureInfo.InvariantCulture); // Logistic(x; μ = 0.42, scale = 1.2)
Constructs a Logistic distribution
with zero location and unit scale.
Constructs a Logistic distribution
with given location and unit scale.
The distribution's location value μ (mu).
Constructs a Logistic distribution
with given location and scale parameters.
The distribution's location value μ (mu).
The distribution's scale value s.
Gets the location value μ (mu).
Gets the location value μ (mu).
The distribution's mean value.
Gets the distribution's scale value (s).
Gets the median for this distribution.
The distribution's median value.
Gets the variance for this distribution.
The distribution's variance.
Gets the mode for this distribution.
In the logistic distribution, the mode is equal
to the distribution value.
The distribution's mode value.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the entropy for this distribution.
In the logistic distribution, the entropy is
equal to ln(s) + 2.
The distribution's entropy.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
A probability value between 0 and 1.
A sample which could original the given probability
value when applied in the .
Gets the first derivative of the
inverse distribution function (icdf) for this distribution evaluated
at probability p.
A probability value between 0 and 1.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
General continuous distribution.
The general continuous distribution provides the automatic calculation for
a variety of distribution functions and measures given only definitions for
the Probability Density Function (PDF) or the Cumulative Distribution Function
(CDF). Values such as the Expected value, Variance, Entropy and others are
computed through numeric integration.
// Let's suppose we have a formula that defines a probability distribution
// but we dont know much else about it. We don't know the form of its cumulative
// distribution function, for example. We would then like to know more about
// it, such as the underlying distribution's moments, characteristics, and
// properties.
// Let's suppose the formula we have is this one:
double mu = 5;
double sigma = 4.2;
Func>double, double> df = x => 1.0 / (sigma * Math.Sqrt(2 * Math.PI))
* Math.Exp(-Math.Pow(x - mu, 2) / (2 * sigma * sigma));
// And for the moment, let's also pretend we don't know it is actually the
// p.d.f. of a Gaussian distribution with mean 5 and std. deviation of 4.2.
// So, let's create a distribution based _solely_ on the formula we have:
var distribution = GeneralContinuousDistribution.FromDensityFunction(df);
// Now, we can check everything that we can know about it:
double mean = distribution.Mean; // 5 (note that all of those have been
double median = distribution.Median; // 5 detected automatically simply from
double var = distribution.Variance; // 17.64 the given density formula through
double mode = distribution.Mode; // 5 numerical methods)
double cdf = distribution.DistributionFunction(x: 1.4); // 0.19568296915377595
double pdf = distribution.ProbabilityDensityFunction(x: 1.4); // 0.065784567984404935
double lpdf = distribution.LogProbabilityDensityFunction(x: 1.4); // -2.7213699972695058
double ccdf = distribution.ComplementaryDistributionFunction(x: 1.4); // 0.80431703084622408
double icdf = distribution.InverseDistributionFunction(p: cdf); // 1.3999999997024655
double hf = distribution.HazardFunction(x: 1.4); // 0.081789351041333558
double chf = distribution.CumulativeHazardFunction(x: 1.4); // 0.21776177055276186
Creates a new with the given PDF and CDF functions.
The distribution's support over the real line.
A probability density function.
A cumulative distribution function.
Creates a new with the given PDF and CDF functions.
A distribution whose properties will be numerically estimated.
Creates a new
from an existing
continuous distribution.
The distribution.
A representing the same
but whose measures and functions are computed
using numerical integration and differentiation.
Creates a new
using only a probability density function definition.
A probability density function.
A created from the
whose measures and functions are computed using
numerical integration and differentiation.
Creates a new
using only a probability density function definition.
The distribution's support over the real line.
A probability density function.
A created from the
whose measures and functions are computed using
numerical integration and differentiation.
Creates a new
using only a cumulative distribution function definition.
A cumulative distribution function.
A created from the
whose measures and functions are computed using
numerical integration and differentiation.
Creates a new
using only a cumulative distribution function definition.
The distribution's support over the real line.
A cumulative distribution function.
A created from the
whose measures and functions are computed using
numerical integration and differentiation.
Creates a new
using only a probability density function definition.
The distribution's support over the real line.
A probability density function.
The integration method to use for numerical computations.
A created from the
whose measures and functions are computed using
numerical integration and differentiation.
Creates a new
using only a cumulative distribution function definition.
The distribution's support over the real line.
A cumulative distribution function.
The integration method to use for numerical computations.
A created from the
whose measures and functions are computed using
numerical integration and differentiation.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the mean for this distribution.
The distribution's mean value.
Gets the variance for this distribution.
The distribution's variance.
Gets the entropy for this distribution.
The distribution's entropy.
Gets the mode for this distribution.
The distribution's mode value.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
The format.
The format provider.
A that represents this instance.
Lévy distribution.
In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous
probability distribution for a non-negative random variable. In spectroscopy, this distribution, with
frequency as the dependent variable, is known as a van der Waals profile. It is a special case of the
inverse-gamma distribution.
It is one of the few distributions that are stable and that have probability density functions that can
be expressed analytically, the others being the normal distribution and the Cauchy distribution. All three
are special cases of the stable distributions, which do not generally have a probability density function
which can be expressed analytically.
References:
-
Wikipedia, The Free Encyclopedia. Lévy distribution. Available on:
https://en.wikipedia.org/wiki/L%C3%A9vy_distribution
This examples shows how to create a Lévy distribution
and how to compute some of its measures and properties.
// Create a new Lévy distribution on 1 with scale 4.2:
var levy = new LevyDistribution(location: 1, scale: 4.2);
double mean = levy.Mean; // +inf
double median = levy.Median; // 10.232059220934481
double mode = levy.Mode; // NaN
double var = levy.Variance; // +inf
double cdf = levy.DistributionFunction(x: 1.4); // 0.0011937454448720029
double pdf = levy.ProbabilityDensityFunction(x: 1.4); // 0.016958939623898304
double lpdf = levy.LogProbabilityDensityFunction(x: 1.4); // -4.0769601727487803
double ccdf = levy.ComplementaryDistributionFunction(x: 1.4); // 0.99880625455512795
double icdf = levy.InverseDistributionFunction(p: cdf); // 1.3999999
double hf = levy.HazardFunction(x: 1.4); // 0.016979208476674869
double chf = levy.CumulativeHazardFunction(x: 1.4); // 0.0011944585265140923
string str = levy.ToString(CultureInfo.InvariantCulture); // Lévy(x; μ = 1, c = 4.2)
Constructs a new
with zero location and unit scale.
Constructs a new in
the given and with unit scale.
The distribution's location.
Constructs a new in the
given and .
The distribution's location.
The distribution's scale.
Gets the location μ (mu) for this distribution.
Gets the location c for this distribution.
Gets the mean for this distribution, which for
the Levy distribution is always positive infinity.
This property always returns Double.PositiveInfinity.
Gets the median for this distribution.
The distribution's median value.
Gets the variance for this distribution, which for
the Levy distribution is always positive infinity.
This property always returns Double.PositiveInfinity.
Gets the mode for this distribution.
The distribution's mode value.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the entropy for this distribution.
The distribution's entropy.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
A probability value between 0 and 1.
A sample which could original the given probability
value when applied in the .
The Inverse Cumulative Distribution Function (ICDF) specifies, for
a given probability, the value which the random variable will be at,
or below, with that probability.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Folded Normal (Gaussian) distribution.
The folded normal distribution is a probability distribution related to the normal
distribution. Given a normally distributed random variable X with mean μ and variance
σ², the random variable Y = |X| has a folded normal distribution. Such a case may be
encountered if only the magnitude of some variable is recorded, but not its sign. The
distribution is called Folded because probability mass to the left of the x = 0 is
"folded" over by taking the absolute value.
The Half-Normal (Gaussian) distribution is a special
case of this distribution and can be created using a named constructor.
References:
-
Wikipedia, The Free Encyclopedia. Folded Normal distribution. Available on:
https://en.wikipedia.org/wiki/Folded_normal_distribution
This examples shows how to create a Folded Normal distribution
and how to compute some of its properties and measures.
// Creates a new Folded Normal distribution based on a Normal
// distribution with mean value 4 and standard deviation 4.2:
//
var fn = new FoldedNormalDistribution(mean: 4, stdDev: 4.2);
double mean = fn.Mean; // 4.765653108337438
double median = fn.Median; // 4.2593565881862734
double mode = fn.Mode; // 2.0806531871308014
double var = fn.Variance; // 10.928550450993715
double cdf = fn.DistributionFunction(x: 1.4); // 0.16867109769018807
double pdf = fn.ProbabilityDensityFunction(x: 1.4); // 0.11998602818182187
double lpdf = fn.LogProbabilityDensityFunction(x: 1.4); // -2.1203799747969523
double ccdf = fn.ComplementaryDistributionFunction(x: 1.4); // 0.83132890230981193
double icdf = fn.InverseDistributionFunction(p: cdf); // 1.4
double hf = fn.HazardFunction(x: 1.4); // 0.14433039420191671
double chf = fn.CumulativeHazardFunction(x: 1.4); // 0.18472977144474392
string str = fn.ToString(CultureInfo.InvariantCulture); // FN(x; μ = 4, σ² = 17.64)
Creates a new
with zero mean and unit standard deviation.
Creates a new with
the given and unit standard deviation.
The mean of the original normal distribution that should be folded.
Creates a new with
the given and
standard deviation
The mean of the original normal distribution that should be folded.
The standard deviation of the original normal distribution that should be folded.
Creates a new Half-normal distribution with the given
standard deviation. The half-normal distribution is a special case
of the when location is zero.
The standard deviation of the original normal distribution that should be folded.
Gets the mean for this distribution.
The distribution's mean value.
Gets the variance for this distribution.
The distribution's variance.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
This method is not supported.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Shift Log-Logistic distribution.
The shifted log-logistic distribution is a probability distribution also known as
the generalized log-logistic or the three-parameter log-logistic distribution. It
has also been called the generalized logistic distribution, but this conflicts with
other uses of the term: see generalized logistic distribution.
References:
-
Wikipedia, The Free Encyclopedia. Shifted log-logistic distribution. Available on:
http://en.wikipedia.org/wiki/Shifted_log-logistic_distribution
This examples shows how to create a Shifted Log-Logistic distribution,
compute some of its properties and generate a number of random samples
from it.
// Create a LLD3 distribution with μ = 0.0, scale = 0.42, and shape = 0.1
var log = new ShiftedLogLogisticDistribution(location: 0, scale: 0.42, shape: 0.1);
double mean = log.Mean; // 0.069891101544818923
double median = log.Median; // 0.0
double mode = log.Mode; // -0.083441677069328604
double var = log.Variance; // 0.62447259946747213
double cdf = log.DistributionFunction(x: 1.4); // 0.94668863559417671
double pdf = log.ProbabilityDensityFunction(x: 1.4); // 0.090123683626808615
double lpdf = log.LogProbabilityDensityFunction(x: 1.4); // -2.4065722895662613
double ccdf = log.ComplementaryDistributionFunction(x: 1.4); // 0.053311364405823292
double icdf = log.InverseDistributionFunction(p: cdf); // 1.4000000037735139
double hf = log.HazardFunction(x: 1.4); // 1.6905154207038875
double chf = log.CumulativeHazardFunction(x: 1.4); // 2.9316057546685061
string str = log.ToString(CultureInfo.InvariantCulture); // LLD3(x; μ = 0, σ = 0.42, ξ = 0.1)
Constructs a Shifted Log-Logistic distribution
with zero location, unit scale, and zero shape.
Constructs a Shifted Log-Logistic distribution
with the given location, unit scale and zero shape.
The distribution's location value μ (mu).
Constructs a Shifted Log-Logistic distribution
with the given location and scale and zero shape.
The distribution's location value μ (mu).
The distribution's scale value σ (sigma).
Constructs a Shifted Log-Logistic distribution
with the given location and scale and zero shape.
The distribution's location value μ (mu).
The distribution's scale value s.
The distribution's shape value ξ (ksi).
Gets the mean for this distribution.
The distribution's mean value.
Gets the distribution's location value μ (mu).
Gets the distribution's scale value (σ).
Gets the distribution's shape value (ξ).
Gets the median for this distribution.
The distribution's median value.
Gets the variance for this distribution.
The distribution's variance.
Gets the mode for this distribution.
The distribution's mode value.
Not supported.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Skew Normal distribution.
In probability theory and statistics, the skew normal distribution is a
continuous probability distribution
that generalises the normal distribution to allow
for non-zero skewness.
References:
-
Wikipedia, The Free Encyclopedia. Skew normal distribution. Available on:
https://en.wikipedia.org/wiki/Skew_normal_distribution
This examples shows how to create a Skew normal distribution
and compute some of its properties and derived measures.
// Create a Skew normal distribution with location 2, scale 3 and shape 4.2
var skewNormal = new SkewNormalDistribution(location: 2, scale: 3, shape: 4.2);
double mean = skewNormal.Mean; // 4.3285611780515953
double median = skewNormal.Median; // 4.0230040653062265
double var = skewNormal.Variance; // 3.5778028400709641
double mode = skewNormal.Mode; // 3.220622226764422
double cdf = skewNormal.DistributionFunction(x: 1.4); // 0.020166854942526125
double pdf = skewNormal.ProbabilityDensityFunction(x: 1.4); // 0.052257431834162059
double lpdf = skewNormal.LogProbabilityDensityFunction(x: 1.4); // -2.9515731621912877
double ccdf = skewNormal.ComplementaryDistributionFunction(x: 1.4); // 0.97983314505747388
double icdf = skewNormal.InverseDistributionFunction(p: cdf); // 1.3999998597203041
double hf = skewNormal.HazardFunction(x: 1.4); // 0.053332990517581239
double chf = skewNormal.CumulativeHazardFunction(x: 1.4); // 0.020372981958858238
string str = skewNormal.ToString(CultureInfo.InvariantCulture); // Sn(x; ξ = 2, ω = 3, α = 4.2)
Constructs a Skew normal distribution with
zero location, unit scale and zero shape.
Constructs a Skew normal distribution with
given location, unit scale and zero skewness.
The distribution's location value ξ (ksi).
Constructs a Skew normal distribution with
given location and scale and zero skewness.
The distribution's location value ξ (ksi).
The distribution's scale value ω (omega).
Constructs a Skew normal distribution
with given mean and standard deviation.
The distribution's location value ξ (ksi).
The distribution's scale value ω (omega).
The distribution's shape value α (alpha).
Gets the skew-normal distribution's location value ξ (ksi).
Gets the skew-normal distribution's scale value ω (omega).
Gets the skew-normal distribution's shape value α (alpha).
Not supported.
Gets the mean for this distribution.
The distribution's mean value.
Gets the variance for this distribution.
The distribution's variance.
Gets the skewness for this distribution.
Gets the excess kurtosis for this distribution.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Create a new that
corresponds to a with
the given mean and standard deviation.
The distribution's mean value μ (mu).
The distribution's standard deviation σ (sigma).
A representing
a with the given parameters.
Trapezoidal distribution.
Trapezoidal distributions have been used in many areas and studied under varying
scopes, such as in the excellent work of (van Dorp and Kotz, 2003), risk analysis
(Pouliquen, 1970) and (Powell and Wilson, 1997), fuzzy set theory (Chen and Hwang,
1992), applied phyisics, and biomedical applications (Flehinger and Kimmel, 1987).
Trapezoidal distributions are appropriate for modeling events that are comprised
by three different stages: one growth stage, where probability grows up until a
plateau is reached; a stability stage, where probability stays more or less the same;
and a decline stage, where probability decreases until zero (van Dorp and Kotz, 2003).
References:
-
J. René van Dorp, Samuel Kotz, Trapezoidal distribution. Available on:
http://www.seas.gwu.edu/~dorpjr/Publications/JournalPapers/Metrika2003VanDorp.pdf
-
Powell MR, Wilson JD (1997). Risk Assessment for National Natural Resource
Conservation Programs, Discussion Paper 97-49. Resources for the Future, Washington
D.C.
-
Chen SJ, Hwang CL (1992). Fuzzy Multiple Attribute Decision-Making: Methods and
Applications, Springer-Verlag, Berlin, New York.
-
Flehinger BJ, Kimmel M (1987). The natural history of lung cancer in periodically
screened population. Biometrics 1987, 43, 127-144.
The following example shows how to create and test the main characteristics
of a Trapezoidal distribution given its parameters:
// Create a new trapezoidal distribution with linear growth between
// 0 and 2, stability between 2 and 8, and decrease between 8 and 10.
//
//
// +-----------+
// /| |\
// / | | \
// / | | \
// -------+---+-----------+---+-------
// ... 0 2 4 6 8 10 ...
//
var trapz = new TrapezoidalDistribution(a: 0, b: 2, c: 8, d: 10, n1: 1, n3: 1);
double mean = trapz.Mean; // 2.25
double median = trapz.Median; // 3.0
double mode = trapz.Mode; // 3.1353457616424696
double var = trapz.Variance; // 17.986666666666665
double cdf = trapz.DistributionFunction(x: 1.4); // 0.13999999999999999
double pdf = trapz.ProbabilityDensityFunction(x: 1.4); // 0.10000000000000001
double lpdf = trapz.LogProbabilityDensityFunction(x: 1.4); // -2.3025850929940455
double ccdf = trapz.ComplementaryDistributionFunction(x: 1.4); // 0.85999999999999999
double icdf = trapz.InverseDistributionFunction(p: cdf); // 1.3999999999999997
double hf = trapz.HazardFunction(x: 1.4); // 0.11627906976744187
double chf = trapz.CumulativeHazardFunction(x: 1.4); // 0.15082288973458366
string str = trapz.ToString(CultureInfo.InvariantCulture); // Trapezoidal(x; a=0, b=2, c=8, d=10, n1=1, n3=1, α = 1)
Creates a new trapezoidal distribution.
The minimum value a.
The beginning of the stability region b.
The end of the stability region c.
The maximum value d.
Creates a new trapezoidal distribution.
The minimum value a.
The beginning of the stability region b.
The end of the stability region c.
The maximum value d.
The growth slope between points and . Default is 2.
The growth slope between points and . Default is 2.
Creates a new trapezoidal distribution.
The minimum value a.
The beginning of the stability region b.
The end of the stability region c.
The maximum value d.
The growth slope between points and . Default is 2.
The growth slope between points and . Default is 2.
The boundary ratio α. Default is 1.
Gets the mean for this distribution.
The distribution's mean value.
Gets the variance for this distribution.
The distribution's variance.
Not supported.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
The 4-parameter Beta distribution.
The generalized beta distribution is a family of continuous probability distributions defined
on any interval (min, max) parameterized by two positive shape parameters and two real location
parameters, typically denoted by α, β, a and b. The beta distribution can be suited to the
statistical modeling of proportions in applications where values of proportions equal to 0 or 1
do not occur. One theoretical case where the beta distribution arises is as the distribution of
the ratio formed by one random variable having a Gamma distribution divided by the sum of it and
another independent random variable also having a Gamma distribution with the same scale parameter
(but possibly different shape parameter).
References:
-
Wikipedia, The Free Encyclopedia. Beta distribution.
Available from: http://en.wikipedia.org/wiki/Beta_distribution
-
Wikipedia, The Free Encyclopedia. Three-point estimation.
Available from: https://en.wikipedia.org/wiki/Three-point_estimation
-
Broadleaf Capital International Pty Ltd. Beta PERT origins.
Available from: http://broadleaf.com.au/resource-material/beta-pert-origins/
-
Malcolm, D. G., Roseboom J. H., Clark C.E., and Fazar, W. Application of a technique of research
and development program evaluation, Operations Research, 7, 646-669, 1959. Available from:
http://mech.vub.ac.be/teaching/info/Ontwerpmethodologie/Appendix%20les%202%20PERT.pdf
-
Clark, C. E. The PERT model for the distribution of an activity time, Operations Research, 10, 405-406,
1962. Available from: http://connection.ebscohost.com/c/articles/18246172/pert-model-distribution-activity-time
Note: Simpler examples are also available at the page.
The following example shows how to create a simpler 2-parameter Beta
distribution and compute some of its properties and measures.
// Using the distribution's parameters
double[] samples = GeneralizedBetaDistribution.Random(alpha: 2, beta: 3, min: 0, max: 1, samples: 1000);
// Using an existing distribution
var b = new GeneralizedBetaDistribution(alpha: 1, beta: 2);
double[] new_samples = b.Generate(1000);
And finally, how to estimate the parameters of a Beta distribution from
a set of observations, using either the Method-of-moments or the Maximum
Likelihood Estimate.
// First we will be drawing 100000 observations from a 4-parameter
// Beta distribution with α = 2, β = 3, min = 10 and max = 15:
double[] samples = GeneralizedBetaDistribution.Random(alpha: 2, beta: 3, min: 10, max: 15, samples: 100000);
// We can estimate a distribution with the known max and min
var B = GeneralizedBetaDistribution.Estimate(samples, 10, 15);
// We can explicitly ask for a Method-of-moments estimation
var mm = GeneralizedBetaDistribution.Estimate(samples, 10, 15,
new GeneralizedBetaOptions { Method = BetaEstimationMethod.Moments });
// or explicitly ask for the Maximum Likelihood estimation
var mle = GeneralizedBetaDistribution.Estimate(samples, 10, 15,
new GeneralizedBetaOptions { Method = BetaEstimationMethod.MaximumLikelihood });
Constructs a Beta distribution defined in the
interval (0,1) with the given parameters α and β.
The shape parameter α (alpha).
The shape parameter β (beta).
Constructs a Beta distribution defined in the
interval (a, b) with parameters α, β, a and b.
The shape parameter α (alpha).
The shape parameter β (beta).
The minimum possible value a.
The maximum possible value b.
Constructs a BetaPERT distribution defined in the interval (a, b)
using Vose's PERT estimation for the parameters a, b, mode and λ.
The minimum possible value a.
The maximum possible value b.
The most likely value m.
A Beta distribution initialized using the Vose's PERT method.
Constructs a BetaPERT distribution defined in the interval (a, b)
using Vose's PERT estimation for the parameters a, b, mode and λ.
The minimum possible value a.
The maximum possible value b.
The most likely value m.
The scale parameter λ (lambda). Default is 4.
A Beta distribution initialized using the Vose's PERT method.
Constructs a BetaPERT distribution defined in the interval (a, b)
using usual PERT estimation for the parameters a, b, mode and λ.
The minimum possible value a.
The maximum possible value b.
The most likely value m.
A Beta distribution initialized using the PERT method.
Constructs a BetaPERT distribution defined in the interval (a, b)
using usual PERT estimation for the parameters a, b, mode and λ.
The minimum possible value a.
The maximum possible value b.
The most likely value m.
The scale parameter λ (lambda). Default is 4.
A Beta distribution initialized using the PERT method.
Constructs a BetaPERT distribution defined in the interval (a, b)
using Golenko-Ginzburg observation that the mode is often at 2/3
of the guessed interval.
The minimum possible value a.
The maximum possible value b.
A Beta distribution initialized using the Golenko-Ginzburg's method.
Constructs a standard Beta distribution defined in the interval (0, 1)
based on the number of successed and trials for an experiment.
The number of success r. Default is 0.
The number of trials n. Default is 1.
A standard Beta distribution initialized using the given parameters.
Gets the minimum value A.
Gets the maximum value B.
Gets the shape parameter α (alpha)
Gets the shape parameter β (beta).
Gets the mean for this distribution,
defined as (a + 4 * m + 6 * b).
The distribution's mean value.
Gets the variance for this distribution,
defined as ((b - a) / (k+2))²
The distribution's variance.
Gets the mode for this distribution.
The beta distribution's mode is given
by (a - 1) / (a + b - 2).
The distribution's mode value.
Gets the distribution support, defined as (, ).
Gets the entropy for this distribution.
The distribution's entropy.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
A probability value between 0 and 1.
A sample which could original the given probability
value when applied in the .
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring in the current distribution.
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting,
such as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting,
such as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against.
The array elements can be either of type double (for univariate data) or type
double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting,
such as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against.
The array elements can be either of type double (for univariate data) or type
double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting,
such as regularization constants and additional parameters.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
A random observations drawn from this distribution.
Generates a random vector of observations from the
Beta distribution with the given parameters.
The shape parameter α (alpha).
The shape parameter β (beta).
The minimum possible value a.
The maximum possible value b.
The number of samples to generate.
An array of double values sampled from the specified Beta distribution.
Generates a random vector of observations from the
Beta distribution with the given parameters.
The shape parameter α (alpha).
The shape parameter β (beta).
The minimum possible value a.
The maximum possible value b.
The number of samples to generate.
The location where to store the samples.
An array of double values sampled from the specified Beta distribution.
Generates a random observation from a
Beta distribution with the given parameters.
The shape parameter α (alpha).
The shape parameter β (beta).
The minimum possible value a.
The maximum possible value b.
A random double value sampled from the specified Beta distribution.
Generates a random vector of observations from the
Beta distribution with the given parameters.
The shape parameter α (alpha).
The shape parameter β (beta).
The minimum possible value a.
The maximum possible value b.
The number of samples to generate.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified Beta distribution.
Generates a random vector of observations from the
Beta distribution with the given parameters.
The shape parameter α (alpha).
The shape parameter β (beta).
The minimum possible value a.
The maximum possible value b.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified Beta distribution.
Generates a random observation from a
Beta distribution with the given parameters.
The shape parameter α (alpha).
The shape parameter β (beta).
The minimum possible value a.
The maximum possible value b.
The random number generator to use as a source of randomness.
Default is to use .
A random double value sampled from the specified Beta distribution.
Estimates a new Beta distribution from a set of observations.
Estimates a new Beta distribution from a set of weighted observations.
Estimates a new Beta distribution from a set of weighted observations.
Estimates a new Beta distribution from a set of observations.
Triangular distribution.
In probability theory and statistics, the triangular distribution is a continuous
probability distribution with lower limit a, upper limit b and mode c, where a <
b and a ≤ c ≤ b.
References:
-
Wikipedia, The Free Encyclopedia. Triangular distribution. Available on:
https://en.wikipedia.org/wiki/Triangular_distribution
This example shows how to create a Triangular distribution
with minimum 1, maximum 6, and most common value 3.
// Create a new Triangular distribution (1, 3, 6).
var trig = new TriangularDistribution(a: 1, b: 6, c: 3);
double mean = trig.Mean; // 3.3333333333333335
double median = trig.Median; // 3.2613872124741694
double mode = trig.Mode; // 3.0
double var = trig.Variance; // 1.0555555555555556
double cdf = trig.DistributionFunction(x: 2); // 0.10000000000000001
double pdf = trig.ProbabilityDensityFunction(x: 2); // 0.20000000000000001
double lpdf = trig.LogProbabilityDensityFunction(x: 2); // -1.6094379124341003
double ccdf = trig.ComplementaryDistributionFunction(x: 2); // 0.90000000000000002
double icdf = trig.InverseDistributionFunction(p: cdf); // 2.0000000655718773
double hf = trig.HazardFunction(x: 2); // 0.22222222222222224
double chf = trig.CumulativeHazardFunction(x: 2); // 0.10536051565782628
string str = trig.ToString(CultureInfo.InvariantCulture); // Triangular(x; a = 1, b = 6, c = 3)
Constructs a Triangular distribution
with the given parameters a, b and c.
The minimum possible value in the distribution (a).
The maximum possible value in the distribution (b).
The most common value in the distribution (c).
Gets the triangular parameter A (the minimum value).
Gets the triangular parameter B (the maximum value).
Gets the mean for this distribution,
defined as (a + b + c) / 3.
The distribution's mean value.
Gets the median for this distribution.
The distribution's median value.
Gets the variance for this distribution, defined
as (a² + b² + c² - ab - ac - bc) / 18.
The distribution's variance.
Gets the mode for this distribution,
also known as the triangular's c.
The distribution's mode value.
Gets the distribution support, defined as (, ).
Gets the entropy for this distribution,
defined as 0.5 + log((max-min)/2)).
The distribution's entropy.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
A random observations drawn from this distribution.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting,
such as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting,
such as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against.
The array elements can be either of type double (for univariate data) or type
double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting,
such as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against.
The array elements can be either of type double (for univariate data) or type
double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting,
such as regularization constants and additional parameters.
Gets the minimum value in a set of weighted observations.
Gets the maximum value in a set of weighted observations.
Finds the index of the last largest value in a set of observations.
Finds the index of the first smallest value in a set of observations.
Finds the index of the first smallest value in a set of weighted observations.
Finds the index of the last largest value in a set of weighted observations.
Gumbel distribution (as known as the Extreme Value Type I distribution).
In probability theory and statistics, the Gumbel distribution is used to model
the distribution of the maximum (or the minimum) of a number of samples of various
distributions. Such a distribution might be used to represent the distribution of
the maximum level of a river in a particular year if there was a list of maximum
values for the past ten years. It is useful in predicting the chance that an extreme
earthquake, flood or other natural disaster will occur.
The potential applicability of the Gumbel distribution to represent the distribution
of maxima relates to extreme value theory which indicates that it is likely to be useful
if the distribution of the underlying sample data is of the normal or exponential type.
The Gumbel distribution is a particular case of the generalized extreme value
distribution (also known as the Fisher-Tippett distribution). It is also known
as the log-Weibull distribution and the double exponential distribution (a term
that is alternatively sometimes used to refer to the Laplace distribution). It
is related to the Gompertz distribution[citation needed]: when its density is
first reflected about the origin and then restricted to the positive half line,
a Gompertz function is obtained.
In the latent variable formulation of the multinomial logit model — common in
discrete choice theory — the errors of the latent variables follow a Gumbel
distribution. This is useful because the difference of two Gumbel-distributed
random variables has a logistic distribution.
The Gumbel distribution is named after Emil Julius Gumbel (1891–1966), based on
his original papers describing the distribution.
References:
-
Wikipedia, The Free Encyclopedia. Gumbel distribution. Available on:
http://en.wikipedia.org/wiki/Gumbel_distribution
The following example shows how to create and test the main characteristics
of an Gumbel distribution given its location and scale parameters:
var gumbel = new GumbelDistribution(location: 4.795, scale: 1 / 0.392);
double mean = gumbel.Mean; // 6.2674889410753387
double median = gumbel.Median; // 5.7299819402593481
double mode = gumbel.Mode; // 4.7949999999999999
double var = gumbel.Variance; // 10.704745853604138
double cdf = gumbel.DistributionFunction(x: 3.4); // 0.17767760424788051
double pdf = gumbel.ProbabilityDensityFunction(x: 3.4); // 0.12033954114322486
double lpdf = gumbel.LogProbabilityDensityFunction(x: 3.4); // -2.1174380222001519
double ccdf = gumbel.ComplementaryDistributionFunction(x: 3.4); // 0.82232239575211952
double icdf = gumbel.InverseDistributionFunction(p: cdf); // 3.3999999904866245
double hf = gumbel.HazardFunction(x: 1.4); // 0.03449691276402958
double chf = gumbel.CumulativeHazardFunction(x: 1.4); // 0.022988793482259906
string str = gumbel.ToString(CultureInfo.InvariantCulture); // Gumbel(x; μ = 4.795, β = 2.55)
Creates a new Gumbel distribution
with location zero and unit scale.
Creates a new Gumbel distribution
with the given location and scale.
The location parameter μ (mu). Default is 0.
The scale parameter β (beta). Default is 1.
Gets the distribution's location parameter mu (μ).
Gets the distribution's scale parameter beta (β).
Gets the mean for this distribution.
The distribution's mean value.
Gets the variance for this distribution.
The distribution's variance.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the median for this distribution.
The distribution's median value.
Gets the mode for this distribution.
The distribution's mode value.
Gets the entropy for this distribution.
The distribution's entropy.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
A single point in the distribution range.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF.
Gets the cumulative hazard function for this
distribution evaluated at point x.
A single point in the distribution range.
The cumulative hazard function H(x)
evaluated at x in the current distribution.
Gets the hazard function, also known as the failure rate or
the conditional failure density function for this distribution
evaluated at point x.
A single point in the distribution range.
The conditional failure density function h(x)
evaluated at x in the current distribution.
The hazard function is the ratio of the probability
density function f(x) to the survival function, S(x).
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
A probability value between 0 and 1.
A sample which could original the given probability
value when applied in the .
The Inverse Cumulative Distribution Function (ICDF) specifies, for
a given probability, the value which the random variable will be at,
or below, with that probability.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Tukey-Lambda distribution.
Formalized by John Tukey, the Tukey lambda distribution is a continuous
probability distribution defined in terms of its quantile function. It is
typically used to identify an appropriate distribution and not used in
statistical models directly.
The Tukey lambda distribution has a single shape parameter λ. As with other
probability distributions, the Tukey lambda distribution can be transformed
with a location parameter, μ, and a scale parameter, σ. Since the general form
of probability distribution can be expressed in terms of the standard distribution,
the subsequent formulas are given for the standard form of the function.
References:
-
Wikipedia, The Free Encyclopedia. Tukey-Lambda distribution. Available on:
http://en.wikipedia.org/wiki/Tukey_lambda_distribution
This examples shows how to create a Tukey distribution and
compute some of its properties .
var tukey = new TukeyLambdaDistribution(lambda: 0.14);
double mean = tukey.Mean; // 0.0
double median = tukey.Median; // 0.0
double mode = tukey.Mode; // 0.0
double var = tukey.Variance; // 2.1102970222144855
double stdDev = tukey.StandardDeviation; // 1.4526861402982014
double cdf = tukey.DistributionFunction(x: 1.4); // 0.83252947230217966
double pdf = tukey.ProbabilityDensityFunction(x: 1.4); // 0.17181242109370659
double lpdf = tukey.LogProbabilityDensityFunction(x: 1.4); // -1.7613519723149427
double ccdf = tukey.ComplementaryDistributionFunction(x: 1.4); // 0.16747052769782034
double icdf = tukey.InverseDistributionFunction(p: cdf); // 1.4000000000000004
double hf = tukey.HazardFunction(x: 1.4); // 1.0219566231014163
double chf = tukey.CumulativeHazardFunction(x: 1.4); // 1.7842102556452939
string str = tukey.ToString(CultureInfo.InvariantCulture); // Tukey(x; λ = 0.14)
Constructs a Tukey-Lambda distribution
with the given lambda (shape) parameter.
Gets the distribution shape parameter lambda (λ).
Gets the mean for this distribution (always zero).
The distribution's mean value.
Gets the median for this distribution (always zero).
The distribution's median value.
Gets the mode for this distribution (always zero).
The distribution's median value.
Gets the entropy for this distribution.
The distribution's entropy.
Gets the variance for this distribution.
The distribution's variance.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
A probability value between 0 and 1.
A sample which could original the given probability
value when applied in the .
Gets the first derivative of the
inverse distribution function (icdf) for this distribution evaluated
at probability p.
A probability value between 0 and 1.
Gets the log of the quantile
density function, which in turn is the first derivative of
the inverse distribution
function (icdf), evaluated at probability p.
A probability value between 0 and 1.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Power Lognormal distribution.
References:
-
NIST/SEMATECH e-Handbook of Statistical Methods. Power Lognormal distribution. Available on:
http://www.itl.nist.gov/div898/handbook/eda/section3/eda366e.htm
This example shows how to create a Power Lognormal
distribution and compute some of its properties.
// Create a Power-Lognormal distribution with p = 4.2 and s = 1.2
var plog = new PowerLognormalDistribution(power: 4.2, shape: 1.2);
double cdf = plog.DistributionFunction(x: 1.4); // 0.98092157745191766
double pdf = plog.ProbabilityDensityFunction(x: 1.4); // 0.046958580233533977
double lpdf = plog.LogProbabilityDensityFunction(x: 1.4); // -3.0584893374471496
double ccdf = plog.ComplementaryDistributionFunction(x: 1.4); // 0.019078422548082351
double icdf = plog.InverseDistributionFunction(p: cdf); // 1.4
double hf = plog.HazardFunction(x: 1.4); // 10.337649063164642
double chf = plog.CumulativeHazardFunction(x: 1.4); // 3.9591972920568446
string str = plog.ToString(CultureInfo.InvariantCulture); // PLD(x; p = 4.2, σ = 1.2)
Constructs a Power Lognormal distribution
with the given power and shape parameters.
The distribution's power p.
The distribution's shape σ.
Gets the distribution's power parameter (p).
Gets the distribution's shape parameter sigma (σ).
Not supported.
Not supported.
Not supported.
Not supported.
Not supported.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
A probability value between 0 and 1.
A sample which could original the given probability
value when applied in the .
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
Gets the hazard function, also known as the failure rate or
the conditional failure density function for this distribution
evaluated at point x.
A single point in the distribution range.
The conditional failure density function h(x)
evaluated at x in the current distribution.
Gets the cumulative hazard function for this
distribution evaluated at point x.
A single point in the distribution range.
The cumulative hazard function H(x)
evaluated at x in the current distribution.
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
A single point in the distribution range.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Generalized Normal distribution (also known as Exponential Power distribution).
The generalized normal distribution or generalized Gaussian distribution
(GGD) is either of two families of parametric continuous probability
distributions on the real line. Both families add a shape parameter to
the normal distribution. To distinguish the two families, they are referred
to below as "version 1" and "version 2". However this is not a standard
nomenclature.
Known also as the exponential power distribution, or the generalized error
distribution, this is a parametric family of symmetric distributions. It includes
all normal and Laplace distributions, and as limiting cases it includes all
continuous uniform distributions on bounded intervals of the real line.
References:
-
Wikipedia, The Free Encyclopedia. Generalized normal distribution. Available on:
https://en.wikipedia.org/wiki/Generalized_normal_distribution
This examples shows how to create a Generalized normal distribution
and compute some of its properties.
// Creates a new generalized normal distribution with the given parameters
var normal = new GeneralizedNormalDistribution(location: 1, scale: 5, shape: 0.42);
double mean = normal.Mean; // 1
double median = normal.Median; // 1
double mode = normal.Mode; // 1
double var = normal.Variance; // 19200.781700666659
double cdf = normal.DistributionFunction(x: 1.4); // 0.51076148867681703
double pdf = normal.ProbabilityDensityFunction(x: 1.4); // 0.024215092283124507
double lpdf = normal.LogProbabilityDensityFunction(x: 1.4); // -3.7207791921441378
double ccdf = normal.ComplementaryDistributionFunction(x: 1.4); // 0.48923851132318297
double icdf = normal.InverseDistributionFunction(p: cdf); // 1.4000000149740108
double hf = normal.HazardFunction(x: 1.4); // 0.049495474543966168
double chf = normal.CumulativeHazardFunction(x: 1.4); // 0.7149051552030572
string str = normal.ToString(CultureInfo.InvariantCulture); // GGD(x; μ = 1, α = 5, β = 0.42)
Constructs a Generalized Normal distribution with the given parameters.
The location parameter μ.
The scale parameter α.
The shape parameter β.
Create an distribution using a
specialization.
The Laplace's location parameter μ (mu).
The Laplace's scale parameter b.
A that provides
a .
Create an distribution using a
specialization.
The Normal's mean parameter μ (mu).
The Normal's standard deviation σ (sigma).
A that provides
a distribution.
Gets the location value μ (mu) for the distribution.
Gets the median for this distribution.
The distribution's median value.
Gets the mode for this distribution.
In the Generalized Normal Distribution, the mode is
equal to the distribution's value.
The distribution's mode value.
Gets the variance for this distribution.
The distribution's variance.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the Entropy for this Normal distribution.
Gets the cumulative distribution function (cdf) for the
Generalized Normal distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
See .
Gets the probability density function (pdf) for
the Normal distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single double value.
For a multivariate distribution, this should be a double array.
The probability of x occurring
in the current distribution.
See .
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Power Normal distribution.
References:
-
NIST/SEMATECH e-Handbook of Statistical Methods. Power Normal distribution. Available on:
http://www.itl.nist.gov/div898/handbook/eda/section3/eda366d.htm
This example shows how to create a Power Normal distribution
and compute some of its properties.
// Create a new Power-Normal distribution with p = 4.2
var pnormal = new PowerNormalDistribution(power: 4.2);
double cdf = pnormal.DistributionFunction(x: 1.4); // 0.99997428721920678
double pdf = pnormal.ProbabilityDensityFunction(x: 1.4); // 0.00020022645890003279
double lpdf = pnormal.LogProbabilityDensityFunction(x: 1.4); // -0.20543269836728234
double ccdf = pnormal.ComplementaryDistributionFunction(x: 1.4); // 0.000025712780793218926
double icdf = pnormal.InverseDistributionFunction(p: cdf); // 1.3999999999998953
double hf = pnormal.HazardFunction(x: 1.4); // 7.7870402470368854
double chf = pnormal.CumulativeHazardFunction(x: 1.4); // 10.568522382550167
string str = pnormal.ToString(); // PND(x; p = 4.2)
Constructs a Power Normal distribution
with given power (shape) parameter.
The distribution's power p.
Gets the distribution shape (power) parameter.
Not supported.
Not supported.
Not supported.
Not supported.
Not supported.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
The Inverse Cumulative Distribution Function (ICDF) specifies, for
a given probability, the value which the random variable will be at,
or below, with that probability.
A probability value between 0 and 1.
A sample which could original the given probability
value when applied in the .
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
The probability of x occurring
in the current distribution.
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
The logarithm of the probability of x
occurring in the current distribution.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
U-quadratic distribution.
In probability theory and statistics, the U-quadratic distribution is a continuous
probability distribution defined by a unique quadratic function with lower limit a
and upper limit b. This distribution is a useful model for symmetric bimodal processes.
Other continuous distributions allow more flexibility, in terms of relaxing the symmetry
and the quadratic shape of the density function, which are enforced in the U-quadratic
distribution - e.g., Beta distribution, Gamma distribution, etc.
References:
-
Wikipedia, The Free Encyclopedia. U-quadratic distribution. Available on:
http://en.wikipedia.org/wiki/U-quadratic_distribution
The following example shows how to create and test the main characteristics
of an U-quadratic distribution given its two parameters:
// Create a new U-quadratic distribution with values
var u2 = new UQuadraticDistribution(a: 0.42, b: 4.2);
double mean = u2.Mean; // 2.3100000000000001
double median = u2.Median; // 2.3100000000000001
double mode = u2.Mode; // 0.8099060089153145
double var = u2.Variance; // 2.1432600000000002
double cdf = u2.DistributionFunction(x: 1.4); // 0.44419041812731797
double pdf = u2.ProbabilityDensityFunction(x: 1.4); // 0.18398763254730335
double lpdf = u2.LogProbabilityDensityFunction(x: 1.4); // -1.6928867380489712
double ccdf = u2.ComplementaryDistributionFunction(x: 1.4); // 0.55580958187268203
double icdf = u2.InverseDistributionFunction(p: cdf); // 1.3999998213768274
double hf = u2.HazardFunction(x: 1.4); // 0.3310263776442936
double chf = u2.CumulativeHazardFunction(x: 1.4); // 0.58732952203701494
string str = u2.ToString(CultureInfo.InvariantCulture); // "UQuadratic(x; a = 0.42, b = 4.2)"
Constructs a new U-quadratic distribution.
Parameter a.
Parameter b.
Gets the mean for this distribution.
The distribution's mean value.
Gets the median for this distribution.
The distribution's median value.
Gets the variance for this distribution.
The distribution's variance.
Not supported.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Wrapped Cauchy Distribution.
In probability theory and directional statistics, a wrapped Cauchy distribution
is a wrapped probability distribution that results from the "wrapping" of the
Cauchy distribution around the unit circle. The Cauchy distribution is sometimes
known as a Lorentzian distribution, and the wrapped Cauchy distribution may
sometimes be referred to as a wrapped Lorentzian distribution.
The wrapped Cauchy distribution is often found in the field of spectroscopy where
it is used to analyze diffraction patterns (e.g. see Fabry–Pérot interferometer).
References:
-
Wikipedia, The Free Encyclopedia. Directional statistics. Available on:
http://en.wikipedia.org/wiki/Directional_statistics
-
Wikipedia, The Free Encyclopedia. Wrapped Cauchy distribution. Available on:
http://en.wikipedia.org/wiki/Wrapped_Cauchy_distribution
// Create a Wrapped Cauchy distribution with μ = 0.42, γ = 3
var dist = new WrappedCauchyDistribution(mu: 0.42, gamma: 3);
// Common measures
double mean = dist.Mean; // 0.42
double var = dist.Variance; // 0.950212931632136
// Probability density functions
double pdf = dist.ProbabilityDensityFunction(x: 0.42); // 0.1758330112785475
double lpdf = dist.LogProbabilityDensityFunction(x: 0.42); // -1.7382205338929015
// String representation
string str = dist.ToString(); // "WrappedCauchy(x; μ = 0,42, γ = 3)"
Initializes a new instance of the class.
The mean resultant parameter μ.
The gamma parameter γ.
Gets the mean for this distribution.
The distribution's mean value.
Gets the variance for this distribution.
The distribution's variance.
Not supported.
Not supported.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the entropy for this distribution.
The distribution's entropy.
Not supported.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Returns a that represents this instance.
A that represents this instance.
Inverse Gamma Distribution.
The inverse gamma distribution is a two-parameter family of continuous probability
distributions on the positive real line, which is the distribution of the reciprocal
of a variable distributed according to the gamma distribution. Perhaps the chief use
of the inverse gamma distribution is in Bayesian statistics, where it serves as the
conjugate prior of the variance of a normal distribution. However, it is common among
Bayesians to consider an alternative parameterization of the normal distribution in
terms of the precision, defined as the reciprocal of the variance, which allows the
gamma distribution to be used directly as a conjugate prior.
References:
-
Wikipedia, The Free Encyclopedia. Inverse Gamma Distribution.
Available from: http://en.wikipedia.org/wiki/Inverse-gamma_distribution
-
John D. Cook. (2008). The Inverse Gamma Distribution.
// Create a new inverse Gamma distribution with α = 0.42 and β = 0.5
var invGamma = new InverseGammaDistribution(shape: 0.42, scale: 0.5);
// Common measures
double mean = invGamma.Mean; // -0.86206896551724133
double median = invGamma.Median; // 3.1072323347401709
double var = invGamma.Variance; // -0.47035626665061164
// Cumulative distribution functions
double cdf = invGamma.DistributionFunction(x: 0.27); // 0.042243552114989695
double ccdf = invGamma.ComplementaryDistributionFunction(x: 0.27); // 0.95775644788501035
double icdf = invGamma.InverseDistributionFunction(p: cdf); // 0.26999994629410995
// Probability density functions
double pdf = invGamma.ProbabilityDensityFunction(x: 0.27); // 0.35679850067181362
double lpdf = invGamma.LogProbabilityDensityFunction(x: 0.27); // -1.0305840804381006
// Hazard (failure rate) functions
double hf = invGamma.HazardFunction(x: 0.27); // 0.3725357333377633
double chf = invGamma.CumulativeHazardFunction(x: 0.27); // 0.043161763098266373
// String representation
string str = invGamma.ToString(); // Γ^(-1)(x; α = 0.42, β = 0.5)
Creates a new Inverse Gamma Distribution.
The shape parameter α (alpha).
The scale parameter β (beta).
Gets the mean for this distribution.
The distribution's mean value.
In the Inverse Gamma distribution, the Mean is given as b / (a - 1).
Gets the mode for this distribution.
The distribution's mode value.
Gets the variance for this distribution.
The distribution's variance.
In the Inverse Gamma distribution, the Variance is given as b² / ((a - 1)² * (a - 2)).
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the entropy for this distribution.
The distribution's entropy.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
In the Inverse Gamma CDF is computed in terms of the
Upper Incomplete Regularized Gamma Function Q as CDF(x) = Q(a, b / x).
See .
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
See .
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
See .
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Laplace's Distribution (as known as the double exponential distribution).
In probability theory and statistics, the Laplace distribution is a continuous
probability distribution named after Pierre-Simon Laplace. It is also sometimes called
the double exponential distribution.
The difference between two independent identically distributed exponential random
variables is governed by a Laplace distribution, as is a Brownian motion evaluated at an
exponentially distributed random time. Increments of Laplace motion or a variance gamma
process evaluated over the time scale also have a Laplace distribution.
The probability density function of the Laplace distribution is also reminiscent of the
normal distribution; however, whereas the normal distribution is expressed in terms of
the squared difference from the mean μ, the Laplace density is expressed in terms of the
absolute difference from the mean. Consequently the Laplace distribution has fatter tails
than the normal distribution.
References:
-
Wikipedia, The Free Encyclopedia. Laplace distribution.
Available from: http://en.wikipedia.org/wiki/Laplace_distribution
// Create a new Laplace distribution with μ = 4 and b = 2
var laplace = new LaplaceDistribution(location: 4, scale: 2);
// Common measures
double mean = laplace.Mean; // 4.0
double median = laplace.Median; // 4.0
double var = laplace.Variance; // 8.0
// Cumulative distribution functions
double cdf = laplace.DistributionFunction(x: 0.27); // 0.077448104942453522
double ccdf = laplace.ComplementaryDistributionFunction(x: 0.27); // 0.92255189505754642
double icdf = laplace.InverseDistributionFunction(p: cdf); // 0.27
// Probability density functions
double pdf = laplace.ProbabilityDensityFunction(x: 0.27); // 0.038724052471226761
double lpdf = laplace.LogProbabilityDensityFunction(x: 0.27); // -3.2512943611198906
// Hazard (failure rate) functions
double hf = laplace.HazardFunction(x: 0.27); // 0.041974931360160776
double chf = laplace.CumulativeHazardFunction(x: 0.27); // 0.080611649844768624
// String representation
string str = laplace.ToString(CultureInfo.InvariantCulture); // Laplace(x; μ = 4, b = 2)
Creates a new Laplace distribution.
The location parameter μ (mu).
The scale parameter b.
Gets the mean for this distribution.
The distribution's mean value.
The Laplace's distribution mean has the
same value as the location parameter μ.
Gets the mode for this distribution (μ).
The Laplace's distribution mode has the
same value as the location parameter μ.
Gets the median for this distribution.
The distribution's median value.
The Laplace's distribution median has the
same value as the location parameter μ.
Gets the variance for this distribution.
The distribution's variance.
The Laplace's variance is computed as 2*b².
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the entropy for this distribution.
The distribution's entropy.
The Laplace's entropy is defined as ln(2*e*b), in which
e is the Euler constant.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
See .
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
See .
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
See .
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Continuity correction to be used when aproximating
discrete values through a continuous distribution.
No correction for continuity should be applied.
The correction for continuity is -0.5 when the statistic is
greater than the mean and +0.5 when it is less than the mean.
This correction is used/described in http://vassarstats.net/textbook/ch12a.html.
The correction for continuity will be -0.5 when computing values at the
right (upper) tail of the
distribution, and +0.5 when computing at the
left (lower) tail.
Mann-Whitney's U statistic distribution.
This is the distribution for Mann-Whitney's U
statistic used in . This distribution is based on
sample statistics.
This is the distribution for the first sample statistic, U1. Some textbooks
(and statistical packages) use alternate definitions for U, which should be
compared with the appropriate statistic tables or alternate distributions.
// Consider the following rank statistics
double[] ranks = { 1, 2, 3, 4, 5 };
// Create a new Mann-Whitney U's distribution with n1 = 2 and n2 = 3
var mannWhitney = new MannWhitneyDistribution(ranks, n1: 2, n2: 3);
// Common measures
double mean = mannWhitney.Mean; // 2.7870954605658511
double median = mannWhitney.Median; // 1.5219615583481305
double var = mannWhitney.Variance; // 18.28163603621158
// Cumulative distribution functions
double cdf = mannWhitney.DistributionFunction(x: 4); // 0.6
double ccdf = mannWhitney.ComplementaryDistributionFunction(x: 4); // 0.4
double icdf = mannWhitney.InverseDistributionFunction(p: cdf); // 3.6666666666666661
// Probability density functions
double pdf = mannWhitney.ProbabilityDensityFunction(x: 4); // 0.2
double lpdf = mannWhitney.LogProbabilityDensityFunction(x: 4); // -1.6094379124341005
// Hazard (failure rate) functions
double hf = mannWhitney.HazardFunction(x: 4); // 0.5
double chf = mannWhitney.CumulativeHazardFunction(x: 4); // 0.916290731874155
// String representation
string str = mannWhitney.ToString(); // MannWhitney(u; n1 = 2, n2 = 3)
Gets the number of observations in the first sample.
Gets the number of observations in the second sample.
Gets or sets the continuity correction
to be applied when using the Normal approximation to this distribution.
Gets whether this distribution computes the exact probabilities
(by searching all possible rank combinations) or gives fast
approximations.
true if this distribution is exact; otherwise, false.
Gets the statistic values for all possible combinations
of ranks. This is used to compute the exact distribution.
Constructs a Mann-Whitney's U-statistic distribution.
The number of observations in the first sample.
The number of observations in the second sample.
Constructs a Mann-Whitney's U-statistic distribution.
The rank statistics.
The number of observations in the first sample.
The number of observations in the second sample.
True to compute the exact distribution. May require a significant
amount of processing power for large samples (n > 30). If left at null, whether to
compute the exact or approximate distribution will depend on the number of samples.
Constructs a Mann-Whitney's U-statistic distribution.
The global rank statistics for the first sample.
The global rank statistics for the second sample.
True to compute the exact distribution. May require a significant
amount of processing power for large samples (n > 30). If left at null, whether to
compute the exact or approximate distribution will depend on the number of samples.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point k.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
A single point in the distribution range.
System.Double.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF.
Gets the Mann-Whitney's U statistic for the first sample.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Gets the mean for this distribution.
The distribution's mean value.
The mean of Mann-Whitney's U distribution
is defined as (n1 * n2) / 2.
Gets the variance for this distribution.
The distribution's variance.
The variance of Mann-Whitney's U distribution
is defined as (n1 * n2 * (n1 + n2 + 1)) / 12.
This method is not supported.
This method is not supported.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the probability density function (pdf) for
this distribution evaluated at point u.
A single point in the distribution range.
The probability of u occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value u will occur.
See .
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of u
occurring in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value u will occur.
See .
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
A probability value between 0 and 1.
A sample which could original the given probability
value when applied in the .
Returns a that represents this instance.
A that represents this instance.
Noncentral t-distribution.
As with other noncentrality parameters, the noncentral t-distribution generalizes
a probability distribution – Student's t-distribution
– using a noncentrality parameter. Whereas the central distribution describes how a
test statistic is distributed when the difference tested is null, the noncentral
distribution also describes how t is distributed when the null is false.
This leads to its use in statistics, especially calculating statistical power. The
noncentral t-distribution is also known as the singly noncentral t-distribution, and
in addition to its primary use in statistical inference, is also used in robust
modeling for data.
References:
-
Wikipedia, The Free Encyclopedia. Noncentral t-distribution. Available on:
http://en.wikipedia.org/wiki/Noncentral_t-distribution
var distribution = new NoncentralTDistribution(
degreesOfFreedom: 4, noncentrality: 2.42);
double mean = distribution.Mean; // 3.0330202123035104
double median = distribution.Median; // 2.6034842414893795
double var = distribution.Variance; // 4.5135883917583683
double cdf = distribution.DistributionFunction(x: 1.4); // 0.15955740661144721
double pdf = distribution.ProbabilityDensityFunction(x: 1.4); // 0.23552141805184526
double lpdf = distribution.LogProbabilityDensityFunction(x: 1.4); // -1.4459534225195116
double ccdf = distribution.ComplementaryDistributionFunction(x: 1.4); // 0.84044259338855276
double icdf = distribution.InverseDistributionFunction(p: cdf); // 1.4000000000123853
double hf = distribution.HazardFunction(x: 1.4); // 0.28023498559521387
double chf = distribution.CumulativeHazardFunction(x: 1.4); // 0.17382662901507062
string str = distribution.ToString(CultureInfo.InvariantCulture); // T(x; df = 4, μ = 2.42)
Gets the degrees of freedom (v) for the distribution.
Gets the noncentrality parameter μ (mu) for the distribution.
Initializes a new instance of the class.
The degrees of freedom v.
The noncentrality parameter μ (mu).
Gets the mean for this distribution.
The noncentral t-distribution's mean is defined in terms of
the Gamma function Γ(x) as
μ * sqrt(v/2) * Γ((v - 1) / 2) / Γ(v / 2) for v > 1.
Gets the variance for this distribution.
The noncentral t-distribution's variance is defined in terms of
the Gamma function Γ(x) as
a - b * c² in which
a = v*(1+μ²) / (v-2),
b = (u² * v) / 2 and
c = Γ((v - 1) / 2) / Γ(v / 2) for v > 2.
Gets the mode for this distribution.
The distribution's mode value.
Not supported.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
See .
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
See .
Not supported.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Computes the cumulative probability at t of the
non-central T-distribution with DF degrees of freedom
and non-centrality parameter.
This function is based on the original work done by
Russell Lent hand John Burkardt, shared under the
LGPL license. Original FORTRAN code can be found at:
http://people.sc.fsu.edu/~jburkardt/f77_src/asa243/asa243.html
Exponential distribution.
In probability theory and statistics, the exponential distribution (a.k.a. negative
exponential distribution) is a family of continuous probability distributions. It
describes the time between events in a Poisson process, i.e. a process in which events
occur continuously and independently at a constant average rate. It is the continuous
analogue of the geometric distribution.
Note that the exponential distribution is not the same as the class of exponential
families of distributions, which is a large class of probability distributions that
includes the exponential distribution as one of its members, but also includes the
normal distribution, binomial distribution, gamma distribution, Poisson, and many
others.
References:
-
Wikipedia, The Free Encyclopedia. Exponential distribution. Available on:
http://en.wikipedia.org/wiki/Exponential_distribution
The following example shows how to create and test the main characteristics
of an Exponential distribution given a lambda (λ) rate of 0.42:
// Create an Exponential distribution with λ = 0.42
var exp = new ExponentialDistribution(rate: 0.42);
// Common measures
double mean = exp.Mean; // 2.3809523809523809
double median = exp.Median; // 1.6503504299046317
double var = exp.Variance; // 5.6689342403628125
// Cumulative distribution functions
double cdf = exp.DistributionFunction(x: 0.27); // 0.10720652870550407
double ccdf = exp.ComplementaryDistributionFunction(x: 0.27); // 0.89279347129449593
double icdf = exp.InverseDistributionFunction(p: cdf); // 0.27
// Probability density functions
double pdf = exp.ProbabilityDensityFunction(x: 0.27); // 0.3749732579436883
double lpdf = exp.LogProbabilityDensityFunction(x: 0.27); // -0.98090056770472311
// Hazard (failure rate) functions
double hf = exp.HazardFunction(x: 0.27); // 0.42
double chf = exp.CumulativeHazardFunction(x: 0.27); // 0.1134
// String representation
string str = exp.ToString(CultureInfo.InvariantCulture); // Exp(x; λ = 0.42)
The following example shows how to generate random samples drawn
from an Exponential distribution and later how to re-estimate a
distribution from the generated samples.
// Create an Exponential distribution with λ = 2.5
var exp = new ExponentialDistribution(rate: 2.5);
// Generate a million samples from this distribution:
double[] samples = target.Generate(1000000);
// Create a default exponential distribution
var newExp = new ExponentialDistribution();
// Fit the samples
newExp.Fit(samples);
// Check the estimated parameters
double rate = newExp.Rate; // 2.5
Creates a new Exponential distribution with the given rate.
Creates a new Exponential distribution with the given rate.
The rate parameter lambda (λ). Default is 1.
Gets the distribution's rate parameter lambda (λ).
Gets the mean for this distribution.
The distribution's mean value.
In the Exponential distribution, the mean is defined as 1/λ.
Gets the variance for this distribution.
The distribution's variance.
In the Exponential distribution, the variance is defined as 1/(λ²)
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the median for this distribution.
The distribution's median value.
In the Exponential distribution, the median is defined as ln(2) / λ.
Gets the mode for this distribution.
The distribution's mode value.
In the Exponential distribution, the median is defined as 0.
Gets the entropy for this distribution.
The distribution's entropy.
In the Exponential distribution, the median is defined as 1 - ln(λ).
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
The Exponential CDF is defined as CDF(x) = 1 - exp(-λ*x).
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
The Exponential PDF is defined as PDF(x) = λ * exp(-λ*x).
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
The Inverse Cumulative Distribution Function (ICDF) specifies, for
a given probability, the value which the random variable will be at,
or below, with that probability.
The Exponential ICDF is defined as ICDF(p) = -ln(1-p)/λ.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Please see .
Estimates a new Exponential distribution from a given set of observations.
Estimates a new Exponential distribution from a given set of observations.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
The random number generator to use as a source of randomness.
Default is to use .
A random observations drawn from this distribution.
Generates a random vector of observations from the
Exponential distribution with the given parameters.
The rate parameter lambda.
The number of samples to generate.
An array of double values sampled from the specified Exponential distribution.
Generates a random vector of observations from the
Exponential distribution with the given parameters.
The rate parameter lambda.
The number of samples to generate.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified Exponential distribution.
Generates a random vector of observations from the
Exponential distribution with the given parameters.
The rate parameter lambda.
The number of samples to generate.
The location where to store the samples.
An array of double values sampled from the specified Exponential distribution.
Generates a random vector of observations from the
Exponential distribution with the given parameters.
The rate parameter lambda.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified Exponential distribution.
Generates a random observation from the
Exponential distribution with the given parameters.
The rate parameter lambda.
A random double value sampled from the specified Exponential distribution.
Generates a random observation from the
Exponential distribution with the given parameters.
The rate parameter lambda.
The random number generator to use as a source of randomness.
Default is to use .
A random double value sampled from the specified Exponential distribution.
Returns a that represents this instance.
A that represents this instance.
Gamma distribution.
The gamma distribution is a two-parameter family of continuous probability
distributions. There are three different parameterizations in common use:
-
With a parameter k and a
parameter θ.
-
With a shape parameter α = k and an inverse scale parameter
β = 1/θ, called a parameter.
-
With a shape parameter k and a
parameter μ = k/β.
In each of these three forms, both parameters are positive real numbers. The
parameterization with k and θ appears to be more common in econometrics and
certain other applied fields, where e.g. the gamma distribution is frequently
used to model waiting times. For instance, in life testing, the waiting time
until death is a random variable that is frequently modeled with a gamma
distribution. This is the default
construction method for this class.
The parameterization with α and β is more common in Bayesian statistics, where
the gamma distribution is used as a conjugate prior distribution for various
types of inverse scale (aka rate) parameters, such as the λ of an exponential
distribution or a Poisson distribution – or for that matter, the β of the gamma
distribution itself. (The closely related inverse gamma distribution is used as
a conjugate prior for scale parameters, such as the variance of a normal distribution.)
In order to create a Gamma distribution using the Bayesian parameterization, you
can use .
If k is an integer, then the distribution represents an Erlang distribution; i.e.,
the sum of k independent exponentially distributed random variables, each of which
has a mean of θ (which is equivalent to a rate parameter of 1/θ).
The gamma distribution is the maximum entropy probability distribution for a random
variable X for which E[X] = kθ = α/β is fixed and greater than zero, and E[ln(X)] =
ψ(k) + ln(θ) = ψ(α) − ln(β) is fixed (ψ is the digamma function).
References:
-
Wikipedia, The Free Encyclopedia. Gamma distribution. Available on:
http://en.wikipedia.org/wiki/Gamma_distribution
The following example shows how to create, test and compute the main
functions of a Gamma distribution given parameters θ = 4 and k = 2:
For n <= 140 and:
- 1/n > x >= 1-1/nUses the Ruben-Gambino formula.
- 1/n < nx² < 0.754693Uses the Durbin matrix algorithm.
- 0.754693 <= nx² < 4Uses the Pomeranz algorithm.
- 4 <= nx² < 18Uses the complementary distribution function.
- nx² >= 18Returns the constant 1.
For 140 < n <= 10^5
- nx² >= 18Returns the constant 1.
- nx^(3/2) < 1.4Durbin matrix algorithm.
- nx^(3/2) > 1.4Pelz-Good asymptotic series.
For n > 10^5
- nx² >= 18Returns the constant 1.
- nx² < 18Pelz-Good asymptotic series.
References:
-
R. Simard and P. L'Ecuyer. (2011). "Computing the Two-Sided Kolmogorov-Smirnov
Distribution", Journal of Statistical Software, Vol. 39, Issue 11, Mar 2011.
Available on: http://www.iro.umontreal.ca/~lecuyer/myftp/papers/ksdist.pdf
-
Marsaglia, G., Tsang, W. W., Wang, J. (2003) "Evaluating Kolmogorov's
Distribution", Journal of Statistical Software, 8 (18), 1–4. jstor.
Available on: http://www.jstatsoft.org/v08/i18/paper
-
Durbin, J. (1972). Distribution Theory for Tests Based on The Sample
Distribution Function, Society for Industrial & Applied Mathematics,
Philadelphia.
The following example shows how to build a Kolmogorov-Smirnov distribution
for 42 samples and compute its main functions and characteristics:
// Create a Kolmogorov-Smirnov distribution with n = 42
var ks = new KolmogorovSmirnovDistribution(samples: 42);
// Common measures
double mean = ks.Mean; // 0.13404812830261556
double median = ks.Median; // 0.12393613519421857
double var = ks.Variance; // 0.019154717445778062
// Cumulative distribution functions
double cdf = ks.DistributionFunction(x: 0.27); // 0.99659863602996079
double ccdf = ks.ComplementaryDistributionFunction(x: 0.27); // 0.0034013639700392062
double icdf = ks.InverseDistributionFunction(p: cdf); // 0.26999997446092017
// Hazard (failure rate) functions
double chf = ks.CumulativeHazardFunction(x: 0.27); // 5.6835787601476619
// String representation
string str = ks.ToString(); // "KS(x; n = 42)"
Gets the number of samples distribution parameter.
Creates a new Kolmogorov-Smirnov distribution.
The number of samples.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the mean for this distribution.
The mean of the K-S distribution for n samples
is computed as Mean = sqrt(π/2) * ln(2) / sqrt(n).
See .
Not supported.
Gets the variance for this distribution.
The variance of the K-S distribution for n samples
is computed as Var = (π² / 12 - mean²) / n, in which
mean is the K-S distribution .
See .
Gets the entropy for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
See .
Not supported.
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF.
See .
Computes the Upper Tail of the P[Dn >= x] distribution.
This function approximates the upper tail of the P[Dn >= x]
distribution using the one-sided Kolmogorov-Smirnov statistic.
Not supported.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Computes the Cumulative Distribution Function (CDF)
for the Kolmogorov-Smirnov statistic's distribution.
The sample size.
The Kolmogorov-Smirnov statistic.
Returns the cumulative probability of the statistic
under a sample size .
This function computes the cumulative probability P[Dn <= x] of
the Kolmogorov-Smirnov distribution using multiple methods as
suggested by Richard Simard (2010).
Simard partitioned the problem of evaluating the CDF using multiple
approximation and asymptotic methods in order to achieve a best compromise
between speed and precision. This function follows the same partitioning as
Simard, which is described in the table below.
For n <= 140 and:
- 1/n > x >= 1-1/nUses the Ruben-Gambino formula.
- 1/n < nx² < 0.754693Uses the Durbin matrix algorithm.
- 0.754693 <= nx² < 4Uses the Pomeranz algorithm.
- 4 <= nx² < 18Uses the complementary distribution function.
- nx² >= 18Returns the constant 1.
For 140 < n <= 10^5
- nx² >= 18Returns the constant 1.
- nx^(3/2) < 1.4Durbin matrix algorithm.
- nx^(3/2) > 1.4Pelz-Good asymptotic series.
For n > 10^5
- nx² >= 18Returns the constant 1.
- nx² < 18Pelz-Good asymptotic series.
Computes the Complementary Cumulative Distribution Function (1-CDF)
for the Kolmogorov-Smirnov statistic's distribution.
The sample size.
The Kolmogorov-Smirnov statistic.
Returns the complementary cumulative probability of the statistic
under a sample size .
Pelz-Good algorithm for computing lower-tail areas
of the Kolmogorov-Smirnov distribution.
As stated in Simard's paper, Pelz and Good (1976) generalized Kolmogorov's
approximation to an asymptotic series in 1/sqrt(n).
References: Wolfgang Pelz and I. J. Good, "Approximating the Lower Tail-Areas of
the Kolmogorov-Smirnov One-Sample Statistic", Journal of the Royal
Statistical Society, Series B. Vol. 38, No. 2 (1976), pp. 152-156
Computes the Upper Tail of the P[Dn >= x] distribution.
This function approximates the upper tail of the P[Dn >= x]
distribution using the one-sided Kolmogorov-Smirnov statistic.
Pomeranz algorithm.
Durbin's algorithm for computing P[Dn < d]
The method presented by Marsaglia (2003), as stated in the paper, is based
on a succession of developments starting with Kolmogorov and culminating in
a masterful treatment by Durbin (1972). Durbin's monograph summarized and
extended many previous works published in the years 1933-73.
This function implements the small C procedure provided by Marsaglia on his
paper with corrections made by Simard (2010). Further optimizations also
have been performed.
References:
- Marsaglia, G., Tsang, W. W., Wang, J. (2003) "Evaluating Kolmogorov's
Distribution", Journal of Statistical Software, 8 (18), 1–4. jstor.
Available on: http://www.jstatsoft.org/v08/i18/paper
- Durbin, J. (1972) Distribution Theory for Tests Based on The Sample
Distribution Function, Society for Industrial & Applied Mathematics,
Philadelphia.
Computes matrix power. Used in the Durbin algorithm.
Initializes the Pomeranz algorithm.
Creates matrix A of the Pomeranz algorithm.
Computes matrix H of the Pomeranz algorithm.
Bernoulli probability distribution.
The Bernoulli distribution is a distribution for a single
binary variable x E {0,1}, representing, for example, the
flipping of a coin. It is governed by a single continuous
parameter representing the probability of an observation
to be equal to 1.
References:
-
Wikipedia, The Free Encyclopedia. Bernoulli distribution. Available on:
http://en.wikipedia.org/wiki/Bernoulli_distribution
-
C. Bishop. “Pattern Recognition and Machine Learning”. Springer. 2006.
// Create a distribution with probability 0.42
var bern = new BernoulliDistribution(mean: 0.42);
// Common measures
double mean = bern.Mean; // 0.42
double median = bern.Median; // 0.0
double var = bern.Variance; // 0.2436
double mode = bern.Mode; // 0.0
// Probability mass functions
double pdf = bern.ProbabilityMassFunction(k: 1); // 0.42
double lpdf = bern.LogProbabilityMassFunction(k: 0); // -0.54472717544167193
// Cumulative distribution functions
double cdf = bern.DistributionFunction(k: 0); // 0.58
double ccdf = bern.ComplementaryDistributionFunction(k: 0); // 0.42
// Quantile functions
int icdf0 = bern.InverseDistributionFunction(p: 0.57); // 0
int icdf1 = bern.InverseDistributionFunction(p: 0.59); // 1
// Hazard / failure rate functions
double hf = bern.HazardFunction(x: 0); // 1.3809523809523814
double chf = bern.CumulativeHazardFunction(x: 0); // 0.86750056770472328
// String representation
string str = bern.ToString(CultureInfo.InvariantCulture); // "Bernoulli(x; p = 0.42, q = 0.58)"
Creates a new Bernoulli distribution.
Creates a new Bernoulli distribution.
The probability of an observation being equal to 1. Default is 0.5
Gets the mean for this distribution.
Gets the median for this distribution.
The distribution's median value.
Gets the mode for this distribution.
The distribution's mode value.
Gets the variance for this distribution.
Gets the entropy for this distribution.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point k.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
A probability value between 0 and 1.
A sample which could original the given probability
value when applied in the .
Gets P(X > k) the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point k.
This function is also known as the Survival function.
A single point in the distribution range.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF.
Gets the probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of k occurring
in the current distribution.
The Probability Mass Function (PMF) describes the
probability that a given value k will occur.
Gets the log-probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of k
occurring in the current distribution.
The Probability Mass Function (PMF) describes the
probability that a given value k will occur.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
The random number generator to use as a source of randomness.
Default is to use .
A random observations drawn from this distribution.
Returns a that represents this instance.
A that represents this instance.
Binomial probability distribution.
The binomial distribution is the discrete probability distribution of the number of
successes in a sequence of >n independent yes/no experiments, each of which
yields success with probability p. Such a success/failure experiment is also
called a Bernoulli experiment or Bernoulli trial; when n = 1, the binomial
distribution is a Bernoulli distribution.
References:
-
Wikipedia, The Free Encyclopedia. Binomial distribution. Available on:
http://en.wikipedia.org/wiki/Binomial_distribution
-
C. Bishop. “Pattern Recognition and Machine Learning”. Springer. 2006.
// Creates a distribution with n = 16 and success probability 0.12
var bin = new BinomialDistribution(trials: 16, probability: 0.12);
// Common measures
double mean = bin.Mean; // 1.92
double median = bin.Median; // 2
double var = bin.Variance; // 1.6896
double mode = bin.Mode; // 2
// Probability mass functions
double pdf = bin.ProbabilityMassFunction(k: 1); // 0.28218979948821621
double lpdf = bin.LogProbabilityMassFunction(k: 0); // -2.0453339441581582
// Cumulative distribution functions
double cdf = bin.DistributionFunction(k: 0); // 0.12933699143209909
double ccdf = bin.ComplementaryDistributionFunction(k: 0); // 0.87066300856790091
// Quantile functions
int icdf0 = bin.InverseDistributionFunction(p: 0.37); // 1
int icdf1 = bin.InverseDistributionFunction(p: 0.50); // 2
int icdf2 = bin.InverseDistributionFunction(p: 0.99); // 5
int icdf3 = bin.InverseDistributionFunction(p: 0.999); // 7
// Hazard (failure rate) functions
double hf = bin.HazardFunction(x: 0); // 1.3809523809523814
double chf = bin.CumulativeHazardFunction(x: 0); // 0.86750056770472328
// String representation
string str = bin.ToString(CultureInfo.InvariantCulture); // "Binomial(x; n = 16, p = 0.12)"
Gets the number of trials n for the distribution.
Gets the success probability p for the distribution.
Constructs a new binomial distribution.
Constructs a new binomial distribution.
The number of trials n.
Constructs a new binomial distribution.
The number of trials n.
The success probability p in each trial.
Gets the mean for this distribution.
The distribution's mean value.
Gets the variance for this distribution.
The distribution's variance.
Gets the mode for this distribution.
The distribution's mode value.
Gets the entropy for this distribution.
The distribution's entropy.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point k.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
The Inverse Cumulative Distribution Function (ICDF) specifies, for
a given probability, the value which the random variable will be at,
or below, with that probability.
Gets the probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of k occurring
in the current distribution.
The Probability Mass Function (PMF) describes the
probability that a given value x will occur.
Gets the log-probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Mass Function (PMF) describes the
probability that a given value k will occur.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Chi-Square (χ²) probability distribution
In probability theory and statistics, the chi-square distribution (also chi-squared
or χ²-distribution) with k degrees of freedom is the distribution of a sum of the
squares of k independent standard normal random variables. It is one of the most
widely used probability distributions in inferential statistics, e.g. in hypothesis
testing, or in construction of confidence intervals.
References:
-
Wikipedia, The Free Encyclopedia. Chi-square distribution. Available on:
http://en.wikipedia.org/wiki/Chi-square_distribution
The following example demonstrates how to create a new χ²
distribution with the given degrees of freedom.
// Create a new χ² distribution with 7 d.f.
var chisq = new ChiSquareDistribution(degreesOfFreedom: 7);
// Common measures
double mean = chisq.Mean; // 7
double median = chisq.Median; // 6.345811195595612
double var = chisq.Variance; // 14
// Cumulative distribution functions
double cdf = chisq.DistributionFunction(x: 6.27); // 0.49139966433823956
double ccdf = chisq.ComplementaryDistributionFunction(x: 6.27); // 0.50860033566176044
double icdf = chisq.InverseDistributionFunction(p: cdf); // 6.2700000000852318
// Probability density functions
double pdf = chisq.ProbabilityDensityFunction(x: 6.27); // 0.11388708001184455
double lpdf = chisq.LogProbabilityDensityFunction(x: 6.27); // -2.1725478476948092
// Hazard (failure rate) functions
double hf = chisq.HazardFunction(x: 6.27); // 0.22392254197721179
double chf = chisq.CumulativeHazardFunction(x: 6.27); // 0.67609276602233315
// String representation
string str = chisq.ToString(); // "χ²(x; df = 7)
Constructs a new Chi-Square distribution
with given degrees of freedom.
Constructs a new Chi-Square distribution
with given degrees of freedom.
The degrees of freedom for the distribution. Default is 1.
Gets the Degrees of Freedom for this distribution.
Gets the probability density function (pdf) for
the χ² distribution evaluated at point x.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
References:
-
Wikipedia, the free encyclopedia. Chi-square distribution.
The probability of x occurring
in the current distribution.
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the cumulative distribution function (cdf) for
the χ² distribution evaluated at point x.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
The χ² distribution function is defined in terms of the
Incomplete Gamma Function Γ(a, x) as CDF(x; df) = Γ(df/2, x/d).
Gets the complementary cumulative distribution function
(ccdf) for the χ² distribution evaluated at point x.
This function is also known as the Survival function.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF.
The χ² complementary distribution function is defined in terms of the
Complemented Incomplete Gamma
Function Γc(a, x) as CDF(x; df) = Γc(df/2, x/d).
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
A probability value between 0 and 1.
A sample which could original the given probability
value when applied in the .
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the mean for this distribution.
The χ² distribution mean is the number of degrees of freedom.
Gets the variance for this distribution.
The χ² distribution variance is twice its degrees of freedom.
Gets the mode for this distribution.
The χ² distribution mode is max(degrees of freedom - 2, 0).
The distribution's mode value.
Gets the entropy for this distribution.
This method is not supported.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
The random number generator to use as a source of randomness.
Default is to use .
A random observations drawn from this distribution.
Generates a random vector of observations from the
Chi-Square distribution with the given parameters.
An array of double values sampled from the specified Chi-Square distribution.
Generates a random observation from the
Chi-Square distribution with the given parameters.
The degrees of freedom for the distribution.
A random double value sampled from the specified Chi-Square distribution.
Generates a random vector of observations from the
Chi-Square distribution with the given parameters.
An array of double values sampled from the specified Chi-Square distribution.
Generates a random vector of observations from the
Chi-Square distribution with the given parameters.
An array of double values sampled from the specified Chi-Square distribution.
Generates a random observation from the
Chi-Square distribution with the given parameters.
The degrees of freedom for the distribution.
The random number generator to use as a source of randomness.
Default is to use .
A random double value sampled from the specified Chi-Square distribution.
Returns a that represents this instance.
A that represents this instance.
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
A probability value between 0 and 1.
The degrees of freedom of the Chi-Square distribution.
A sample which could original the given probability
value when applied in the .
Abstract class for univariate discrete probability distributions.
A probability distribution identifies either the probability of each value of an
unidentified random variable (when the variable is discrete), or the probability
of the value falling within a particular interval (when the variable is continuous).
The probability distribution describes the range of possible values that a random
variable can attain and the probability that the value of the random variable is
within any (measurable) subset of that range.
The function describing the probability that a given discrete value will
occur is called the probability function (or probability mass function,
abbreviated PMF), and the function describing the cumulative probability
that a given value or any value smaller than it will occur is called the
distribution function (or cumulative distribution function, abbreviated CDF).
References:
-
Wikipedia, The Free Encyclopedia. Probability distribution. Available on:
http://en.wikipedia.org/wiki/Probability_distribution
-
Weisstein, Eric W. "Statistical Distribution." From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/StatisticalDistribution.html
Constructs a new UnivariateDistribution class.
Gets the mean for this distribution.
The distribution's mean value.
Gets the variance for this distribution.
The distribution's variance.
Gets the entropy for this distribution.
The distribution's entropy.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the mode for this distribution.
The distribution's mode value.
Gets the median for this distribution.
The distribution's median value.
Gets the Standard Deviation (the square root of
the variance) for the current distribution.
The distribution's standard deviation.
Gets the Quartiles for this distribution.
A object containing the first quartile
(Q1) as its minimum value, and the third quartile (Q2) as the maximum.
Gets the distribution range within a given percentile.
If 0.25 is passed as the argument,
this function returns the same as the function.
The percentile at which the distribution ranges will be returned.
A object containing the minimum value
for the distribution value, and the third quartile (Q2) as the maximum.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the distribution range within a given percentile.
If 0.25 is passed as the argument,
this function returns the same as the function.
The percentile at which the distribution ranges will be returned.
A object containing the minimum value
for the distribution value, and the third quartile (Q2) as the maximum.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
The probability of x occurring
in the current distribution.
Gets P(X ≤ k), the cumulative distribution function
(cdf) for this distribution evaluated at point k.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets P(X ≤ k), the cumulative distribution function
(cdf) for this distribution evaluated at point k.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets P(X ≤ k) or P(X < k), the cumulative distribution function
(cdf) for this distribution evaluated at point k, depending
on the value of the parameter.
A single point in the distribution range.
True to return P(X ≤ x), false to return P(X < x). Default is true.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
// Compute P(X = k)
double equal = dist.ProbabilityMassFunction(k: 1);
// Compute P(X < k)
double less = dist.DistributionFunction(k: 1, inclusive: false);
// Compute P(X ≤ k)
double lessThanOrEqual = dist.DistributionFunction(k: 1, inclusive: true);
// Compute P(X > k)
double greater = dist.ComplementaryDistributionFunction(k: 1);
// Compute P(X ≥ k)
double greaterThanOrEqual = dist.ComplementaryDistributionFunction(k: 1, inclusive: true);
Gets the cumulative distribution function (cdf) for this
distribution in the semi-closed interval (a; b] given as
P(a < X ≤ b).
The start of the semi-closed interval (a; b].
The end of the semi-closed interval (a; b].
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
The Inverse Cumulative Distribution Function (ICDF) specifies, for
a given probability, the value which the random variable will be at,
or below, with that probability.
A probability value between 0 and 1.
A sample which could original the given probability
value when applied in the .
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
The Inverse Cumulative Distribution Function (ICDF) specifies, for
a given probability, the value which the random variable will be at,
or below, with that probability.
A probability value between 0 and 1.
A sample which could original the given probability
value when applied in the .
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p using a numerical
approximation based on binary search.
The Inverse Cumulative Distribution Function (ICDF) specifies, for
a given probability, the value which the random variable will be at,
or below, with that probability.
A probability value between 0 and 1.
A sample which could original the given probability
value when applied in the .
Computes the cumulative distribution function by summing the outputs of the
for all elements in the distribution domain. Note that this method should not be used in case there is a more
efficient formula for computing the CDF of a distribution.
A single point in the distribution range.
Gets the first derivative of the
inverse distribution function (icdf) for this distribution evaluated
at probability p.
A probability value between 0 and 1.
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point k.
This function is also known as the Survival function.
A single point in the distribution range.
True to return P(X >= x), false to return P(X > x). Default is false.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF.
// Compute P(X = k)
double equal = dist.ProbabilityMassFunction(k: 1);
// Compute P(X < k)
double less = dist.DistributionFunction(k: 1, inclusive: false);
// Compute P(X ≤ k)
double lessThanOrEqual = dist.DistributionFunction(k: 1, inclusive: true);
// Compute P(X > k)
double greater = dist.ComplementaryDistributionFunction(k: 1);
// Compute P(X ≥ k)
double greaterThanOrEqual = dist.ComplementaryDistributionFunction(k: 1, inclusive: true);
Gets P(X > k) the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point k.
This function is also known as the Survival function.
A single point in the distribution range.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF.
Gets P(X > k) the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point k.
This function is also known as the Survival function.
A single point in the distribution range.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF.
Gets the probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Probability Mass Function (PMF) describes the
probability that a given value x will occur.
The probability of k occurring
in the current distribution.
Gets the probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Probability Mass Function (PMF) describes the
probability that a given value x will occur.
The probability of k occurring
in the current distribution.
Gets the log-probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Probability Mass Function (PMF) describes the
probability that a given value k will occur.
The logarithm of the probability of x
occurring in the current distribution.
Gets the log-probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Probability Mass Function (PMF) describes the
probability that a given value k will occur.
The logarithm of the probability of x
occurring in the current distribution.
Gets the hazard function, also known as the failure rate or
the conditional failure density function for this distribution
evaluated at point x.
The hazard function is the ratio of the probability
density function f(x) to the survival function, S(x).
A single point in the distribution range.
The conditional failure density function h(x)
evaluated at x in the current distribution.
Gets the cumulative hazard function for this
distribution evaluated at point x.
A single point in the distribution range.
The cumulative hazard function H(x)
evaluated at x in the current distribution.
Gets the log-cumulative hazard function for this
distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the cumulative hazard function H(x)
evaluated at x in the current distribution.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
A random observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
The random number generator to use as a source of randomness.
Default is to use .
A random observations drawn from this distribution.
Wilcoxon's W statistic distribution.
This is the distribution for the positive side statistic W+ of the Wilcoxon
test. Some textbooks (and statistical packages) use alternate definitions for
W, which should be compared with the appropriate statistic tables or alternate
distributions.
The Wilcoxon signed-rank test is a non-parametric statistical hypothesis test
used when comparing two related samples, matched samples, or repeated measurements
on a single sample to assess whether their population mean ranks differ (i.e. it
is a paired difference test). It can be used as an alternative to the paired
Student's t-test, t-test for matched pairs, or the t-test for dependent samples
when the population cannot be assumed to be normally distributed.
References:
-
Wikipedia, The Free Encyclopedia. Wilcoxon signed-rank test. Available on:
http://en.wikipedia.org/wiki/Wilcoxon_signed-rank_test
// Compute some rank statistics (see other examples below)
double[] ranks = { 1, 2, 3, 4, 5.5, 5.5, 7, 8, 9, 10, 11, 12 };
// Create a new Wilcoxon's W distribution
WilcoxonDistribution W = new WilcoxonDistribution(ranks);
// Common measures
double mean = W.Mean; // 39.0
double median = W.Median; // 38.5
double var = W.Variance; // 162.5
// Probability density functions
double pdf = W.ProbabilityDensityFunction(w: 42); // 0.38418508862319295
double lpdf = W.LogProbabilityDensityFunction(w: 42); // 0.38418508862319295
// Cumulative distribution functions
double cdf = W.DistributionFunction(w: 42); // 0.60817384423279575
double ccdf = W.ComplementaryDistributionFunction(x: 42); // 0.39182615576720425
// Quantile function
double icdf = W.InverseDistributionFunction(p: cdf); // 42
// Hazard (failure rate) functions
double hf = W.HazardFunction(x: 42); // 0.98049883339449373
double chf = W.CumulativeHazardFunction(x: 42); // 0.936937017743799
// String representation
string str = W.ToString(); // "W+(x; R)"
The following example shows how to compute the W+ statistic
given a sample. The Statsstics is given as the sum of all
positive signed ranks
in a sample.
// Suppose we have computed a vector of differences between
// samples and an hypothesized value (as in Wilcoxon's test).
double[] differences = ... // differences between samples and an hypothesized median
// Compute the ranks of the absolute differences and their sign
double[] ranks = Measures.Rank(differences.Abs());
int[] signs = Accord.Math.Matrix.Sign(differences).ToInt32();
// Compute the W+ statistics from the signed ranks
double W = WilcoxonDistribution.WPositive(Signs, ranks);
Gets the number of effective samples.
Gets whether this distribution computes the exact probabilities
(by searching all possible sign combinations) or gives fast
approximations.
true if this distribution is exact; otherwise, false.
Gets the statistic values for all possible combinations
of ranks. This is used to compute the exact distribution.
Gets or sets the continuity correction
to be applied when using the Normal approximation to this distribution.
Creates a new Wilcoxon's W+ distribution.
The number of observations.
Creates a new Wilcoxon's W+ distribution.
The rank statistics for the samples.
True to compute the exact distribution. May require a significant
amount of processing power for large samples (n > 12). If left at null, whether to
compute the exact or approximate distribution will depend on the number of samples.
Computes the Wilcoxon's W+ statistic.
The W+ statistic is computed as the sum of all positive signed ranks.
Computes the Wilcoxon's W- statistic.
The W- statistic is computed as the sum of all negative signed ranks.
Computes the Wilcoxon's W statistic (equivalent to
Mann-Whitney U when used in two-sample tests).
The W statistic is computed as the minimum of the W+ and W- statistics.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Gets the mean for this distribution.
The distribution's mean value.
Gets the variance for this distribution.
The distribution's variance.
Gets the mode for this distribution.
The distribution's mode value. In the current
implementation, returns the same as the .
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the entropy for this distribution.
The distribution's entropy.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point k.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
A single point in the distribution range.
System.Double.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF.
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
A probability value between 0 and 1.
A sample which could original the given probability
value when applied in the .
Gets the probability density function (pdf) for
this distribution evaluated at point w.
A single point in the distribution range.
The probability of w occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the log-probability density function (pdf) for
this distribution evaluated at point w.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Returns a that represents this instance.
A that represents this instance.
Degenerate distribution.
In mathematics, a degenerate distribution or deterministic distribution is
the probability distribution of a random variable which only takes a single
value. Examples include a two-headed coin and rolling a die whose sides all
show the same number. While this distribution does not appear random in the
everyday sense of the word, it does satisfy the definition of random variable.
The degenerate distribution is localized at a point k0 on the real line. The
probability mass function is a Delta function at k0.
References:
-
Wikipedia, The Free Encyclopedia. Degenerate distribution. Available on:
http://en.wikipedia.org/wiki/Degenerate_distribution
This example shows how to create a Degenerate distribution
and compute some of its properties.
var dist = new DegenerateDistribution(value: 2);
double mean = dist.Mean; // 2
double median = dist.Median; // 2
double mode = dist.Mode; // 2
double var = dist.Variance; // 1
double cdf1 = dist.DistributionFunction(k: 1); // 0
double cdf2 = dist.DistributionFunction(k: 2); // 1
double pdf1 = dist.ProbabilityMassFunction(k: 1); // 0
double pdf2 = dist.ProbabilityMassFunction(k: 2); // 1
double pdf3 = dist.ProbabilityMassFunction(k: 3); // 0
double lpdf = dist.LogProbabilityMassFunction(k: 2); // 0
double ccdf = dist.ComplementaryDistributionFunction(k: 2); // 0.0
int icdf1 = dist.InverseDistributionFunction(p: 0.0); // 1
int icdf2 = dist.InverseDistributionFunction(p: 0.5); // 3
int icdf3 = dist.InverseDistributionFunction(p: 1.0); // 2
double hf = dist.HazardFunction(x: 0); // 0.0
double chf = dist.CumulativeHazardFunction(x: 0); // 0.0
string str = dist.ToString(CultureInfo.InvariantCulture); // Degenerate(x; k0 = 2)
Gets the unique value whose probability is different from zero.
Initializes a new instance of the class.
Initializes a new instance of the class.
The only value whose probability is different from zero. Default is zero.
Gets the mean for this distribution.
In the Degenerate distribution, the mean is equal to the
unique value within its domain.
The distribution's mean value, which should equal .
Gets the median for this distribution, which should equal .
In the Degenerate distribution, the mean is equal to the
unique value within its domain.
The distribution's median value.
Gets the variance for this distribution, which should equal 0.
In the Degenerate distribution, the variance equals 0.
The distribution's variance.
Gets the mode for this distribution, which should equal .
In the Degenerate distribution, the mean is equal to the
unique value within its domain.
The distribution's mode value.
Gets the entropy for this distribution, which is zero.
The distribution's entropy.
Gets the support interval for this distribution.
The degenerate distribution's support is given only on the
point interval (, ).
A containing
the support interval for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point k.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of k occurring
in the current distribution.
The Probability Mass Function (PMF) describes the
probability that a given value x will occur.
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
A probability value between 0 and 1.
A sample which could original the given probability
value when applied in the .
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
The does not support fitting.
Symmetric Geometric Distribution.
The Symmetric Geometric Distribution can be seen as a discrete case
of the .
Gets the success probability for the distribution.
Creates a new symmetric geometric distribution.
The success probability.
Gets the support interval for this distribution, which
in the case of the Symmetric Geometric is [-inf, +inf].
A containing
the support interval for this distribution.
Gets the mean for this distribution, which in
the case of the Symmetric Geometric is zero.
The distribution's mean value.
Gets the variance for this distribution.
The distribution's variance.
Not supported.
Not supported.
Gets the probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of k occurring
in the current distribution.
The Probability Mass Function (PMF) describes the
probability that a given value x will occur.
Gets the log-probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Mass Function (PMF) describes the
probability that a given value k will occur.
Generates a random observation from the current distribution.
The random number generator to use as a source of randomness.
Default is to use .
A random observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Negative Binomial distribution.
The negative binomial distribution is a discrete probability distribution of the number
of successes in a sequence of Bernoulli trials before a specified (non-random) number of
failures (denoted r) occur. For example, if one throws a die repeatedly until the third
time “1” appears, then the probability distribution of the number of non-“1”s that had
appeared will be negative binomial.
References:
-
Wikipedia, The Free Encyclopedia. Negative Binomial distribution.
Available from: http://en.wikipedia.org/wiki/Negative_binomial_distribution
// Create a new Negative Binomial distribution with r = 7 and p = 0.42
var dist = new NegativeBinomialDistribution(failures: 7, probability: 0.42);
// Common measures
double mean = dist.Mean; // 5.068965517241379
double median = dist.Median; // 5.0
double var = dist.Variance; // 8.7395957193816862
// Cumulative distribution functions
double cdf = dist.DistributionFunction(k: 2); // 0.19605133020527743
double ccdf = dist.ComplementaryDistributionFunction(k: 2); // 0.80394866979472257
// Probability mass functions
double pmf1 = dist.ProbabilityMassFunction(k: 4); // 0.054786846293416853
double pmf2 = dist.ProbabilityMassFunction(k: 5); // 0.069908015870399909
double pmf3 = dist.ProbabilityMassFunction(k: 6); // 0.0810932984096639
double lpmf = dist.LogProbabilityMassFunction(k: 2); // -2.3927801721315989
// Quantile function
int icdf1 = dist.InverseDistributionFunction(p: 0.17); // 2
int icdf2 = dist.InverseDistributionFunction(p: 0.46); // 4
int icdf3 = dist.InverseDistributionFunction(p: 0.87); // 8
// Hazard (failure rate) functions
double hf = dist.HazardFunction(x: 4); // 0.10490438293398294
double chf = dist.CumulativeHazardFunction(x: 4); // 0.64959916255036043
// String representation
string str = dist.ToString(CultureInfo.InvariantCulture); // "NegativeBinomial(x; r = 7, p = 0.42)"
Creates a new Negative Binomial distribution.
Number of failures r.
Success probability in each experiment.
Gets the number of failures.
The number of failures.
Gets the probability of success.
The probability of success in each experiment.
Gets the mean for this distribution.
The distribution's mean value.
Gets the variance for this distribution.
The distribution's variance.
Gets the entropy for this distribution.
The distribution's entropy.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets P( X<= k), the cumulative distribution function
(cdf) for this distribution evaluated at point k.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of k occurring
in the current distribution.
The Probability Mass Function (PMF) describes the
probability that a given value x will occur.
Gets the log-probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Mass Function (PMF) describes the
probability that a given value k will occur.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Returns a that represents this instance.
A that represents this instance.
Pareto's Distribution.
The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law
probability distribution that coincides with social, scientific, geophysical, actuarial,
and many other types of observable phenomena. Outside the field of economics it is sometimes
referred to as the Bradford distribution.
References:
-
Wikipedia, The Free Encyclopedia. Pareto distribution.
Available from: http://en.wikipedia.org/wiki/Pareto_distribution
// Creates a new Pareto's distribution with xm = 0.42, α = 3
var pareto = new ParetoDistribution(scale: 0.42, shape: 3);
// Common measures
double mean = pareto.Mean; // 0.63
double median = pareto.Median; // 0.52916684095584676
double var = pareto.Variance; // 0.13229999999999997
// Cumulative distribution functions
double cdf = pareto.DistributionFunction(x: 1.4); // 0.973
double ccdf = pareto.ComplementaryDistributionFunction(x: 1.4); // 0.027000000000000024
double icdf = pareto.InverseDistributionFunction(p: cdf); // 1.4000000446580794
// Probability density functions
double pdf = pareto.ProbabilityDensityFunction(x: 1.4); // 0.057857142857142857
double lpdf = pareto.LogProbabilityDensityFunction(x: 1.4); // -2.8497783609309111
// Hazard (failure rate) functions
double hf = pareto.HazardFunction(x: 1.4); // 2.142857142857141
double chf = pareto.CumulativeHazardFunction(x: 1.4); // 3.6119184129778072
// String representation
string str = pareto.ToString(CultureInfo.InvariantCulture); // Pareto(x; xm = 0.42, α = 3)
Creates new Pareto distribution.
Creates new Pareto distribution.
The scale parameter xm. Default is 1.
The shape parameter α (alpha). Default is 1.
Gets the scale parameter xm for this distribution.
Gets the shape parameter α (alpha) for this distribution.
Gets the mean for this distribution.
The distribution's mean value.
The Pareto distribution's mean is defined as
α * xm / (α - 1).
Gets the variance for this distribution.
The distribution's variance.
The Pareto distribution's mean is defined as
α * xm² / ((α - 1)² * (α - 2).
Gets the entropy for this distribution.
The distribution's entropy.
The Pareto distribution's Entropy is defined as
ln(xm / α) + 1 / α + 1.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the mode for this distribution.
The distribution's mode value.
Gets the median for this distribution.
The distribution's median value.
The Pareto distribution's median is defined
as xm * 2^(1 / α).
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
See .
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
See .
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
See .
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
A random observations drawn from this distribution.
Discrete uniform distribution.
In probability theory and statistics, the discrete uniform distribution is a
symmetric probability distribution whereby a finite number of values are equally
likely to be observed; every one of n values has equal probability 1/n. Another
way of saying "discrete uniform distribution" would be "a known, finite number of
outcomes equally likely to happen".
A simple example of the discrete uniform distribution is throwing a fair die.
The possible values are 1, 2, 3, 4, 5, 6, and each time the die is thrown the
probability of a given score is 1/6. If two dice are thrown and their values
added, the resulting distribution is no longer uniform since not all sums have
equal probability.
The discrete uniform distribution itself is inherently non-parametric. It is
convenient, however, to represent its values generally by an integer interval
[a,b], so that a,b become the main parameters of the distribution (often one
simply considers the interval [1,n] with the single parameter n).
References:
-
Wikipedia, The Free Encyclopedia. Uniform distribution (discrete). Available on:
http://en.wikipedia.org/wiki/Uniform_distribution_(discrete)
// Create an uniform (discrete) distribution in [2, 6]
var dist = new UniformDiscreteDistribution(a: 2, b: 6);
// Common measures
double mean = dist.Mean; // 4.0
double median = dist.Median; // 4.0
double var = dist.Variance; // 1.3333333333333333
// Cumulative distribution functions
double cdf = dist.DistributionFunction(k: 2); // 0.2
double ccdf = dist.ComplementaryDistributionFunction(k: 2); // 0.8
// Probability mass functions
double pmf1 = dist.ProbabilityMassFunction(k: 4); // 0.2
double pmf2 = dist.ProbabilityMassFunction(k: 5); // 0.2
double pmf3 = dist.ProbabilityMassFunction(k: 6); // 0.2
double lpmf = dist.LogProbabilityMassFunction(k: 2); // -1.6094379124341003
// Quantile function
int icdf1 = dist.InverseDistributionFunction(p: 0.17); // 2
int icdf2 = dist.InverseDistributionFunction(p: 0.46); // 4
int icdf3 = dist.InverseDistributionFunction(p: 0.87); // 6
// Hazard (failure rate) functions
double hf = dist.HazardFunction(x: 4); // 0.5
double chf = dist.CumulativeHazardFunction(x: 4); // 0.916290731874155
// String representation
string str = dist.ToString(CultureInfo.InvariantCulture); // "U(x; a = 2, b = 6)"
Creates a discrete uniform distribution defined in the interval [a;b].
The starting (minimum) value a.
The ending (maximum) value b.
Gets the minimum value of the distribution (a).
Gets the maximum value of the distribution (b).
Gets the length of the distribution (b - a + 1).
Gets the mean for this distribution.
Gets the variance for this distribution.
Gets the entropy for this distribution.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point k.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of k occurring
in the current distribution.
The Probability Mass Function (PMF) describes the
probability that a given value k will occur.
Gets the log-probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of k
occurring in the current distribution.
The Probability Mass Function (PMF) describes the
probability that a given value k will occur.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Generates a random vector of observations from the
Uniform distribution with the given parameters.
The starting number a.
The ending number b.
The number of samples to generate.
An array of double values sampled from the specified Uniform distribution.
Generates a random vector of observations from the
Uniform distribution with the given parameters.
The starting number a.
The ending number b.
The number of samples to generate.
The location where to store the samples.
An array of double values sampled from the specified Uniform distribution.
Generates a random observation from the Uniform
distribution defined in the interval 0 and MAXVALUE.
The number of samples to generate.
An array of double values sampled from the specified Uniform distribution.
Generates a random observation from the Uniform
distribution defined in the interval 0 and MAXVALUE.
The number of samples to generate.
The location where to store the samples.
An array of double values sampled from the specified Uniform distribution.
Generates a random observation from the Uniform
distribution defined in the interval 0 and MAXVALUE.
A random double value sampled from the specified Uniform distribution.
Generates a random observation from the
Uniform distribution with the given parameters.
The starting number a.
The ending number b.
A random double value sampled from the specified Uniform distribution.
Generates a random vector of observations from the
Uniform distribution with the given parameters.
The starting number a.
The ending number b.
The number of samples to generate.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified Uniform distribution.
Generates a random vector of observations from the
Uniform distribution with the given parameters.
The starting number a.
The ending number b.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified Uniform distribution.
Generates a random observation from the Uniform
distribution defined in the interval 0 and MAXVALUE.
The number of samples to generate.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified Uniform distribution.
Generates a random observation from the Uniform
distribution defined in the interval 0 and MAXVALUE.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified Uniform distribution.
Generates a random observation from the Uniform
distribution defined in the interval 0 and MAXVALUE.
A random double value sampled from the specified Uniform distribution.
Generates a random observation from the
Uniform distribution with the given parameters.
The starting number a.
The ending number b.
The random number generator to use as a source of randomness.
Default is to use .
A random double value sampled from the specified Uniform distribution.
Returns a that represents this instance.
A that represents this instance.
Generates a random observation from the current distribution.
The random number generator to use as a source of randomness.
Default is to use .
A random observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
(Shifted) Geometric Distribution.
This class represents the shifted version of the Geometric distribution
with support on { 0, 1, 2, 3, ... }. This is the probability distribution
of the number Y = X − 1 of failures before the first success, supported
on the set { 0, 1, 2, 3, ... }.
References:
-
Wikipedia, The Free Encyclopedia. Geometric distribution. Available on:
http://en.wikipedia.org/wiki/Geometric_distribution
// Create a Geometric distribution with 42% success probability
var dist = new GeometricDistribution(probabilityOfSuccess: 0.42);
// Common measures
double mean = dist.Mean; // 1.3809523809523812
double median = dist.Median; // 1.0
double var = dist.Variance; // 3.2879818594104315
double mode = dist.Mode; // 0.0
// Cumulative distribution functions
double cdf = dist.DistributionFunction(k: 2); // 0.80488799999999994
double ccdf = dist.ComplementaryDistributionFunction(k: 2); // 0.19511200000000006
// Probability mass functions
double pdf1 = dist.ProbabilityMassFunction(k: 0); // 0.42
double pdf2 = dist.ProbabilityMassFunction(k: 1); // 0.2436
double pdf3 = dist.ProbabilityMassFunction(k: 2); // 0.141288
double lpdf = dist.LogProbabilityMassFunction(k: 2); // -1.956954918588067
// Quantile functions
int icdf1 = dist.InverseDistributionFunction(p: 0.17); // 0
int icdf2 = dist.InverseDistributionFunction(p: 0.46); // 1
int icdf3 = dist.InverseDistributionFunction(p: 0.87); // 3
// Hazard (failure rate) functions
double hf = dist.HazardFunction(x: 0); // 0.72413793103448265
double chf = dist.CumulativeHazardFunction(x: 0); // 0.54472717544167193
// String representation
string str = dist.ToString(CultureInfo.InvariantCulture); // "Geometric(x; p = 0.42)"
Gets the success probability for the distribution.
Creates a new (shifted) geometric distribution.
The success probability.
Gets the mean for this distribution.
The distribution's mean value.
Gets the mode for this distribution.
The distribution's mode value.
Gets the median for this distribution.
The distribution's median value.
Gets the variance for this distribution.
The distribution's variance.
Gets the entropy for this distribution.
The distribution's entropy.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point k.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of k occurring
in the current distribution.
The Probability Mass Function (PMF) describes the
probability that a given value x will occur.
Gets the log-probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Mass Function (PMF) describes the
probability that a given value k will occur.
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
A probability value between 0 and 1.
A sample which could original the given probability
value when applied in the .
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Generates a random observation from the current distribution.
The random number generator to use as a source of randomness.
Default is to use .
A random observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
The probability of success.
A random observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The probability of success.
The number of samples to generate.
The location where to store the samples.
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The probability of success.
The number of samples to generate.
The location where to store the samples.
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
The probability of success.
The random number generator to use as a source of randomness.
Default is to use .
A random observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The probability of success.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The probability of success.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Hypergeometric probability distribution.
The hypergeometric distribution is a discrete probability distribution that
describes the probability of k successes in n draws from a finite population
without replacement. This is in contrast to the
binomial distribution, which describes the probability of k successes
in n draws with replacement.
References:
-
Wikipedia, The Free Encyclopedia. Hypergeometric distribution. Available on:
http://en.wikipedia.org/wiki/Hypergeometric_distribution
// Distribution parameters
int populationSize = 15; // population size N
int success = 7; // number of successes in the sample
int samples = 8; // number of samples drawn from N
// Create a new Hypergeometric distribution with N = 15, n = 8, and s = 7
var dist = new HypergeometricDistribution(populationSize, success, samples);
// Common measures
double mean = dist.Mean; // 1.3809523809523812
double median = dist.Median; // 4.0
double var = dist.Variance; // 3.2879818594104315
double mode = dist.Mode; // 4.0
// Cumulative distribution functions
double cdf = dist.DistributionFunction(k: 2); // 0.80488799999999994
double ccdf = dist.ComplementaryDistributionFunction(k: 2); // 0.19511200000000006
// Probability mass functions
double pdf1 = dist.ProbabilityMassFunction(k: 4); // 0.38073038073038074
double pdf2 = dist.ProbabilityMassFunction(k: 5); // 0.18275058275058276
double pdf3 = dist.ProbabilityMassFunction(k: 6); // 0.030458430458430458
double lpdf = dist.LogProbabilityMassFunction(k: 2); // -2.3927801721315989
// Quantile function
int icdf1 = dist.InverseDistributionFunction(p: 0.17); // 3
int icdf2 = dist.InverseDistributionFunction(p: 0.46); // 4
int icdf3 = dist.InverseDistributionFunction(p: 0.87); // 5
// Hazard (failure rate) functions
double hf = dist.HazardFunction(x: 4); // 1.7753623188405792
double chf = dist.CumulativeHazardFunction(x: 4); // 1.5396683418789763
// String representation
string str = dist.ToString(CultureInfo.InvariantCulture); // "HyperGeometric(x; N = 15, m = 7, n = 8)"
Gets the size N of the
population for this distribution.
Gets the size n of the sample drawn
from N.
Gets the count of success trials in the
population for this distribution. This
is often referred as m.
Constructs a new Hypergeometric distribution. To build from
the number of failures and succcesses like in R, please see .
Size N of the population.
The number m of successes in the population.
The number n of samples drawn from the population.
Creates a new from the number of successes, failures,
and the number of samples drawn. This is the same parameterization used in R.
The number m of successes in the population.
The number m of failures in the population.
The number n of samples drawn from the population.
Gets the mean for this distribution.
The distribution's mean value.
Gets the variance for this distribution.
The distribution's variance.
Gets the entropy for this distribution.
The distribution's entropy.
Gets the mode for this distribution.
The distribution's mode value.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point k.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of k occurring
in the current distribution.
The Probability Mass Function (PMF) describes the
probability that a given value x will occur.
Gets the log-probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Mass Function (PMF) describes the
probability that a given value k will occur.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Inverse Gaussian (Normal) Distribution, also known as the Wald distribution.
The Inverse Gaussian distribution is a two-parameter family of continuous probability
distributions with support on (0,∞). As λ tends to infinity, the inverse Gaussian distribution
becomes more like a normal (Gaussian) distribution. The inverse Gaussian distribution has
several properties analogous to a Gaussian distribution. The name can be misleading: it is
an "inverse" only in that, while the Gaussian describes a Brownian Motion's level at a fixed
time, the inverse Gaussian describes the distribution of the time a Brownian Motion with positive
drift takes to reach a fixed positive level.
References:
-
Wikipedia, The Free Encyclopedia. Inverse Gaussian distribution. Available on:
http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution
// Create a new inverse Gaussian distribution with μ = 0.42 and λ = 1.2
var invGaussian = new InverseGaussianDistribution(mean: 0.42, shape: 1.2);
// Common measures
double mean = invGaussian.Mean; // 0.42
double median = invGaussian.Median; // 0.35856861093990083
double var = invGaussian.Variance; // 0.061739999999999989
// Cumulative distribution functions
double cdf = invGaussian.DistributionFunction(x: 0.27); // 0.30658791274125458
double ccdf = invGaussian.ComplementaryDistributionFunction(x: 0.27); // 0.69341208725874548
double icdf = invGaussian.InverseDistributionFunction(p: cdf); // 0.26999999957543408
// Probability density functions
double pdf = invGaussian.ProbabilityDensityFunction(x: 0.27); // 2.3461495925760354
double lpdf = invGaussian.LogProbabilityDensityFunction(x: 0.27); // 0.85277551314980737
// Hazard (failure rate) functions
double hf = invGaussian.HazardFunction(x: 0.27); // 3.383485283406336
double chf = invGaussian.CumulativeHazardFunction(x: 0.27); // 0.36613081401302111
// String representation
string str = invGaussian.ToString(CultureInfo.InvariantCulture); // "N^-1(x; μ = 0.42, λ = 1.2)"
Constructs a new Inverse Gaussian distribution.
The mean parameter mu.
The shape parameter lambda.
Gets the shape parameter (lambda)
for this distribution.
The distribution's lambda value.
Gets the mean for this distribution.
The distribution's mean value.
Gets the variance for this distribution.
The distribution's variance.
Gets the mode for this distribution.
The distribution's mode value.
Gets the entropy for this distribution.
The distribution's entropy.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
See .
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
See .
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
See .
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Estimates a new Normal distribution from a given set of observations.
Estimates a new Normal distribution from a given set of observations.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
The random number generator to use as a source of randomness.
Default is to use .
A random observations drawn from this distribution.
Generates a random observation from the
Inverse Gaussian distribution with the given parameters.
The mean parameter mu.
The shape parameter lambda.
A random double value sampled from the specified Uniform distribution.
Generates a random observation from the
Inverse Gaussian distribution with the given parameters.
The mean parameter mu.
The shape parameter lambda.
The random number generator to use as a source of randomness.
Default is to use .
A random double value sampled from the specified Uniform distribution.
Generates a random vector of observations from the
Inverse Gaussian distribution with the given parameters.
The mean parameter mu.
The shape parameter lambda.
The number of samples to generate.
An array of double values sampled from the specified Uniform distribution.
Generates a random vector of observations from the
Inverse Gaussian distribution with the given parameters.
The mean parameter mu.
The shape parameter lambda.
The number of samples to generate.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified Uniform distribution.
Generates a random vector of observations from the
Inverse Gaussian distribution with the given parameters.
The mean parameter mu.
The shape parameter lambda.
The number of samples to generate.
The location where to store the samples.
An array of double values sampled from the inverse Gaussian distribution.
Generates a random vector of observations from the
Inverse Gaussian distribution with the given parameters.
The mean parameter mu.
The shape parameter lambda.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the inverse Gaussian distribution.
Returns a that represents this instance.
The format.
The format provider.
A that represents this instance.
Nakagami distribution.
The Nakagami distribution has been used in the modeling of wireless
signal attenuation while traversing multiple paths.
References:
-
Wikipedia, The Free Encyclopedia. Nakagami distribution. Available on:
http://en.wikipedia.org/wiki/Nakagami_distribution
-
Laurenson, Dave (1994). "Nakagami Distribution". Indoor Radio Channel Propagation
Modeling by Ray Tracing Techniques.
-
R. Kolar, R. Jirik, J. Jan (2004) "Estimator Comparison of the Nakagami-m Parameter
and Its Application in Echocardiography", Radioengineering, 13 (1), 8–12
var nakagami = new NakagamiDistribution(shape: 2.4, spread: 4.2);
double mean = nakagami.Mean; // 1.946082119049118
double median = nakagami.Median; // 1.9061151110206338
double var = nakagami.Variance; // 0.41276438591729486
double cdf = nakagami.DistributionFunction(x: 1.4); // 0.20603416752368109
double pdf = nakagami.ProbabilityDensityFunction(x: 1.4); // 0.49253215371343023
double lpdf = nakagami.LogProbabilityDensityFunction(x: 1.4); // -0.708195533773302
double ccdf = nakagami.ComplementaryDistributionFunction(x: 1.4); // 0.79396583247631891
double icdf = nakagami.InverseDistributionFunction(p: cdf); // 1.400000000131993
double hf = nakagami.HazardFunction(x: 1.4); // 0.62034426869133652
double chf = nakagami.CumulativeHazardFunction(x: 1.4); // 0.23071485080660473
string str = nakagami.ToString(CultureInfo.InvariantCulture); // Nakagami(x; μ = 2,4, ω = 4,2)"
Initializes a new instance of the class.
The shape parameter μ (mu).
The spread parameter ω (omega).
Gets the distribution's shape parameter μ (mu).
The shape parameter μ (mu).
Gets the distribution's spread parameter ω (omega).
The spread parameter ω (omega).
Gets the mean for this distribution.
The distribution's mean value.
Nakagami's mean is defined in terms of the
Gamma function Γ(x) as (Γ(μ + 0.5) / Γ(μ)) * sqrt(ω / μ).
Gets the mode for this distribution.
The distribution's mode value.
Gets the variance for this distribution.
The distribution's variance.
Nakagami's variance is defined in terms of the
Gamma function Γ(x) as ω * (1 - (1 / μ) * (Γ(μ + 0.5) / Γ(μ))².
This method is not supported.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
The Nakagami's distribution CDF is defined in terms of the
Lower incomplete regularized
Gamma function P(a, x) as CDF(x) = P(μ, μ / ω) * x².
See .
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Nakagami's PDF is defined as
PDF(x) = c * x^(2 * μ - 1) * exp(-(μ / ω) * x²)
in which c = 2 * μ ^ μ / (Γ(μ) * ω ^ μ))
See .
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Nakagami's PDF is defined as
PDF(x) = c * x^(2 * μ - 1) * exp(-(μ / ω) * x²)
in which c = 2 * μ ^ μ / (Γ(μ) * ω ^ μ))
See .
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Estimates a new Nakagami distribution from a given set of observations.
Estimates a new Nakagami distribution from a given set of observations.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
The random number generator to use as a source of randomness.
Default is to use .
A random observations drawn from this distribution.
Generates a random vector of observations from the
Nakagami distribution with the given parameters.
The shape parameter μ.
The spread parameter ω.
The number of samples to generate.
An array of double values sampled from the specified Nakagami distribution.
Generates a random vector of observations from the
Nakagami distribution with the given parameters.
The shape parameter μ.
The spread parameter ω.
The number of samples to generate.
The location where to store the samples.
An array of double values sampled from the specified Nakagami distribution.
Generates a random observation from the
Nakagami distribution with the given parameters.
The shape parameter μ.
The spread parameter ω.
A random double value sampled from the specified Nakagami distribution.
Generates a random vector of observations from the
Nakagami distribution with the given parameters.
The shape parameter μ.
The spread parameter ω.
The number of samples to generate.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified Nakagami distribution.
Generates a random vector of observations from the
Nakagami distribution with the given parameters.
The shape parameter μ.
The spread parameter ω.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified Nakagami distribution.
Generates a random observation from the
Nakagami distribution with the given parameters.
The shape parameter μ.
The spread parameter ω.
The random number generator to use as a source of randomness.
Default is to use .
A random double value sampled from the specified Nakagami distribution.
Returns a that represents this instance.
The format.
The format provider.
A that represents this instance.
Rayleigh distribution.
In probability theory and statistics, the Rayleigh distribution is a continuous
probability distribution. A Rayleigh distribution is often observed when the overall
magnitude of a vector is related to its directional components.
One example where the Rayleigh distribution naturally arises is when wind speed
is analyzed into its orthogonal 2-dimensional vector components. Assuming that the
magnitude of each component is uncorrelated and normally distributed with equal variance,
then the overall wind speed (vector magnitude) will be characterized by a Rayleigh
distribution.
References:
-
Wikipedia, The Free Encyclopedia. Rayleigh distribution. Available on:
http://en.wikipedia.org/wiki/Rayleigh_distribution
// Create a new Rayleigh's distribution with σ = 0.42
var rayleigh = new RayleighDistribution(sigma: 0.42);
// Common measures
double mean = rayleigh.Mean; // 0.52639193767251
double median = rayleigh.Median; // 0.49451220943852386
double var = rayleigh.Variance; // 0.075711527953380237
// Cumulative distribution functions
double cdf = rayleigh.DistributionFunction(x: 1.4); // 0.99613407986052716
double ccdf = rayleigh.ComplementaryDistributionFunction(x: 1.4); // 0.0038659201394728449
double icdf = rayleigh.InverseDistributionFunction(p: cdf); // 1.4000000080222026
// Probability density functions
double pdf = rayleigh.ProbabilityDensityFunction(x: 1.4); // 0.030681905868831811
double lpdf = rayleigh.LogProbabilityDensityFunction(x: 1.4); // -3.4840821835248961
// Hazard (failure rate) functions
double hf = rayleigh.HazardFunction(x: 1.4); // 7.9365079365078612
double chf = rayleigh.CumulativeHazardFunction(x: 1.4); // 5.5555555555555456
// String representation
string str = rayleigh.ToString(CultureInfo.InvariantCulture); // Rayleigh(x; σ = 0.42)
Creates a new Rayleigh distribution.
The scale parameter σ (sigma).
Gets the mean for this distribution.
The distribution's mean value.
The Rayleight's mean value is defined
as mean = σ * sqrt(π / 2).
Gets the Rayleight's scale parameter σ (sigma)
Gets the variance for this distribution.
The distribution's variance.
The Rayleight's variance value is defined
as var = (4 - π) / 2 * σ².
Gets the mode for this distribution.
In the Rayleigh distribution, the mode equals σ (sigma).
The distribution's mode value.
Gets the entropy for this distribution.
The distribution's entropy.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
See .
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
See .
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
See .
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Estimates a new Gamma distribution from a given set of observations.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
The random number generator to use as a source of randomness.
Default is to use .
A random observations drawn from this distribution.
Generates a random vector of observations from the
Rayleigh distribution with the given parameters.
The Rayleigh distribution's sigma.
The number of samples to generate.
An array of double values sampled from the specified Rayleigh distribution.
Generates a random vector of observations from the
Rayleigh distribution with the given parameters.
The Rayleigh distribution's sigma.
The number of samples to generate.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified Rayleigh distribution.
Generates a random vector of observations from the
Rayleigh distribution with the given parameters.
The Rayleigh distribution's sigma.
The number of samples to generate.
The location where to store the samples.
An array of double values sampled from the specified Rayleigh distribution.
Generates a random vector of observations from the
Rayleigh distribution with the given parameters.
The Rayleigh distribution's sigma.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified Rayleigh distribution.
Generates a random observation from the
Rayleigh distribution with the given parameters.
The Rayleigh distribution's sigma.
A random double value sampled from the specified Rayleigh distribution.
Generates a random observation from the
Rayleigh distribution with the given parameters.
The Rayleigh distribution's sigma.
The random number generator to use as a source of randomness.
Default is to use .
A random double value sampled from the specified Rayleigh distribution.
Returns a that represents this instance.
A that represents this instance.
Student's t-distribution.
In probability and statistics, Student's t-distribution (or simply the
t-distribution) is a family of continuous probability distributions that
arises when estimating the mean of a normally distributed population in
situations where the sample size is small and population standard deviation
is unknown. It plays a role in a number of widely used statistical analyses,
including the Student's t-test for assessing the statistical significance of
the difference between two sample means, the construction of confidence intervals
for the difference between two population means, and in linear regression
analysis. The Student's t-distribution also arises in the Bayesian analysis of
data from a normal family.
If we take k samples from a normal distribution with fixed unknown mean and
variance, and if we compute the sample mean and sample variance for these k
samples, then the t-distribution (for k) can be defined as the distribution
of the location of the true mean, relative to the sample mean and divided by
the sample standard deviation, after multiplying by the normalizing term
sqrt(n), where n is the sample size. In this way the t-distribution
can be used to estimate how likely it is that the true mean lies in any given
range.
The t-distribution is symmetric and bell-shaped, like the normal distribution,
but has heavier tails, meaning that it is more prone to producing values that
fall far from its mean. This makes it useful for understanding the statistical
behavior of certain types of ratios of random quantities, in which variation in
the denominator is amplified and may produce outlying values when the denominator
of the ratio falls close to zero. The Student's t-distribution is a special case
of the generalized hyperbolic distribution.
References:
-
Wikipedia, The Free Encyclopedia. Student's t-distribution. Available on:
http://en.wikipedia.org/wiki/Student's_t-distribution
// Create a new Student's T distribution with d.f = 4.2
TDistribution t = new TDistribution(degreesOfFreedom: 4.2);
// Common measures
double mean = t.Mean; // 0.0
double median = t.Median; // 0.0
double var = t.Variance; // 1.9090909090909089
// Cumulative distribution functions
double cdf = t.DistributionFunction(x: 1.4); // 0.88456136730659074
double pdf = t.ProbabilityDensityFunction(x: 1.4); // 0.13894002185341031
double lpdf = t.LogProbabilityDensityFunction(x: 1.4); // -1.9737129364307417
// Probability density functions
double ccdf = t.ComplementaryDistributionFunction(x: 1.4); // 0.11543863269340926
double icdf = t.InverseDistributionFunction(p: cdf); // 1.4000000000000012
// Hazard (failure rate) functions
double hf = t.HazardFunction(x: 1.4); // 1.2035833984833988
double chf = t.CumulativeHazardFunction(x: 1.4); // 2.1590162088918525
// String representation
string str = t.ToString(CultureInfo.InvariantCulture); // T(x; df = 4.2)
Gets the degrees of freedom for the distribution.
Initializes a new instance of the class.
The degrees of freedom.
Gets the mean for this distribution.
The distribution's mean value.
In the T Distribution, the mean is zero if the number of degrees
of freedom is higher than 1. Otherwise, it is undefined.
Gets the mode for this distribution (always zero).
The distribution's mode value (zero).
Gets the variance for this distribution.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Not supported.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
See .
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
See .
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
See .
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
The Inverse Cumulative Distribution Function (ICDF) specifies, for
a given probability, the value which the random variable will be at,
or below, with that probability.
See .
Not supported.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Gets the inverse of the cumulative distribution function (icdf) for
the left tail T-distribution evaluated at probability p.
Based on the stdtril function from the Cephes Math Library
Release 2.8, adapted with permission of Stephen L. Moshier.
Continuous Uniform Distribution.
The continuous uniform distribution or rectangular distribution is a family of
symmetric probability distributions such that for each member of the family, all
intervals of the same length on the distribution's support are equally probable.
The support is defined by the two parameters, a and b, which are its minimum and
maximum values. The distribution is often abbreviated U(a,b). It is the maximum
entropy probability distribution for a random variate X under no constraint other
than that it is contained in the distribution's support.
References:
-
Wikipedia, The Free Encyclopedia. Uniform Distribution (continuous). Available on:
http://en.wikipedia.org/wiki/Uniform_distribution_(continuous)
The following example demonstrates how to create an uniform
distribution defined over the interval [0.42, 1.1].
// Create a new uniform continuous distribution from 0.42 to 1.1
var uniform = new UniformContinuousDistribution(a: 0.42, b: 1.1);
// Common measures
double mean = uniform.Mean; // 0.76
double median = uniform.Median; // 0.76
double var = uniform.Variance; // 0.03853333333333335
// Cumulative distribution functions
double cdf = uniform.DistributionFunction(x: 0.9); // 0.70588235294117641
double ccdf = uniform.ComplementaryDistributionFunction(x: 0.9); // 0.29411764705882359
double icdf = uniform.InverseDistributionFunction(p: cdf); // 0.9
// Probability density functions
double pdf = uniform.ProbabilityDensityFunction(x: 0.9); // 1.4705882352941173
double lpdf = uniform.LogProbabilityDensityFunction(x: 0.9); // 0.38566248081198445
// Hazard (failure rate) functions
double hf = uniform.HazardFunction(x: 0.9); // 4.9999999999999973
double chf = uniform.CumulativeHazardFunction(x: 0.9); // 1.2237754316221154
// String representation
string str = uniform.ToString(CultureInfo.InvariantCulture); // "U(x; a = 0.42, b = 1.1)"
Creates a new uniform distribution defined in the interval [0;1].
Creates a new uniform distribution defined in the interval [min;max].
The output range for the uniform distribution.
Creates a new uniform distribution defined in the interval [a;b].
The starting number a.
The ending number b.
Gets the minimum value of the distribution (a).
Gets the maximum value of the distribution (b).
Gets the length of the distribution (b-a).
Gets the mean for this distribution.
Gets the variance for this distribution.
Gets the mode for this distribution.
The mode of the uniform distribution is any value contained
in the interval of the distribution. The framework return
the same value as the .
The distribution's mode value.
Gets the entropy for this distribution.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Gets the Standard Uniform Distribution,
starting at zero and ending at one (a=0, b=1).
Estimates a new uniform distribution from a given set of observations.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
The random number generator to use as a source of randomness.
Default is to use .
A random observations drawn from this distribution.
Generates a random vector of observations from the
Uniform distribution with the given parameters.
The starting number a.
The ending number b.
The number of samples to generate.
An array of double values sampled from the specified Uniform distribution.
Generates a random vector of observations from the
Uniform distribution with the given parameters.
The starting number a.
The ending number b.
The number of samples to generate.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified Uniform distribution.
Generates a random vector of observations from the
Uniform distribution with the given parameters.
The starting number a.
The ending number b.
The number of samples to generate.
The location where to store the samples.
An array of double values sampled from the specified Uniform distribution.
Generates a random vector of observations from the
Uniform distribution with the given parameters.
The starting number a.
The ending number b.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified Uniform distribution.
Generates a random observation from the Uniform
distribution defined in the interval 0 and 1.
The number of samples to generate.
An array of double values sampled from the specified Uniform distribution.
Generates a random observation from the Uniform
distribution defined in the interval 0 and 1.
The number of samples to generate.
The location where to store the samples.
An array of double values sampled from the specified Uniform distribution.
Generates a random observation from the Uniform
distribution defined in the interval 0 and 1.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified Uniform distribution.
Generates a random observation from the Uniform
distribution defined in the interval 0 and 1.
A random double value sampled from the specified Uniform distribution.
Generates a random observation from the Uniform
distribution defined in the interval 0 and 1.
A random double value sampled from the specified Uniform distribution.
Generates a random observation from the
Uniform distribution with the given parameters.
The starting number a.
The ending number b.
A random double value sampled from the specified Uniform distribution.
Generates a random observation from the
Uniform distribution with the given parameters.
The starting number a.
The ending number b.
The random number generator to use as a source of randomness.
Default is to use .
A random double value sampled from the specified Uniform distribution.
Returns a that represents this instance.
A that represents this instance.
Log-Normal (Galton) distribution.
The log-normal distribution is a probability distribution of a random
variable whose logarithm is normally distributed.
References:
-
Wikipedia, The Free Encyclopedia. Log-normal distribution.
Available on: http://en.wikipedia.org/wiki/Log-normal_distribution
-
NIST/SEMATECH e-Handbook of Statistical Methods. Lognormal Distribution.
Available on: http://www.itl.nist.gov/div898/handbook/eda/section3/eda3669.htm
-
Weisstein, Eric W. "Normal Distribution Function." From MathWorld--A Wolfram Web
Resource. http://mathworld.wolfram.com/NormalDistributionFunction.html
// Create a new Log-normal distribution with μ = 2.79 and σ = 1.10
var log = new LognormalDistribution(location: 0.42, shape: 1.1);
// Common measures
double mean = log.Mean; // 2.7870954605658511
double median = log.Median; // 1.5219615583481305
double var = log.Variance; // 18.28163603621158
// Cumulative distribution functions
double cdf = log.DistributionFunction(x: 0.27); // 0.057961222885664958
double ccdf = log.ComplementaryDistributionFunction(x: 0.27); // 0.942038777114335
double icdf = log.InverseDistributionFunction(p: cdf); // 0.26999997937815973
// Probability density functions
double pdf = log.ProbabilityDensityFunction(x: 0.27); // 0.39035530085982068
double lpdf = log.LogProbabilityDensityFunction(x: 0.27); // -0.94069792674674835
// Hazard (failure rate) functions
double hf = log.HazardFunction(x: 0.27); // 0.41437285846720867
double chf = log.CumulativeHazardFunction(x: 0.27); // 0.059708840588116374
// String representation
string str = log.ToString("N2", CultureInfo.InvariantCulture); // Lognormal(x; μ = 2.79, σ = 1.10)
Constructs a Log-Normal (Galton) distribution
with zero location and unit shape.
Constructs a Log-Normal (Galton) distribution
with given location and unit shape.
The distribution's location value μ (mu).
Constructs a Log-Normal (Galton) distribution
with given mean and standard deviation.
The distribution's location value μ (mu).
The distribution's shape deviation σ (sigma).
Shape parameter σ (sigma) of
the log-normal distribution.
Squared shape parameter σ² (sigma-squared)
of the log-normal distribution.
Location parameter μ (mu) of the log-normal distribution.
Gets the Mean for this Log-Normal distribution.
The Lognormal distribution's mean is
defined as exp(μ + σ²/2).
Gets the Variance (the square of the standard
deviation) for this Log-Normal distribution.
The Lognormal distribution's variance is
defined as (exp(σ²) - 1) * exp(2*μ + σ²).
Gets the mode for this distribution.
The distribution's mode value.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the Entropy for this Log-Normal distribution.
Gets the cumulative distribution function (cdf) for
the this Log-Normal distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
The calculation is computed through the relationship to the error function
as erfc(-z/sqrt(2)) / 2. See
[Weisstein] for more details.
References:
-
Weisstein, Eric W. "Normal Distribution Function." From MathWorld--A Wolfram Web
Resource. http://mathworld.wolfram.com/NormalDistributionFunction.html
See .
Gets the probability density function (pdf) for
the Normal distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
See .
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
See .
Gets the Standard Log-Normal Distribution,
with location set to zero and unit shape.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Estimates a new Log-Normal distribution from a given set of observations.
Estimates a new Log-Normal distribution from a given set of observations.
Estimates a new Log-Normal distribution from a given set of observations.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
A random observations drawn from this distribution.
Generates a random observation from the
Lognormal distribution with the given parameters.
The distribution's location value.
The distribution's shape deviation.
A random double value sampled from the specified Lognormal distribution.
Generates a random vector of observations from the
Lognormal distribution with the given parameters.
The distribution's location value.
The distribution's shape deviation.
The number of samples to generate.
An array of double values sampled from the specified Lognormal distribution.
Generates a random vector of observations from the
Lognormal distribution with the given parameters.
The distribution's location value.
The distribution's shape deviation.
The number of samples to generate.
The location where to store the samples.
An array of double values sampled from the specified Lognormal distribution.
Generates a random observation from the
Lognormal distribution with the given parameters.
The distribution's location value.
The distribution's shape deviation.
The random number generator to use as a source of randomness.
Default is to use .
A random double value sampled from the specified Lognormal distribution.
Generates a random vector of observations from the
Lognormal distribution with the given parameters.
The distribution's location value.
The distribution's shape deviation.
The number of samples to generate.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified Lognormal distribution.
Generates a random vector of observations from the
Lognormal distribution with the given parameters.
The distribution's location value.
The distribution's shape deviation.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified Lognormal distribution.
Returns a that represents this instance.
A that represents this instance.
Empirical distribution.
Empirical distributions are based solely on the data. This class
uses the empirical distribution function and the Gaussian kernel
density estimation to provide an univariate continuous distribution
implementation which depends only on sampled data.
References:
-
Wikipedia, The Free Encyclopedia. Empirical Distribution Function. Available on:
http://en.wikipedia.org/wiki/Empirical_distribution_function
-
PlanetMath. Empirical Distribution Function. Available on:
http://planetmath.org/encyclopedia/EmpiricalDistributionFunction.html
-
Wikipedia, The Free Encyclopedia. Kernel Density Estimation. Available on:
http://en.wikipedia.org/wiki/Kernel_density_estimation
-
Bishop, Christopher M.; Pattern Recognition and Machine Learning.
Springer; 1st ed. 2006.
The following example shows how to build an empirical distribution directly from a sample:
// Consider the following univariate samples
double[] samples = { 5, 5, 1, 4, 1, 2, 2, 3, 3, 3, 4, 3, 3, 3, 4, 3, 2, 3 };
// Create a non-parametric, empirical distribution using those samples:
EmpiricalDistribution distribution = new EmpiricalDistribution(samples);
// Common measures
double mean = distribution.Mean; // 3
double median = distribution.Median; // 2.9999993064186787
double var = distribution.Variance; // 1.2941176470588236
// Cumulative distribution function
double cdf = distribution.DistributionFunction(x: 4.2); // 0.88888888888888884
double ccdf = distribution.ComplementaryDistributionFunction(x: 4.2); //0.11111111111111116
double icdf = distribution.InverseDistributionFunction(p: cdf); // 4.1999999999999993
// Probability density functions
double pdf = distribution.ProbabilityDensityFunction(x: 4.2); // 0.15552784414141974
double lpdf = distribution.LogProbabilityDensityFunction(x: 4.2); // -1.8609305013898356
// Hazard (failure rate) functions
double hf = distribution.HazardFunction(x: 4.2); // 1.3997505972727771
double chf = distribution.CumulativeHazardFunction(x: 4.2); // 2.1972245773362191
// Automatically estimated smooth parameter (gamma)
double smoothing = distribution.Smoothing; // 1.9144923416414432
// String representation
string str = distribution.ToString(CultureInfo.InvariantCulture); // Fn(x; S)
Creates a new Empirical Distribution from the data samples.
The data samples.
Creates a new Empirical Distribution from the data samples.
The data samples.
The kernel smoothing or bandwidth to be used in density estimation.
By default, the normal distribution approximation will be used.
Creates a new Empirical Distribution from the data samples.
The data samples.
The fractional weights to use for the samples.
The weights must sum up to one.
Creates a new Empirical Distribution from the data samples.
The data samples.
The number of repetition counts for each sample.
Creates a new Empirical Distribution from the data samples.
The data samples.
The fractional weights to use for the samples.
The weights must sum up to one.
The kernel smoothing or bandwidth to be used in density estimation.
By default, the normal distribution approximation will be used.
Creates a new Empirical Distribution from the data samples.
The data samples.
The kernel smoothing or bandwidth to be used in density estimation.
By default, the normal distribution approximation will be used.
The number of repetition counts for each sample.
Gets the samples giving this empirical distribution.
Gets the fractional weights associated with each sample. Note that
changing values on this array will not result int any effect in
this distribution. The distribution must be computed from scratch
with new values in case new weights needs to be used.
Gets the repetition counts associated with each sample. Note that
changing values on this array will not result int any effect in
this distribution. The distribution must be computed from scratch
with new values in case new weights needs to be used.
Gets the total number of samples in this distribution.
Gets the bandwidth smoothing parameter
used in the kernel density estimation.
Gets the mean for this distribution.
See .
Gets the mode for this distribution.
The distribution's mode value.
Gets the variance for this distribution.
See .
Gets the entropy for this distribution.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
See .
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
See .
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Gets the default estimative of the smoothing parameter.
This method is based on the practical estimation of the bandwidth as
suggested in Wikipedia: http://en.wikipedia.org/wiki/Kernel_density_estimation
The observations for the empirical distribution.
An estimative of the smoothing parameter.
Gets the default estimative of the smoothing parameter.
This method is based on the practical estimation of the bandwidth as
suggested in Wikipedia: http://en.wikipedia.org/wiki/Kernel_density_estimation
The observations for the empirical distribution.
The fractional importance for each sample. Those values must sum up to one.
An estimative of the smoothing parameter.
Gets the default estimative of the smoothing parameter.
This method is based on the practical estimation of the bandwidth as
suggested in Wikipedia: http://en.wikipedia.org/wiki/Kernel_density_estimation
The observations for the empirical distribution.
The number of times each sample should be repeated.
An estimative of the smoothing parameter.
Gets the default estimative of the smoothing parameter.
This method is based on the practical estimation of the bandwidth as
suggested in Wikipedia: http://en.wikipedia.org/wiki/Kernel_density_estimation
The observations for the empirical distribution.
The fractional importance for each sample. Those values must sum up to one.
The number of times each sample should be repeated.
An estimative of the smoothing parameter.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
The random number generator to use as a source of randomness.
Default is to use .
A random observations drawn from this distribution.
F (Fisher-Snedecor) distribution.
In probability theory and statistics, the F-distribution is a continuous
probability distribution. It is also known as Snedecor's F distribution
or the Fisher-Snedecor distribution (after R.A. Fisher and George W. Snedecor).
The F-distribution arises frequently as the null distribution of a test statistic,
most notably in the analysis of variance; see .
References:
-
Wikipedia, The Free Encyclopedia. F-distribution. Available on:
http://en.wikipedia.org/wiki/F-distribution
The following example shows how to construct a Fisher-Snedecor's F-distribution
with 8 and 5 degrees of freedom, respectively.
// Create a Fisher-Snedecor's F distribution with 8 and 5 d.f.
FDistribution F = new FDistribution(degrees1: 8, degrees2: 5);
// Common measures
double mean = F.Mean; // 1.6666666666666667
double median = F.Median; // 1.0545096252132447
double var = F.Variance; // 7.6388888888888893
// Cumulative distribution functions
double cdf = F.DistributionFunction(x: 0.27); // 0.049463408057268315
double ccdf = F.ComplementaryDistributionFunction(x: 0.27); // 0.95053659194273166
double icdf = F.InverseDistributionFunction(p: cdf); // 0.27
// Probability density functions
double pdf = F.ProbabilityDensityFunction(x: 0.27); // 0.45120469723580559
double lpdf = F.LogProbabilityDensityFunction(x: 0.27); // -0.79583416831212883
// Hazard (failure rate) functions
double hf = F.HazardFunction(x: 0.27); // 0.47468419528555084
double chf = F.CumulativeHazardFunction(x: 0.27); // 0.050728620222091653
// String representation
string str = F.ToString(CultureInfo.InvariantCulture); // F(x; df1 = 8, df2 = 5)
Constructs a F-distribution with
the given degrees of freedom.
Constructs a F-distribution with
the given degrees of freedom.
The first degree of freedom. Default is 1.
The second degree of freedom. Default is 1.
Gets the first degree of freedom.
Gets the second degree of freedom.
Gets the mean for this distribution.
Gets the variance for this distribution.
Gets the mode for this distribution.
The distribution's mode value.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the entropy for this distribution.
Gets the cumulative distribution function (cdf) for
the F-distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
The F-distribution CDF is computed in terms of the
Incomplete Beta function Ix(a,b) as CDF(x) = Iu(d1/2, d2/2) in which
u is given as u = (d1 * x) / (d1 * x + d2).
Gets the complementary cumulative distribution
function evaluated at point x.
The F-distribution complementary CDF is computed in terms of the
Incomplete Beta function Ix(a,b) as CDFc(x) = Iu(d2/2, d1/2) in which
u is given as u = (d2 * x) / (d2 * x + d1).
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
The Inverse Cumulative Distribution Function (ICDF) specifies, for
a given probability, the value which the random variable will be at,
or below, with that probability.
Gets the probability density function (pdf) for
the F-distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Not available.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
A random observations drawn from this distribution.
Generates a random vector of observations from the
F-distribution with the given parameters.
The first degree of freedom.
The second degree of freedom.
The number of samples to generate.
An array of double values sampled from the specified F-distribution.
Generates a random vector of observations from the
F-distribution with the given parameters.
The first degree of freedom.
The second degree of freedom.
The number of samples to generate.
The location where to store the samples.
An array of double values sampled from the specified F-distribution.
Generates a random observation from the
F-distribution with the given parameters.
The first degree of freedom.
The second degree of freedom.
A random double value sampled from the specified F-distribution.
Generates a random vector of observations from the
F-distribution with the given parameters.
The first degree of freedom.
The second degree of freedom.
The number of samples to generate.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified F-distribution.
Generates a random vector of observations from the
F-distribution with the given parameters.
The first degree of freedom.
The second degree of freedom.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified F-distribution.
Generates a random observation from the
F-distribution with the given parameters.
The first degree of freedom.
The second degree of freedom.
The random number generator to use as a source of randomness.
Default is to use .
A random double value sampled from the specified F-distribution.
Returns a that represents this instance.
A that represents this instance.
Outcome status for survival methods. A sample can
enter the experiment, exit the experiment while still
alive or exit the experiment due to failure.
Observation started. The observation was left censored before
the current time and has now entered the experiment. This is
equivalent to R's censoring code -1.
Failure happened. This is equivalent to R's censoring code 1.
The sample was right-censored. This is equivalent to R's censoring code 0.
Estimators for estimating parameters of Hazard distributions.
Breslow-Nelson-Aalen estimator (default).
Kaplan-Meier estimator.
Methods for handling ties in hazard/survival estimation algorithms.
Efron's method for ties (default).
Breslow's method for ties.
Estimators for Survival distribution functions.
Fleming-Harrington estimator (default).
Kaplan-Meier estimator.
Empirical Hazard Distribution.
The Empirical Hazard (or Survival) Distribution can be used as an
estimative of the true Survival function for a dataset which does
not relies on distribution or model assumptions about the data.
The most direct use for this class is in Survival Analysis, such as when
using or creating
Cox's Proportional Hazards models.
// references
http://www.statsdirect.com/help/default.htm#survival_analysis/kaplan_meier.htm
The following example shows how to construct an empirical hazards
function from a set of hazard values at the given time instants.
// Consider the following observations, occurring at the given time steps
double[] times = { 11, 10, 9, 8, 6, 5, 4, 2 };
double[] values = { 0.22, 0.67, 1.00, 0.18, 1.00, 1.00, 1.00, 0.55 };
// Create a new empirical distribution function given the observations and event times
EmpiricalHazardDistribution distribution = new EmpiricalHazardDistribution(times, values);
// Common measures
double mean = distribution.Mean; // 2.1994135014183138
double median = distribution.Median; // 3.9999999151458066
double var = distribution.Variance; // 4.2044065839577112
// Cumulative distribution functions
double cdf = distribution.DistributionFunction(x: 4.2); // 0.7877520261732569
double ccdf = distribution.ComplementaryDistributionFunction(x: 4.2); // 0.21224797382674304
double icdf = distribution.InverseDistributionFunction(p: cdf); // 4.3304819115496436
// Probability density functions
double pdf = distribution.ProbabilityDensityFunction(x: 4.2); // 0.21224797382674304
double lpdf = distribution.LogProbabilityDensityFunction(x: 4.2); // -1.55
// Hazard (failure rate) functions
double hf = distribution.HazardFunction(x: 4.2); // 1.0
double chf = distribution.CumulativeHazardFunction(x: 4.2); // 1.55
// String representation
string str = distribution.ToString(); // H(x; v, t)
Gets the time steps of the hazard density values.
Gets the hazard rate values at each time step.
Gets the survival values at each time step.
Gets the survival function estimator being used in this distribution.
Initializes a new instance of the class.
Initializes a new instance of the class.
The time steps.
The hazard rates at the time steps.
Initializes a new instance of the class.
The time steps.
The hazard rates at the time steps.
The survival function estimator to be used. Default is
Initializes a new instance of the class.
The survival function estimator to be used. Default is
Gets the mean for this distribution.
The distribution's mean value.
Gets the variance for this distribution.
The distribution's variance.
This method is not supported.
This method is not supported.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
A single point in the distribution range.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF. In the Empirical Hazard Distribution, this function
is computed using the Fleming-Harrington estimator.
Gets the cumulative hazard function for this
distribution evaluated at point x.
A single point in the distribution range.
The cumulative hazard function H(x)
evaluated at x in the current distribution.
Gets the hazard function, also known as the failure rate or
the conditional failure density function for this distribution
evaluated at point x.
A single point in the distribution range.
The conditional failure density function h(x)
evaluated at x in the current distribution.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring in the current distribution.
In the Empirical Hazard Distribution, the PDF is defined
as the product of the hazard function h(x) and survival
function S(x), as PDF(x) = h(x) * S(x).
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Sorts time-censored events considering their time of occurrence and the type of event.
Events are first sorted in decreased order of occurrence, and then with failures coming
before censoring.
The time of occurrence for the event.
The outcome at the time of event (failure or censored).
The indices of the new sorting.
Sorts time-censored events considering their time of occurrence and the type of event.
Events are first sorted in decreased order of occurrence, and then with failures coming
before censoring.
The time of occurrence for the event.
The outcome at the time of event (failure or censored).
The input vector associated with the event.
The indices of the new sorting.
Sorts time-censored events considering their time of occurrence and the type of event.
Events are first sorted in decreased order of occurrence, and then with failures coming
before censoring.
The time of occurrence for the event.
The outcome at the time of event (failure or censored).
The weights associated with each event.
The indices of the new sorting.
Returns a that represents this instance.
The format.
The format provider.
A that represents this instance.
Estimates an Empirical Hazards distribution considering event times and the outcome of the
observed sample at the time of event, plus additional parameters for the hazard estimation.
The time of occurrence for the event.
The weights associated with each event.
The hazard estimator to use. Default is .
The survival estimator to use. Default is .
The method for handling event ties. Default is .
The estimated from the given data.
Estimates an Empirical Hazards distribution considering event times and the outcome of the
observed sample at the time of event, plus additional parameters for the hazard estimation.
The time of occurrence for the event.
The outcome at the time of event (failure or censored).
The weights associated with each event.
The hazard estimator to use. Default is .
The survival estimator to use. Default is .
The method for handling event ties. Default is .
The estimated from the given data.
Estimates an Empirical Hazards distribution considering event times and the outcome of the
observed sample at the time of event, plus additional parameters for the hazard estimation.
The time of occurrence for the event.
The outcome at the time of event (failure or censored).
The weights associated with each event.
The hazard estimator to use. Default is .
The survival estimator to use. Default is .
The method for handling event ties. Default is .
The estimated from the given data.
Gompertz distribution.
The Gompertz distribution is a continuous probability distribution. The
Gompertz distribution is often applied to describe the distribution of
adult lifespans by demographers and actuaries. Related fields of science
such as biology and gerontology also considered the Gompertz distribution
for the analysis of survival. More recently, computer scientists have also
started to model the failure rates of computer codes by the Gompertz
distribution. In marketing science, it has been used as an individual-level
model of customer lifetime.
References:
-
Wikipedia, The Free Encyclopedia. Gompertz distribution. Available on:
http://en.wikipedia.org/wiki/Gompertz_distribution
The following example shows how to construct a Gompertz
distribution with η = 4.2 and b = 1.1.
// Create a new Gompertz distribution with η = 4.2 and b = 1.1
GompertzDistribution dist = new GompertzDistribution(eta: 4.2, b: 1.1);
// Common measures
double median = dist.Median; // 0.13886469671401389
// Cumulative distribution functions
double cdf = dist.DistributionFunction(x: 0.27); // 0.76599768199799145
double ccdf = dist.ComplementaryDistributionFunction(x: 0.27); // 0.23400231800200855
double icdf = dist.InverseDistributionFunction(p: cdf); // 0.26999999999766749
// Probability density functions
double pdf = dist.ProbabilityDensityFunction(x: 0.27); // 1.4549484164912097
double lpdf = dist.LogProbabilityDensityFunction(x: 0.27); // 0.37497044741163688
// Hazard (failure rate) functions
double hf = dist.HazardFunction(x: 0.27); // 6.2176666834502088
double chf = dist.CumulativeHazardFunction(x: 0.27); // 1.4524242576820101
// String representation
string str = dist.ToString(CultureInfo.InvariantCulture); // "Gompertz(x; η = 4.2, b = 1.1)"
Initializes a new instance of the class.
The shape parameter η.
The scale parameter b.
Not supported.
Not supported.
Gets the mode for this distribution.
The distribution's mode value.
Gets the median for this distribution.
The distribution's median value.
Not supported.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Generalized Pareto distribution (three parameters).
In statistics, the generalized Pareto distribution (GPD) is a family of
continuous probability distributions. It is often used to model the tails
of another distribution. It is specified by three parameters: location μ,
scale σ, and shape ξ. Sometimes it is specified by only scale and shape
and sometimes only by its shape parameter. Some references give the shape
parameter as κ = − ξ.
References:
-
Wikipedia, The Free Encyclopedia. Generalized Pareto distribution.
Available from: https://en.wikipedia.org/wiki/Generalized_Pareto_distribution
-
Wikipedia, The Free Encyclopedia. Mixture density. Available on:
http://en.wikipedia.org/wiki/Mixture_density
The type of the univariate component distributions.
// Create a new mixture containing two Normal distributions
Mixture<NormalDistribution> mix = new Mixture<NormalDistribution>(
new NormalDistribution(2, 1), new NormalDistribution(5, 1));
// Common measures
double mean = mix.Mean; // 3.5
double median = mix.Median; // 3.4999998506015895
double var = mix.Variance; // 3.25
// Cumulative distribution functions
double cdf = mix.DistributionFunction(x: 4.2); // 0.59897597553494908
double ccdf = mix.ComplementaryDistributionFunction(x: 4.2); // 0.40102402446505092
// Probability mass functions
double pmf1 = mix.ProbabilityDensityFunction(x: 1.2); // 0.14499174984363708
double pmf2 = mix.ProbabilityDensityFunction(x: 2.3); // 0.19590437513747333
double pmf3 = mix.ProbabilityDensityFunction(x: 3.7); // 0.13270883471234715
double lpmf = mix.LogProbabilityDensityFunction(x: 4.2); // -1.8165661905848629
// Quantile function
double icdf1 = mix.InverseDistributionFunction(p: 0.17); // 1.5866611690305095
double icdf2 = mix.InverseDistributionFunction(p: 0.46); // 3.1968506765456883
double icdf3 = mix.InverseDistributionFunction(p: 0.87); // 5.6437596300843076
// Hazard (failure rate) functions
double hf = mix.HazardFunction(x: 4.2); // 0.40541978256972522
double chf = mix.CumulativeHazardFunction(x: 4.2); // 0.91373394208601633
// String representation:
// Mixture(x; 0.5 * N(x; μ = 5, σ² = 1) + 0.5 * N(x; μ = 5, σ² = 1))
string str = mix.ToString(CultureInfo.InvariantCulture);
The following example shows how to estimate (fit) a Mixture of Normal distributions
from weighted data:
// Randomly initialize some mixture components
NormalDistribution[] components = new NormalDistribution[2];
components[0] = new NormalDistribution(2, 1);
components[1] = new NormalDistribution(5, 1);
// Create an initial mixture
var mixture = new Mixture<NormalDistribution>(components);
// Now, suppose we have a weighted data
// set. Those will be the input points:
double[] points = { 0, 3, 1, 7, 3, 5, 1, 2, -1, 2, 7, 6, 8, 6 }; // (14 points)
// And those are their respective unnormalized weights:
double[] weights = { 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 3, 1, 1 }; // (14 weights)
// Let's normalize the weights so they sum up to one:
weights = weights.Divide(weights.Sum());
// Now we can fit our model to the data:
mixture.Fit(points, weights); // done!
// Our model will be:
double mean1 = mixture.Components[0].Mean; // 1.41126
double mean2 = mixture.Components[1].Mean; // 6.53301
// With mixture weights
double pi1 = mixture.Coefficients[0]; // 0.51408
double pi2 = mixture.Coefficients[0]; // 0.48591
// If we need the GaussianMixtureModel functionality, we can
// use the estimated mixture to initialize a new model:
GaussianMixtureModel gmm = new GaussianMixtureModel(mixture);
mean1 = gmm.Gaussians[0].Mean[0]; // 1.41126 (same)
mean2 = gmm.Gaussians[1].Mean[0]; // 6.53301 (same)
p1 = gmm.Gaussians[0].Proportion; // 0.51408 (same)
p2 = gmm.Gaussians[1].Proportion; // 0.48591 (same)
Initializes a new instance of the class.
The mixture distribution components.
Initializes a new instance of the class.
The mixture weight coefficients.
The mixture distribution components.
Gets the mixture components.
Gets the weight coefficients.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the probability density function (pdf) for one of
the component distributions evaluated at point x.
The index of the desired component distribution.
A single point in the distribution range.
The probability of x occurring in the component distribution,
computed as the PDF of the component distribution times its mixture
coefficient.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the log-probability density function (pdf) for one
of the component distributions evaluated at point x.
The index of the desired component distribution.
A single point in the distribution range.
The logarithm of the probability of x occurring in the
component distribution, computed as the PDF of the component
distribution times its mixture coefficient.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the cumulative distribution function (cdf) for one
component of this distribution evaluated at point x.
The component distribution's index.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Computes the log-likelihood of the distribution
for a given set of observations.
Computes the log-likelihood of the distribution
for a given set of observations.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Gets the mean for this distribution.
Gets the variance for this distribution.
References: Lidija Trailovic and Lucy Y. Pao, Variance Estimation and
Ranking of Gaussian Mixture Distributions in Target Tracking
Applications, Department of Electrical and Computer Engineering
This method is not supported.
This method is not supported.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Estimates a new mixture model from a given set of observations.
A set of observations.
The initial components of the mixture model.
Returns a new Mixture fitted to the given observations.
Estimates a new mixture model from a given set of observations.
A set of observations.
The initial mixture coefficients.
The initial components of the mixture model.
Returns a new Mixture fitted to the given observations.
Estimates a new mixture model from a given set of observations.
A set of observations.
The initial mixture coefficients.
The convergence threshold for the Expectation-Maximization estimation.
The initial components of the mixture model.
Returns a new Mixture fitted to the given observations.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
A random observations drawn from this distribution.
Returns a that represents this instance.
A that represents this instance.
Univariate general discrete distribution, also referred as the
Categorical distribution.
An univariate categorical distribution is a statistical distribution
whose variables can take on only discrete values. Each discrete value
defined within the interval of the distribution has an associated
probability value indicating its frequency of occurrence.
The discrete uniform distribution is a special case of a generic
discrete distribution whose probability values are constant.
// Create a Categorical distribution for 3 symbols, in which
// the first and second symbol have 25% chance of appearing,
// and the third symbol has 50% chance of appearing.
// 1st 2nd 3rd
double[] probabilities = { 0.25, 0.25, 0.50 };
// Create the categorical with the given probabilities
var dist = new GeneralDiscreteDistribution(probabilities);
// Common measures
double mean = dist.Mean; // 1.25
double median = dist.Median; // 1.00
double var = dist.Variance; // 0.6875
// Cumulative distribution functions
double cdf = dist.DistributionFunction(k: 2); // 1.0
double ccdf = dist.ComplementaryDistributionFunction(k: 2); // 0.0
// Probability mass functions
double pdf1 = dist.ProbabilityMassFunction(k: 0); // 0.25
double pdf2 = dist.ProbabilityMassFunction(k: 1); // 0.25
double pdf3 = dist.ProbabilityMassFunction(k: 2); // 0.50
double lpdf = dist.LogProbabilityMassFunction(k: 2); // -0.69314718055994529
// Quantile function
int icdf1 = dist.InverseDistributionFunction(p: 0.17); // 0
int icdf2 = dist.InverseDistributionFunction(p: 0.39); // 1
int icdf3 = dist.InverseDistributionFunction(p: 0.56); // 2
// Hazard (failure rate) functions
double hf = dist.HazardFunction(x: 0); // 0.33333333333333331
double chf = dist.CumulativeHazardFunction(x: 0); // 0.2876820724517809
// String representation
string str = dist.ToString(CultureInfo.InvariantCulture); // "Categorical(x; p = { 0.25, 0.25, 0.5 })"
Constructs a new generic discrete distribution.
The frequency of occurrence for each integer value in the
distribution. The distribution is assumed to begin in the
interval defined by start up to size of this vector.
True if the distribution should be represented using logarithms; false otherwise.
Constructs a new generic discrete distribution.
The integer value where the distribution starts, also
known as the offset value. Default value is 0.
The frequency of occurrence for each integer value in the
distribution. The distribution is assumed to begin in the
interval defined by start up to size of this vector.
Constructs a new uniform discrete distribution.
The integer value where the distribution starts, also
known as the offset value. Default value is 0.
The number of discrete values within the distribution.
The distribution is assumed to belong to the interval
[start, start + symbols].
True if the distribution should be represented using logarithms; false otherwise.
Constructs a new generic discrete distribution.
The frequency of occurrence for each integer value in the
distribution. The distribution is assumed to begin in the
interval defined by start up to size of this vector.
Constructs a new uniform discrete distribution.
The number of discrete values within the distribution.
The distribution is assumed to belong to the interval
[start, start + symbols].
True if the distribution should be represented using logarithms; false otherwise.
Constructs a new uniform discrete distribution.
The integer value where the distribution starts, also
known as a. Default value is 0.
The integer value where the distribution ends, also
known as b.
Gets the probability value associated with the symbol .
The symbol's index.
The probability of the given symbol.
Gets the integer value where the
discrete distribution starts.
Gets the integer value where the
discrete distribution ends.
Gets the number of symbols in the distribution.
Gets the probabilities associated with each discrete variable value.
Note: if the frequencies in this property are manually changed, the
rest of the class properties (Mode, Mean, ...) will not be automatically
updated to reflect the actual inserted values.
Gets the mean for this distribution.
Gets the variance for this distribution.
Gets the mode for this distribution.
Gets the entropy for this distribution.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The Probability Density Function (PDF) describes the
probability that a given value k will occur.
The probability of k occurring
in the current distribution.
Gets the probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Probability Mass Function (PMF) describes the
probability that a given value x will occur.
The probability of x occurring
in the current distribution.
Gets the log-probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of k
occurring in the current distribution.
The Probability Mass Function (PMF) describes the
probability that a given value x will occur.
Generates a random observation from the current distribution.
The random number generator to use as a source of randomness.
Default is to use .
A random observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a random sample within the given symbol probabilities.
The probabilities for the discrete symbols.
The number of samples to generate.
A random sample within the given probabilities.
Returns a random sample within the given symbol probabilities.
The probabilities for the discrete symbols.
The number of samples to generate.
The random number generator to use as a source of randomness.
Default is to use .
A random sample within the given probabilities.
Returns a random sample within the given symbol probabilities.
The probabilities for the discrete symbols.
The number of samples to generate.
The location where to store the samples.
Pass true if the vector
contains log-probabilities instead of standard probabilities.
A random sample within the given probabilities.
Returns a random sample within the given symbol probabilities.
The probabilities for the discrete symbols.
The number of samples to generate.
The location where to store the samples.
Pass true if the vector
contains log-probabilities instead of standard probabilities.
The random number generator to use as a source of randomness.
Default is to use .
A random sample within the given probabilities.
Returns a random symbol within the given symbol probabilities.
The probabilities for the discrete symbols.
Pass true if the vector
contains log-probabilities instead of standard probabilities.
A random symbol within the given probabilities.
Returns a random symbol within the given symbol probabilities.
The probabilities for the discrete symbols.
Pass true if the vector
contains log-probabilities instead of standard probabilities.
The random number generator to use as a source of randomness.
Default is to use .
A random symbol within the given probabilities.
Returns a that represents this instance.
A that represents this instance.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Creates general discrete distributions given a matrix of symbol probabilities.
Creates general discrete distributions given a matrix of symbol probabilities.
Abstract class for univariate continuous probability Distributions.
A probability distribution identifies either the probability of each value of an
unidentified random variable (when the variable is discrete), or the probability
of the value falling within a particular interval (when the variable is continuous).
The probability distribution describes the range of possible values that a random
variable can attain and the probability that the value of the random variable is
within any (measurable) subset of that range.
The function describing the probability that a given value will occur is called
the probability function (or probability density function, abbreviated PDF), and
the function describing the cumulative probability that a given value or any value
smaller than it will occur is called the distribution function (or cumulative
distribution function, abbreviated CDF).
References:
-
Wikipedia, The Free Encyclopedia. Probability distribution. Available on:
http://en.wikipedia.org/wiki/Probability_distribution
-
Weisstein, Eric W. "Statistical Distribution." From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/StatisticalDistribution.html
Constructs a new UnivariateDistribution class.
Gets the mean for this distribution.
The distribution's mean value.
Gets the variance for this distribution.
The distribution's variance.
Gets the entropy for this distribution.
The distribution's entropy.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the mode for this distribution.
The distribution's mode value.
Gets the Quartiles for this distribution.
A object containing the first quartile
(Q1) as its minimum value, and the third quartile (Q2) as the maximum.
Gets the distribution range within a given percentile.
If 0.25 is passed as the argument,
this function returns the same as the function.
The percentile at which the distribution ranges will be returned.
A object containing the minimum value
for the distribution value, and the third quartile (Q2) as the maximum.
Gets the median for this distribution.
The distribution's median value.
Gets the Standard Deviation (the square root of
the variance) for the current distribution.
The distribution's standard deviation.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
The probability of x occurring
in the current distribution.
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
The probability of x occurring
in the current distribution.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
The logarithm of the probability of x
occurring in the current distribution.
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
The logarithm of the probability of x
occurring in the current distribution.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the cumulative distribution function (cdf) for this
distribution in the semi-closed interval (a; b] given as
P(a < X ≤ b).
The start of the semi-closed interval (a; b].
The end of the semi-closed interval (a; b].
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
A single point in the distribution range.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF.
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
A single point in the distribution range.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF.
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
The Inverse Cumulative Distribution Function (ICDF) specifies, for
a given probability, the value which the random variable will be at,
or below, with that probability.
A probability value between 0 and 1.
A sample which could original the given probability
value when applied in the .
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
The Inverse Cumulative Distribution Function (ICDF) specifies, for
a given probability, the value which the random variable will be at,
or below, with that probability.
A probability value between 0 and 1.
A sample which could original the given probability
value when applied in the .
Gets the first derivative of the
inverse distribution function (icdf) for this distribution evaluated
at probability p.
A probability value between 0 and 1.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
The probability of x occurring
in the current distribution.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
The probability of x occurring
in the current distribution.
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
The logarithm of the probability of x
occurring in the current distribution.
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
The logarithm of the probability of x
occurring in the current distribution.
Gets the hazard function, also known as the failure rate or
the conditional failure density function for this distribution
evaluated at point x.
The hazard function is the ratio of the probability
density function f(x) to the survival function, S(x).
A single point in the distribution range.
The conditional failure density function h(x)
evaluated at x in the current distribution.
Gets the cumulative hazard function for this
distribution evaluated at point x.
A single point in the distribution range.
The cumulative hazard function H(x)
evaluated at x in the current distribution.
Gets the log of the cumulative hazard function for this
distribution evaluated at point x.
A single point in the distribution range.
The cumulative hazard function H(x)
evaluated at x in the current distribution.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
The following example shows how to fit a using
the method. However, any other kind of
distribution could be fit in the exactly same way. Please consider the code below
as an example only:
-
Wikipedia, The Free Encyclopedia. Normal distribution. Available on:
https://en.wikipedia.org/wiki/Normal_distribution
This examples shows how to create a Normal distribution,
compute some of its properties and generate a number of
random samples from it.
// Create a normal distribution with mean 2 and sigma 3
var normal = new NormalDistribution(mean: 2, stdDev: 3);
// In a normal distribution, the median and
// the mode coincide with the mean, so
double mean = normal.Mean; // 2
double mode = normal.Mode; // 2
double median = normal.Median; // 2
// The variance is the square of the standard deviation
double variance = normal.Variance; // 3² = 9
// Let's check what is the cumulative probability of
// a value less than 3 occurring in this distribution:
double cdf = normal.DistributionFunction(3); // 0.63055
// Finally, let's generate 1000 samples from this distribution
// and check if they have the specified mean and standard devs
double[] samples = normal.Generate(1000);
double sampleMean = samples.Mean(); // 1.92
double sampleDev = samples.StandardDeviation(); // 3.00
This example further demonstrates how to compute
derived measures from a Normal distribution:
var normal = new NormalDistribution(mean: 4, stdDev: 4.2);
double mean = normal.Mean; // 4.0
double median = normal.Median; // 4.0
double mode = normal.Mode; // 4.0
double var = normal.Variance; // 17.64
double cdf = normal.DistributionFunction(x: 1.4); // 0.26794249453351904
double pdf = normal.ProbabilityDensityFunction(x: 1.4); // 0.078423391448155175
double lpdf = normal.LogProbabilityDensityFunction(x: 1.4); // -2.5456330358182586
double ccdf = normal.ComplementaryDistributionFunction(x: 1.4); // 0.732057505466481
double icdf = normal.InverseDistributionFunction(p: cdf); // 1.4
double hf = normal.HazardFunction(x: 1.4); // 0.10712736480747137
double chf = normal.CumulativeHazardFunction(x: 1.4); // 0.31189620872601354
string str = normal.ToString(CultureInfo.InvariantCulture); // N(x; μ = 4, σ² = 17.64)
Constructs a Normal (Gaussian) distribution
with zero mean and unit standard deviation.
Constructs a Normal (Gaussian) distribution
with given mean and unit standard deviation.
The distribution's mean value μ (mu).
Constructs a Normal (Gaussian) distribution
with given mean and standard deviation.
The distribution's mean value μ (mu).
The distribution's standard deviation σ (sigma).
Gets the Mean value μ (mu) for this Normal distribution.
Gets the median for this distribution.
The normal distribution's median value
equals its value μ.
The distribution's median value.
Gets the Variance σ² (sigma-squared), which is the square
of the standard deviation σ for this Normal distribution.
Gets the Standard Deviation σ (sigma), which is the
square root of the variance for this Normal distribution.
Gets the mode for this distribution.
The normal distribution's mode value
equals its value μ.
The distribution's mode value.
Gets the skewness for this distribution. In
the Normal distribution, this is always 0.
Gets the excess kurtosis for this distribution.
In the Normal distribution, this is always 0.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the Entropy for this Normal distribution.
Gets the cumulative distribution function (cdf) for
the this Normal distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
The calculation is computed through the relationship to the error function
as erfc(-z/sqrt(2)) / 2.
References:
-
Weisstein, Eric W. "Normal Distribution." From MathWorld--A Wolfram Web Resource.
Available on: http://mathworld.wolfram.com/NormalDistribution.html
-
Wikipedia, The Free Encyclopedia. Normal distribution. Available on:
http://en.wikipedia.org/wiki/Normal_distribution#Cumulative_distribution_function
See .
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
A single point in the distribution range.
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
The Inverse Cumulative Distribution Function (ICDF) specifies, for
a given probability, the value which the random variable will be at,
or below, with that probability.
The Normal distribution's ICDF is defined in terms of the
standard normal inverse cumulative
distribution function I as ICDF(p) = μ + σ * I(p).
See .
Gets the probability density function (pdf) for
the Normal distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
The Normal distribution's PDF is defined as
PDF(x) = c * exp((x - μ / σ)²/2).
See .
Gets the probability density function (pdf) for
the Normal distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
See .
Gets the Z-Score for a given value.
Gets the Standard Gaussian Distribution, with zero mean and unit variance.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Estimates a new Normal distribution from a given set of observations.
Estimates a new Normal distribution from a given set of observations.
Estimates a new Normal distribution from a given set of observations.
Converts this univariate distribution into a
1-dimensional multivariate distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
The random number generator to use as a source of randomness.
Default is to use .
A observation drawn from this distribution.
Generates a single random observation from the
Normal distribution with the given parameters.
The mean value μ (mu).
The standard deviation σ (sigma).
An double value sampled from the specified Normal distribution.
Generates a single random observation from the
Normal distribution with the given parameters.
The mean value μ (mu).
The standard deviation σ (sigma).
The random number generator to use as a source of randomness.
Default is to use .
An double value sampled from the specified Normal distribution.
Generates a random vector of observations from the
Normal distribution with the given parameters.
The mean value μ (mu).
The standard deviation σ (sigma).
The number of samples to generate.
An array of double values sampled from the specified Normal distribution.
Generates a random vector of observations from the
Normal distribution with the given parameters.
The mean value μ (mu).
The standard deviation σ (sigma).
The number of samples to generate.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified Normal distribution.
Generates a random vector of observations from the
Normal distribution with the given parameters.
The mean value μ (mu).
The standard deviation σ (sigma).
The number of samples to generate.
The location where to store the samples.
An array of double values sampled from the specified Normal distribution.
Generates a random vector of observations from the
Normal distribution with the given parameters.
The mean value μ (mu).
The standard deviation σ (sigma).
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified Normal distribution.
Generates a random vector of observations from the standard
Normal distribution (zero mean and unit standard deviation).
The number of samples to generate.
The location where to store the samples.
An array of double values sampled from the specified Normal distribution.
Generates a random vector of observations from the standard
Normal distribution (zero mean and unit standard deviation).
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified Normal distribution.
Generates a random value from a standard Normal
distribution (zero mean and unit standard deviation).
Generates a random value from a standard Normal
distribution (zero mean and unit standard deviation).
Poisson probability distribution.
The Poisson distribution is a discrete probability distribution that
expresses the probability of a number of events occurring in a fixed
period of time if these events occur with a known average rate and
independently of the time since the last event.
References:
-
Wikipedia, The Free Encyclopedia. Poisson distribution. Available on:
http://en.wikipedia.org/wiki/Poisson_distribution
The following example shows how to instantiate a new Poisson distribution
with a given rate λ and how to compute its measures and associated functions.
// Create a new Poisson distribution with
var dist = new PoissonDistribution(lambda: 4.2);
// Common measures
double mean = dist.Mean; // 4.2
double median = dist.Median; // 4.0
double var = dist.Variance; // 4.2
// Cumulative distribution functions
double cdf1 = dist.DistributionFunction(k: 2); // 0.21023798702309743
double cdf2 = dist.DistributionFunction(k: 4); // 0.58982702131057763
double cdf3 = dist.DistributionFunction(k: 7); // 0.93605666027257894
double ccdf = dist.ComplementaryDistributionFunction(k: 2); // 0.78976201297690252
// Probability mass functions
double pmf1 = dist.ProbabilityMassFunction(k: 4); // 0.19442365170822165
double pmf2 = dist.ProbabilityMassFunction(k: 5); // 0.1633158674349062
double pmf3 = dist.ProbabilityMassFunction(k: 6); // 0.11432110720443435
double lpmf = dist.LogProbabilityMassFunction(k: 2); // -2.0229781299813
// Quantile function
int icdf1 = dist.InverseDistributionFunction(p: cdf1); // 2
int icdf2 = dist.InverseDistributionFunction(p: cdf2); // 4
int icdf3 = dist.InverseDistributionFunction(p: cdf3); // 7
// Hazard (failure rate) functions
double hf = dist.HazardFunction(x: 4); // 0.47400404660843515
double chf = dist.CumulativeHazardFunction(x: 4); // 0.89117630901575073
// String representation
string str = dist.ToString(CultureInfo.InvariantCulture); // "Poisson(x; λ = 4.2)"
This example shows hows to call the distribution function
to compute different types of probabilities.
// Create a new Poisson distribution
var dist = new PoissonDistribution(lambda: 4.2);
// P(X = 1) = 0.0629814226460064
double equal = dist.ProbabilityMassFunction(k: 1);
// P(X < 1) = 0.0149955768204777
double less = dist.DistributionFunction(k: 1, inclusive: false);
// P(X ≤ 1) = 0.0779769994664841
double lessThanOrEqual = dist.DistributionFunction(k: 1, inclusive: true);
// P(X > 1) = 0.922023000533516
double greater = dist.ComplementaryDistributionFunction(k: 1);
// P(X ≥ 1) = 0.985004423179522
double greaterThanOrEqual = dist.ComplementaryDistributionFunction(k: 1, inclusive: true);
Creates a new Poisson distribution with λ = 1.
Creates a new Poisson distribution with the given λ (lambda).
The Poisson's λ (lambda) parameter. Default is 1.
Gets the Poisson's parameter λ (lambda).
Gets the mean for this distribution.
Gets the variance for this distribution.
Gets the entropy for this distribution.
A closed form expression for the entropy of a Poisson
distribution is unknown. This property returns an approximation
for large lambda.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point k.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the probability mass function (pmf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Probability Mass Function (PMF) describes the
probability that a given value k will occur.
The probability of x occurring
in the current distribution.
Gets the log-probability mass function (pmf) for
this distribution evaluated at point k.
A single point in the distribution range.
The logarithm of the probability of k
occurring in the current distribution.
The Probability Mass Function (PMF) describes the
probability that a given value k will occur.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
A random observations drawn from this distribution.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Gets the standard Poisson distribution,
with lambda (rate) equal to 1.
von-Mises (Circular Normal) distribution.
The von Mises distribution (also known as the circular normal distribution
or Tikhonov distribution) is a continuous probability distribution on the circle.
It may be thought of as a close approximation to the wrapped normal distribution,
which is the circular analogue of the normal distribution.
The wrapped normal distribution describes the distribution of an angle that
is the result of the addition of many small independent angular deviations, such as
target sensing, or grain orientation in a granular material. The von Mises distribution
is more mathematically tractable than the wrapped normal distribution and is the
preferred distribution for many applications.
References:
-
Wikipedia, The Free Encyclopedia. Von-Mises distribution. Available on:
http://en.wikipedia.org/wiki/Von_Mises_distribution
-
Suvrit Sra, "A short note on parameter approximation for von Mises-Fisher distributions:
and a fast implementation of $I_s(x)$". (revision of Apr. 2009). Computational Statistics (2011).
Available on: http://www.kyb.mpg.de/publications/attachments/vmfnote_7045%5B0%5D.pdf
-
Zheng Sun. M.Sc. Comparing measures of fit for circular distributions. Master thesis, 2006.
Available on: https://dspace.library.uvic.ca:8443/bitstream/handle/1828/2698/zhengsun_master_thesis.pdf
// Create a new von-Mises distribution with μ = 0.42 and κ = 1.2
var vonMises = new VonMisesDistribution(mean: 0.42, concentration: 1.2);
// Common measures
double mean = vonMises.Mean; // 0.42
double median = vonMises.Median; // 0.42
double var = vonMises.Variance; // 0.48721760532782921
// Cumulative distribution functions
double cdf = vonMises.DistributionFunction(x: 1.4); // 0.81326928491589345
double ccdf = vonMises.ComplementaryDistributionFunction(x: 1.4); // 0.18673071508410655
double icdf = vonMises.InverseDistributionFunction(p: cdf); // 1.3999999637927665
// Probability density functions
double pdf = vonMises.ProbabilityDensityFunction(x: 1.4); // 0.2228112141141676
double lpdf = vonMises.LogProbabilityDensityFunction(x: 1.4); // -1.5014304395467863
// Hazard (failure rate) functions
double hf = vonMises.HazardFunction(x: 1.4); // 1.1932220899695576
double chf = vonMises.CumulativeHazardFunction(x: 1.4); // 1.6780877262500649
// String representation
string str = vonMises.ToString(CultureInfo.InvariantCulture); // VonMises(x; μ = 0.42, κ = 1.2)
Constructs a von-Mises distribution with zero mean and unit concentration.
Constructs a von-Mises distribution with zero mean.
The concentration value κ (kappa). Default is 1.
Constructs a von-Mises distribution.
The mean value μ (mu). Default is 0.
The concentration value κ (kappa). Default is 1.
Gets the mean value μ (mu) for this distribution.
Gets the median value μ (mu) for this distribution.
Gets the mode value μ (mu) for this distribution.
Gets the concentration κ (kappa) for this distribution.
Gets the variance for this distribution.
The von-Mises Variance is defined in terms of the
Bessel function of the first
kind In(x) as var = 1 - I(1, κ) / I(0, κ)
Gets the entropy for this distribution.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Creates a new circular uniform distribution by creating a
new with zero kappa.
The mean value μ (mu).
A with zero kappa, which
is equivalent to creating an uniform circular distribution.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Returns a that represents this instance.
A that represents this instance.
Estimates a new von-Mises distribution from a given set of angles.
Estimates a new von-Mises distribution from a given set of angles.
Estimates a new von-Mises distribution from a given set of angles.
von-Mises cumulative distribution function.
This method implements the Von-Mises CDF calculation code
as given by Geoffrey Hill on his original FORTRAN code and
shared under the GNU LGPL license.
References:
- Geoffrey Hill, ACM TOMS Algorithm 518,
Incomplete Bessel Function I0: The von Mises Distribution,
ACM Transactions on Mathematical Software, Volume 3, Number 3,
September 1977, pages 279-284.
The point where to calculate the CDF.
The location parameter μ (mu).
The concentration parameter κ (kappa).
The value of the von-Mises CDF at point .
Weibull distribution.
In probability theory and statistics, the Weibull distribution is a
continuous probability distribution. It is named after Waloddi Weibull,
who described it in detail in 1951, although it was first identified by
Fréchet (1927) and first applied by Rosin and Rammler (1933) to describe a
particle size distribution.
The Weibull distribution is related to a number of other probability distributions;
in particular, it interpolates between the
exponential distribution (for k = 1) and the
Rayleigh distribution (when k = 2).
If the quantity x is a "time-to-failure", the Weibull distribution gives a
distribution for which the failure rate is proportional to a power of time.
The shape parameter, k, is that power plus one, and so this parameter can be
interpreted directly as follows:
-
A value of k < 1 indicates that the failure rate decreases over time. This
happens if there is significant "infant mortality", or defective items failing
early and the failure rate decreasing over time as the defective items are
weeded out of the population.
-
A value of k = 1 indicates that the failure rate is constant over time. This
might suggest random external events are causing mortality, or failure.
-
A value of k > 1 indicates that the failure rate increases with time. This
happens if there is an "aging" process, or parts that are more likely to fail
as time goes on.
In the field of materials science, the shape parameter k of a distribution
of strengths is known as the Weibull modulus.
References:
-
Wikipedia, The Free Encyclopedia. Weibull distribution. Available on:
http://en.wikipedia.org/wiki/Weibull_distribution
// Create a new Weibull distribution with λ = 0.42 and k = 1.2
var weilbull = new WeibullDistribution(scale: 0.42, shape: 1.2);
// Common measures
double mean = weilbull.Mean; // 0.39507546046784414
double median = weilbull.Median; // 0.30945951550913292
double var = weilbull.Variance; // 0.10932249666369542
double mode = weilbull.Mode; // 0.094360430821809421
// Cumulative distribution functions
double cdf = weilbull.DistributionFunction(x: 1.4); // 0.98560487188700052
double pdf = weilbull.ProbabilityDensityFunction(x: 1.4); // 0.052326687031379278
double lpdf = weilbull.LogProbabilityDensityFunction(x: 1.4); // -2.9502487697674415
// Probability density functions
double ccdf = weilbull.ComplementaryDistributionFunction(x: 1.4); // 0.22369885565908001
double icdf = weilbull.InverseDistributionFunction(p: cdf); // 1.400000001051205
// Hazard (failure rate) functions
double hf = weilbull.HazardFunction(x: 1.4); // 1.1093328057258516
double chf = weilbull.CumulativeHazardFunction(x: 1.4); // 1.4974545260150962
// String representation
string str = weilbull.ToString(CultureInfo.InvariantCulture); // Weibull(x; λ = 0.42, k = 1.2)
Initializes a new instance of the class.
The scale parameter λ (lambda).
The shape parameter k.
Gets the shape parameter k.
The value for this distribution's shape parameter k.
Gets the scale parameter λ (lambda).
The value for this distribution's scale parameter λ (lambda).
Gets the mean for this distribution.
The distribution's mean value.
Gets the variance for this distribution.
The distribution's variance.
Gets the median for this distribution.
The distribution's median value.
Gets the mode for this distribution.
The distribution's mode value.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the entropy for this distribution.
The distribution's entropy.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The probability of x occurring
in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range.
The logarithm of the probability of x
occurring in the current distribution.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
Gets the hazard function, also known as the failure rate or
the conditional failure density function for this distribution
evaluated at point x.
A single point in the distribution range.
The conditional failure density function h(x)
evaluated at x in the current distribution.
Gets the cumulative hazard function for this
distribution evaluated at point x.
A single point in the distribution range.
The cumulative hazard function H(x)
evaluated at x in the current distribution.
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
A single point in the distribution range.
Gets the inverse of the .
The inverse complementary distribution function is also known as the
inverse survival Function.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Generates a random vector of observations from the current distribution.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
A random vector of observations drawn from this distribution.
Generates a random observation from the current distribution.
The random number generator to use as a source of randomness.
Default is to use .
A random observations drawn from this distribution.
Generates a random vector of observations from the
Weibull distribution with the given parameters.
The scale parameter lambda.
The shape parameter k.
The number of samples to generate.
An array of double values sampled from the specified Weibull distribution.
Generates a random vector of observations from the
Weibull distribution with the given parameters.
The scale parameter lambda.
The shape parameter k.
The number of samples to generate.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified Weibull distribution.
Generates a random vector of observations from the
Weibull distribution with the given parameters.
The scale parameter lambda.
The shape parameter k.
The number of samples to generate.
The location where to store the samples.
An array of double values sampled from the specified Weibull distribution.
Generates a random vector of observations from the
Weibull distribution with the given parameters.
The scale parameter lambda.
The shape parameter k.
The number of samples to generate.
The location where to store the samples.
The random number generator to use as a source of randomness.
Default is to use .
An array of double values sampled from the specified Weibull distribution.
Generates a random observation from the
Weibull distribution with the given parameters.
The scale parameter lambda.
The shape parameter k.
A random double value sampled from the specified Weibull distribution.
Generates a random observation from the
Weibull distribution with the given parameters.
The scale parameter lambda.
The shape parameter k.
The random number generator to use as a source of randomness.
Default is to use .
A random double value sampled from the specified Weibull distribution.
Returns a that represents this instance.
A that represents this instance.
Common interface for univariate probability distributions.
This interface is implemented by both univariate
Discrete Distributions and Continuous
Distributions. However, unlike , this interface
has a generic parameter that allows to define the type of the distribution values (i.e.
).
For Multivariate distributions, see .
Gets the mean value for the distribution.
The distribution's mean.
Gets the variance value for the distribution.
The distribution's variance.
Gets the median value for the distribution.
The distribution's median.
Gets the mode value for the distribution.
The distribution's mode.
Gets entropy of the distribution.
The distribution's entropy.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
The Inverse Cumulative Distribution Function (ICDF) specifies, for
a given probability, the value which the random variable will be at,
or below, with that probability.
A probability value between 0 and 1.
A sample which could original the given probability
value when applied in the .
Gets the hazard function, also known as the failure rate or
the conditional failure density function for this distribution
evaluated at point x.
The hazard function is the ratio of the probability
density function f(x) to the survival function, S(x).
A single point in the distribution range.
The conditional failure density function h(x)
evaluated at x in the current distribution.
Gets the cumulative hazard function for this
distribution evaluated at point x.
A single point in the distribution range.
The cumulative hazard function H(x)
evaluated at x in the current distribution.
Common interface for univariate probability distributions.
This interface is implemented by both univariate
Discrete Distributions and Continuous
Distributions.
For Multivariate distributions, see .
Gets the mean value for the distribution.
The distribution's mean.
Gets the variance value for the distribution.
The distribution's variance.
Gets the median value for the distribution.
The distribution's median.
Gets the mode value for the distribution.
The distribution's mode.
Gets entropy of the distribution.
The distribution's entropy.
Gets the support interval for this distribution.
A containing
the support interval for this distribution.
Gets the Quartiles for this distribution.
A object containing the first quartile
(Q1) as its minimum value, and the third quartile (Q2) as the maximum.
Gets the distribution range within a given percentile.
If 0.25 is passed as the argument,
this function returns the same as the function.
The percentile at which the distribution ranges will be returned.
A object containing the minimum value
for the distribution value, and the third quartile (Q2) as the maximum.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the cumulative distribution function (cdf) for this
distribution in the semi-closed interval (a; b] given as
P(a < X ≤ b).
The start of the semi-closed interval (a; b].
The end of the semi-closed interval (a; b].
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
The probability of x occurring
in the current distribution.
Gets the log-probability density function (pdf)
for this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The logarithm of the probability of x
occurring in the current distribution.
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
The Inverse Cumulative Distribution Function (ICDF) specifies, for
a given probability, the value which the random variable will be at,
or below, with that probability.
A probability value between 0 and 1.
A sample which could original the given probability
value when applied in the .
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
A single point in the distribution range.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF.
Gets the hazard function, also known as the failure rate or
the conditional failure density function for this distribution
evaluated at point x.
The hazard function is the ratio of the probability
density function f(x) to the survival function, S(x).
A single point in the distribution range.
The conditional failure density function h(x)
evaluated at x in the current distribution.
Gets the cumulative hazard function for this
distribution evaluated at point x.
A single point in the distribution range.
The cumulative hazard function H(x)
evaluated at x in the current distribution.
Gets the cumulative hazard function for this
distribution evaluated at point x.
A single point in the distribution range.
The cumulative hazard function H(x)
evaluated at x in the current distribution.
Gets the first derivative of the
inverse distribution function (icdf) for this distribution evaluated
at probability p.
A probability value between 0 and 1.
Common interface for mixture distributions.
The type of the mixture distribution, if either univariate or multivariate.
Gets the mixture coefficients (component weights).
Gets the mixture components.
Base class for statistical distribution implementations.
Returns a that represents this instance.
A that represents this instance.
Returns a that represents this instance.
A that represents this instance.
Returns a that represents this instance.
A that represents this instance.
Returns a that represents this instance.
The format.
The format provider.
A that represents this instance.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Common interface for probability distributions.
This interface is implemented by all probability distributions in the framework, including
s and s. This
includes
,
,
, and
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
The probability of x occurring
in the current distribution.
Gets the log-probability density function (pdf)
for this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The logarithm of the probability of x
occurring in the current distribution.
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Fits the underlying distribution to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Although both double[] and double[][] arrays are supported,
providing a double[] for a multivariate distribution or a
double[][] for a univariate distribution may have a negative
impact in performance.
Common interface for probability distributions.
This interface is implemented by all generic probability distributions in the framework, including
s and s.
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
The Cumulative Distribution Function (CDF) describes the cumulative
probability that a given value or any value smaller than it will occur.
Gets the probability density function (pdf) for
this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The Probability Density Function (PDF) describes the
probability that a given value x will occur.
The probability of x occurring
in the current distribution.
Gets the log-probability density function (pdf)
for this distribution evaluated at point x.
A single point in the distribution range. For a
univariate distribution, this should be a single
double value. For a multivariate distribution,
this should be a double array.
The logarithm of the probability of x
occurring in the current distribution.
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
The Complementary Cumulative Distribution Function (CCDF) is
the complement of the Cumulative Distribution Function, or 1
minus the CDF.
Base abstract class for the Data Table preprocessing filters.
The column options type.
The filter type to whom these options should belong to.
Gets or sets whether this filter is active. An inactive
filter will repass the input table as output unchanged.
Gets the collection of filter options.
Gets or sets a cancellation token that can be used to
stop the learning algorithm while it is running.
The token.
Creates a new DataTable Filter Base.
Applies the Filter to a .
The source .
The name of the columns that should be processed.
The processed .
Applies the Filter to a .
The source .
The processed .
Processes the current filter.
Gets options associated with a given variable (data column).
The name of the variable.
Gets options associated with a given variable (data column).
The column's index for the variable.
Gets the number of inputs accepted by the model.
The number of inputs.
Returns an enumerator that iterates through the collection.
An enumerator that can be used to iterate through the collection.
Returns an enumerator that iterates through a collection.
An object that can be used to iterate through the collection.
Add a new column options definition to the collection.
Called when a new column options definition is being added.
Can be used to validate or modify these options beforehand.
The column options being added.
Column option collection.
The type of the filter that this collection belongs to.
The type of the column options that will be used by the
to determine how to process a particular column.
Occurs when a new is being
added to the collection. Handlers of this event can prevent a column
options from being added by throwing an exception.
Compatibility event args for the event. This
is only required and used for the .NET 3.5 version of the framework.
Extracts the key from the specified column options.
Adds a new column options definition to the collection.
The column options to be added.
The added column options.
Gets the associated options for the given column name.
The name of the column whose options should be retrieved.
The retrieved options.
True if the options was contained in the collection; false otherwise.
Column options for filter which have per-column settings.
Gets or sets the filter to which these options belong to.
The owner filter.
Gets or sets the name of the column that the options will apply to.
Gets or sets a user-determined object associated with this column.
Gets or sets a cancellation token that can be used to
stop the learning algorithm while it is running.
The token.
Constructs the base class for Column Options.
Column's name.
Returns a that represents this instance.
A that represents this instance.
Data processing interface for in-place filters.
Applies the filter to a ,
modifying the table in place.
Source table to apply filter to.
The method modifies the source table in place.
Indicates that a column filter supports automatic initialization.
Auto detects the column options by analyzing a given .
The column to analyze.
Indicates that a filter supports automatic initialization.
Auto detects the filter options by analyzing a given .
Branching filter.
The branching filter allows for different filter sequences to be
applied to different subsets of a data table. For instance, consider
a data table whose first column, "IsStudent", is an indicator variable:
a value of 1 indicates the row contains information about a student, and
a value of 0 indicates the row contains information about someone who is
not currently a student. Using the branching filter, it becomes possible
to apply a different set of filters for the rows that represent students
and different filters for rows that represent non-students.
Suppose we have the following data table. In this table, each row represents
a person, an indicator variable tell us whether this person is a smoker, and
the last column indicates the age of each person. Let's say we would like to
convert the age of smokers to a scale from -1 to 0, and the age of non-smokers
to a scale from 0 to 1.
object[,] data =
{
{ "Id", "IsSmoker", "Age" },
{ 0, 1, 10 },
{ 1, 1, 15 },
{ 2, 0, 40 },
{ 3, 1, 20 },
{ 4, 0, 70 },
{ 5, 0, 55 },
};
// Create a DataTable from data
DataTable input = data.ToTable();
// We will create two filters, one to operate on the smoking
// branch of the data, and other in the non-smoking subjects.
//
var smoker = new LinearScaling();
var common = new LinearScaling();
// for the smokers, we will convert the age to [-1; 0]
smoker.Columns.Add(new LinearScaling.Options("Age")
{
SourceRange = new DoubleRange(10, 20),
OutputRange = new DoubleRange(-1, 0)
});
// for non-smokers, we will convert the age to [0; +1]
common.Columns.Add(new LinearScaling.Options("Age")
{
SourceRange = new DoubleRange(40, 70),
OutputRange = new DoubleRange(0, 1)
});
// We now configure and create the branch filter
var settings = new Branching.Options("IsSmoker");
settings.Filters.Add(1, smoker);
settings.Filters.Add(0, common);
Branching branching = new Branching(settings);
// Finally, we can process the input data:
DataTable actual = branching.Apply(input);
// As result, the generated table will
// then contain the following entries:
// { "Id", "IsSmoker", "Age" },
// { 0, 1, -1.0 },
// { 1, 1, -0.5 },
// { 2, 0, 0.0 },
// { 3, 1, 0.0 },
// { 4, 0, 1.0 },
// { 5, 0, 0.5 },
Initializes a new instance of the class.
Initializes a new instance of the class.
The columns to use as filters.
Initializes a new instance of the class.
The columns to use as filters.
Processes the current filter.
Column options for the branching filter.
Gets the collection of filters associated with a given label value.
Initializes a new instance of the class.
The column name.
Initializes a new instance of the class.
Auto detects the column options by analyzing a given .
The column to analyze.
Codification Filter class.
The object type that needs to be codified. Default is string.
The codification filter performs an integer codification of classes in
given in a string form. An unique integer identifier will be assigned
for each of the string classes.
Every Learn() method in the framework expects the class labels to be contiguous and zero-indexed,
meaning that if there is a classification problem with n classes, all class labels must be numbers
ranging from 0 to n-1. However, not every dataset might be in this format and sometimes we will
have to pre-process the data to be in this format. The example below shows how to use the
Codification class to perform such pre-processing.
Number of elements in (number of distinct elements).
-
Number of elements in (number of distinct elements).
-
Number of elements in (number of distinct elements).
-
1 (there are no symbols to be encoded).
-
1 (there are no symbols to be encoded).
Gets the codification type that should be used for this variable.
Gets the number of inputs accepted by the model (value will be 1).
The number of inputs.
Gets the number of outputs generated by the model. See remarks for details.
The number of outputs.
This method returns the following table of values:
Number of elements in (number of distinct elements).
-
Number of elements in minus 1 (number of distinct elements, except the baseline, which is encoded as the absence of other values).
-
1 (there is just one single ordinal variable).
-
1 (there is just one single continuous variable).
-
1 (there is just one single discrete variable).
Gets the values associated with each symbol, in the order of the symbols.
Gets or sets the number of classes expected and recognized by the classifier.
The number of classes.
Determines whether the given object denotes a missing value.
Forces the given key to have a specific symbol value.
The key.
The value that should be associated with this key.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
The output generated by applying this transformation to the given input.
Applies the transformation to a set of input vectors,
producing an associated set of output vectors.
The input data to which
the transformation should be applied.
The output generated by applying this
transformation to the given input.
Applies the transformation to a set of input vectors,
producing an associated set of output vectors.
The input data to which
the transformation should be applied.
The location to where to store the
result of this transformation.
The output generated by applying this
transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
The output generated by applying this transformation to the given input.
Applies the transformation to an input, producing an associated output.
The input data to which the transformation should be applied.
The output generated by applying this transformation to the given input.
Applies the transformation to a set of input vectors,
producing an associated set of output vectors.
The input data to which
the transformation should be applied.
The location to where to store the
result of this transformation.
The output generated by applying this
transformation to the given input.
Reverts the transformation to a set of output vectors,
producing an associated set of input vectors.
The input data to which
the transformation should be reverted.
Reverts the transformation to a set of output vectors,
producing an associated set of input vectors.
The input data to which
the transformation should be reverted.
Reverts the transformation to a set of output vectors,
producing an associated set of input vectors.
The input data to which
the transformation should be reverted.
The location to where to store the
result of this transformation.
The input generated by reverting this
transformation to the given output.
Learns a model that can map the given inputs to the desired outputs.
The model inputs.
The weight of importance for each input sample.
A model that has learned how to produce suitable outputs
given the input data .
Weights are not supported and should be null.
Learns a model that can map the given inputs to the desired outputs.
The model inputs.
The weight of importance for each input sample.
A model that has learned how to produce suitable outputs
given the input data .
Weights are not supported and should be null.
Learns a model that can map the given inputs to the desired outputs.
The model inputs.
The weight of importance for each input sample.
A model that has learned how to produce suitable outputs
given the input data .
Weights are not supported and should be null.
Computes a class-label decision for a given .
The input vector that should be classified into
one of the possible classes.
A class-label that best described according
to this classifier.
Computes class-label decisions for each vector in the given .
The input vectors that should be classified into
one of the possible classes.
The class-labels that best describe each
vectors according to this classifier.
Computes class-label decisions for each vector in the given .
The input vectors that should be classified into
one of the possible classes.
The location where to store the class-labels.
The class-labels that best describe each
vectors according to this classifier.
Applies the transformation to a set of input vectors,
producing an associated set of output vectors.
The input data to which
the transformation should be applied.
The location to where to store the
result of this transformation.
The output generated by applying this
transformation to the given input.
Constructs a new Options object.
Constructs a new Options object for the given column.
The name of the column to create this options for.
Constructs a new Options object for the given column.
The name of the column to create this options for.
The type of the variable in the column.
Constructs a new Options object for the given column.
The name of the column to create this options for.
The type of the variable in the column.
The order of the variables in the mapping. The first variable
will be assigned to position (symbol) 1, the second to position 2, and so on.
Constructs a new Options object for the given column.
The name of the column to create this options for.
The type of the variable in the column.
The baseline value to be used in conjunction with
. The baseline
value will be treated as absolute zero in a otherwise one-hot representation.
Constructs a new Options object for the given column.
The name of the column to create this options for.
The initial mapping for this column.
Value discretization preprocessing filter.
This filter converts ranges of values into a different representation
according to a set of rules. Please see the examples below to see how
this filter can be used in practice.
The discretization filter can be used to convert any range of values into another representation. For example,
let's say we have a dataset where a column represents percentages using floating point numbers, but we would
like to discretize those numbers into more descriptive labels:
// Show the start data
DataGridBox.Show(table);
// Create a new data projection (column) filter
var filter = new Codification(table, "Category");
// Apply the filter and get the result
DataTable result = filter.Apply(table);
// Show it
DataGridBox.Show(result);
The following more elaborated examples show how to use the filter without
necessarily handling System.Data.DataTables.
// Show the start data
DataGridBox.Show(table);
// Create a new data projection (column) filter
var filter = new Discretization("Cost (M)");
// Apply the filter and get the result
DataTable result = filter.Apply(table);
// Show it
DataGridBox.Show(result);
Creates a new Discretization filter.
Creates a new Discretization filter.
Processes the current filter.
Auto detects the filter options by analyzing a given .
Options for the discretization filter.
Gets or sets the threshold for the discretization filter.
Gets or sets whether the discretization threshold is symmetric.
If a symmetric threshold of 0.4 is used, for example, a real value of
0.5 will be rounded to 1.0 and a real value of -0.5 will be rounded to
-1.0.
If a non-symmetric threshold of 0.4 is used, a real value of 0.5
will be rounded towards 1.0, but a real value of -0.5 will be rounded
to 0.0 (because |-0.5| is higher than the threshold of 0.4).
Constructs a new Options class for the discretization filter.
Constructs a new Options object.
Sequence of table processing filters.
Initializes a new instance of the class.
Initializes a new instance of the class.
Sequence of filters to apply.
Applies the sequence of filters to a given table.
Sample processing filter interface.
The interface defines the set of methods which should be
provided by all table processing filters. Methods of this interface should
keep the source table unchanged and return the result of data processing
filter as new data table.
Applies the filter to a .
Source table to apply filter to.
Returns filter's result obtained by applying the filter to
the source table.
The method keeps the source table unchanged and returns the
the result of the table processing filter as new data table.
Data normalization preprocessing filter.
The normalization filter is able to transform numerical data into
Z-Scores, subtracting the mean for each variable and dividing by
their standard deviation. The filter is able to distinguish
numerical columns automatically, leaving other columns unaffected.
It is also possible to control which columns should be processed
by the filter.
Suppose we have a data table relating the age of a person and its
categorical classification, as in "child", "adult" or "elder".
The normalization filter can be used to transform the "Age" column
into Z-scores, as shown below:
// Create the aforementioned sample table
DataTable table = new DataTable("Sample data");
table.Columns.Add("Age", typeof(double));
table.Columns.Add("Label", typeof(string));
// age label
table.Rows.Add(10, "child");
table.Rows.Add(07, "child");
table.Rows.Add(04, "child");
table.Rows.Add(21, "adult");
table.Rows.Add(27, "adult");
table.Rows.Add(12, "child");
table.Rows.Add(79, "elder");
table.Rows.Add(40, "adult");
table.Rows.Add(30, "adult");
// The filter will ignore non-real (continuous) data
Normalization normalization = new Normalization(table);
double mean = normalization["Age"].Mean; // 25.55
double sdev = normalization["Age"].StandardDeviation; // 23.29
// Now we can process another table at once:
DataTable result = normalization.Apply(table);
// The result will be a table with the same columns, but
// in which any column named "Age" will have been normalized
// using the previously detected mean and standard deviation:
DataGridBox.Show(result);
The resulting data is shown below:
Creates a new data normalization filter.
Creates a new data normalization filter.
Creates a new data normalization filter.
Processes the current filter.
Applies the Filter to a .
The source .
The processed .
Applies the Filter to a .
The source .
The processed .
Auto detects the filter options by analyzing a given .
Auto detects the filter options by analyzing a given matrix.
Options for normalizing a column.
Gets or sets the mean of the data contained in the column.
Gets or sets the standard deviation of the data contained in the column.
Gets or sets if the column's data should be standardized to Z-Scores.
Constructs a new Options object.
Constructs a new Options object for the given column.
The name of the column to create this options for.
Constructs a new Options object for the given column.
The name of the column to create this options for.
The mean value for normalization.
The standard deviation value for standardization.
Principal component projection filter.
Gets or sets the analysis associated with the filter.
Creates a new Principal Component Projection filter.
Creates a new data normalization filter.
Processes the filter.
The data.
Auto detects the filter options by analyzing a given .
Options for normalizing a column.
Initializes a new instance of the class.
Initializes a new instance of the class.
Name of the column.
Relational-algebra projection filter.
This filter is able to selectively remove columns from tables, and keep
only the columns of interest.
// Show the start data
DataGridBox.Show(table);
// Create a new data projection (column) filter
var filter = new Projection("Floors", "Finished");
// Apply the filter and get the result
DataTable result = filter.Apply(table);
// Show it
DataGridBox.Show(result);
List of columns to keep in the projection.
Creates a new projection filter.
Creates a new projection filter.
Creates a new projection filter.
Applies the filter to the DataTable.
Linear Scaling Filter
Creates a new Linear Scaling Filter.
Creates a new Linear Scaling Filter.
Creates a new Linear Scaling Filter.
Creates a new Linear Scaling Filter.
Applies the filter to the DataTable.
Auto detects the filter options by analyzing a given .
Auto detects the filter options by analyzing a given .
Options for the Linear Scaling filter.
Range of the input values
Target range of the output values after scaling.
Creates a new column options.
Constructs a new Options object.
Relational-algebra selection filter.
Gets or sets the eSQL filter expression for the filter.
Gets or sets the ordering to apply for the filter.
Constructs a new Selection Filter.
The filtering criteria.
The desired sort order.
Constructs a new Selection Filter.
The filtering criteria.
Constructs a new Selection Filter.
Applies the filter to the current data.
Additive combination of kernels.
Gets the combination of kernels to use.
Gets the weight array to use in the weighted kernel sum.
Constructs a new additive kernel.
Kernels to combine.
Constructs a new additive kernel.
Kernels to combine.
Weight values for each of the kernels.
Default is to assign equal weights.
Additive Kernel Combination function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
ANOVA (ANalysis Of VAriance) Kernel.
The ANOVA kernel is a graph kernel, which can be
computed using dynamic programming tables.
References:
- http://www.cse.ohio-state.edu/mlss09/mlss09_talks/1.june-MON/jst_tutorial.pdf
Constructs a new ANOVA Kernel.
Length of the input vector.
Length of the subsequences for the ANOVA decomposition.
ANOVA Kernel function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Kernel function interface.
In Machine Learning and statistics, a Kernel is a function that returns
the value of the dot product between the images of the two arguments.
k(x,y) = ‹S(x),S(y)›
References:
-
http://www.support-vector.net/icml-tutorial.pdf
Elementwise addition of a and b, storing in result.
The first vector to add.
The second vector to add.
An array to store the result.
The same vector passed as result.
Elementwise multiplication of scalar a and vector b, accumulating in result.
The scalar to be multiplied.
The vector to be multiplied.
An array to store the result.
Elementwise multiplication of vector a and vector b, accumulating in result.
The vector to be multiplied.
The vector to be multiplied.
An array to store the result.
Compress a set of support vectors and weights into a single
parameter vector.
The weights associated with each support vector.
The support vectors.
The constant (bias) value.
A single parameter vector.
The kernel function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Gets the number of parameters in the input vectors.
Creates an input vector from the given double values.
Converts the input vectors to a double-precision representation.
Kernel function interface.
In Machine Learning and statistics, a Kernel is a function that returns
the value of the dot product between the images of the two arguments.
k(x,y) = ‹S(x),S(y)›
References:
-
http://www.support-vector.net/icml-tutorial.pdf
Kernel function interface.
In Machine Learning and statistics, a Kernel is a function that returns
the value of the dot product between the images of the two arguments.
k(x,y) = ‹S(x),S(y)›
References:
-
http://www.support-vector.net/icml-tutorial.pdf
The kernel function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Interface for Radial Basis Function kernels.
A radial basis function (RBF) is a real-valued function whose value depends only
on the distance from the origin, so that ϕ(x) = ϕ(||x||); or alternatively
on the distance from some other point c, called a center, so that
ϕ(x,c) = ϕ(||x−c||). Any function ϕ that satisfies the property
ϕ(x) = ϕ(||x||) is a radial function. The norm is usually Euclidean distance,
although other distance functions are also possible.
Examples of radial basis kernels include:
References:
-
Wikipedia, The Free Encyclopedia. Radial basis functions. Available on:
https://en.wikipedia.org/wiki/Radial_basis_function
-
Bart Hamers, Kernel Models for Large Scale Applications. Doctoral thesis.
Available on: ftp://ftp.esat.kuleuven.ac.be/pub/SISTA/hamers/PhD_bhamers.pdf
Gets or sets the B-Spline order.
Constructs a new B-Spline Kernel.
B-Spline Kernel Function
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Cauchy Kernel.
The Cauchy kernel comes from the Cauchy distribution (Basak, 2008). It is a
long-tailed kernel and can be used to give long-range influence and sensitivity
over the high dimension space.
Gets or sets the kernel's sigma value.
Constructs a new Cauchy Kernel.
The value for sigma.
Cauchy Kernel Function
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Cauchy Kernel Function
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Computes the squared distance in feature space
between two points given in input space.
Vector x in input space.
Vector y in input space.
Squared distance between x and y in feature (kernel) space.
Cauchy Kernel Function
Distance z in input space.
Dot product in feature (kernel) space.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Chi-Square Kernel.
The Chi-Square kernel comes from the Chi-Square distribution.
Constructs a new Chi-Square kernel.
Chi-Square Kernel Function
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Circular Kernel.
The circular kernel comes from a statistics perspective. It is an example
of an isotropic stationary kernel and is positive definite in R^2.
Gets or sets the kernel's sigma value.
Constructs a new Circular Kernel.
Value for sigma.
Circular Kernel Function
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Circular Kernel Function
Distance z in input space.
Dot product in feature (kernel) space.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Radial Basis Function Dynamic Time Warping Sequence Kernel.
The Dynamic Time Warping Sequence Kernel is a sequence kernel, accepting
vector sequences of variable size as input. Despite the sequences being
variable in size, the vectors contained in such sequences should have its
size fixed and should be informed at the construction of this kernel.
The conversion of the DTW global distance to a dot product uses a combination
of a technique known as spherical normalization and the polynomial kernel. The
degree of the polynomial kernel and the alpha for the spherical normalization
should be given at the construction of the kernel. For more information,
please see the referenced papers shown below.
The use of a cache is highly advisable
when using this kernel.
References:
-
V. Wan, J. Carmichael; Polynomial Dynamic Time Warping Kernel Support
Vector Machines for Dysarthric Speech Recognition with Sparse Training
Data. Interspeech'2005 - Eurospeech - 9th European Conference on Speech
Communication and Technology. Lisboa, 2005.
The following example demonstrates how to create and learn a Support Vector
Machine (SVM) to recognize sequences of univariate observations using the
Dynamic Time Warping kernel.
-
Chaudhuri et al, A Comparative Study of Kernels for the Multi-class Support Vector
Machine, 2008. Available on: http://www.computer.org/portal/web/csdl/doi/10.1109/ICNC.2008.803
Constructs a new Symmetric Triangle Kernel
Gets or sets the gamma value for the kernel.
Symmetric Triangle Kernel function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Symmetric Triangle Kernel function.
Distance z in input space.
Dot product in feature (kernel) space.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Squared Sinc Kernel.
References:
-
Chaudhuri et al, A Comparative Study of Kernels for the Multi-class Support Vector
Machine, 2008. Available on: http://www.computer.org/portal/web/csdl/doi/10.1109/ICNC.2008.803
Constructs a new Squared Sinc Kernel
Gets or sets the gamma value for the kernel.
Squared Sine Kernel function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Squared Sine Kernel function.
Distance z in input space.
Dot product in feature (kernel) space.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Custom Kernel.
Constructs a new Custom kernel.
Custom kernel function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Dirichlet Kernel.
References:
-
A Tutorial on Support Vector Machines (1998). Available on: http://www.umiacs.umd.edu/~joseph/support-vector-machines4.pdf
Constructs a new Dirichlet Kernel
Gets or sets the dimension for the kernel.
Dirichlet Kernel function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Dynamic Time Warping Sequence Kernel.
The Dynamic Time Warping Sequence Kernel is a sequence kernel, accepting
vector sequences of variable size as input. Despite the sequences being
variable in size, the vectors contained in such sequences should have its
size fixed and should be informed at the construction of this kernel.
The conversion of the DTW global distance to a dot product uses a combination
of a technique known as spherical normalization and the polynomial kernel. The
degree of the polynomial kernel and the alpha for the spherical normalization
should be given at the construction of the kernel. For more information,
please see the referenced papers shown below.
The use of a cache is highly advisable
when using this kernel.
References:
-
V. Wan, J. Carmichael; Polynomial Dynamic Time Warping Kernel Support
Vector Machines for Dysarthric Speech Recognition with Sparse Training
Data. Interspeech'2005 - Eurospeech - 9th European Conference on Speech
Communication and Technology. Lisboa, 2005.
The following example demonstrates how to create and learn a Support Vector
Machine (SVM) to recognize sequences of univariate observations using the
Dynamic Time Warping kernel.
-
P. F. Evangelista, M. J. Embrechts, and B. K. Szymanski. Some Properties
of the Gaussian Kernel for One Class Learning. Available on:
http://www.cs.rpi.edu/~szymansk/papers/icann07.pdf
Constructs a new Gaussian Kernel with a given sigma value. To
construct from a gamma value, use the
named constructor instead.
The kernel's sigma parameter.
Gets or sets the sigma value for the kernel. When setting
sigma, gamma gets updated accordingly (gamma = 0.5/sigma^2).
Gets or sets the sigma² value for the kernel. When setting
sigma², gamma gets updated accordingly (gamma = 0.5/sigma²).
Gets or sets the gamma value for the kernel. When setting
gamma, sigma gets updated accordingly (gamma = 0.5/sigma^2).
Gaussian Kernel function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Gaussian Kernel function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Gaussian Kernel function.
Distance z in input space.
Dot product in feature (kernel) space.
Computes the squared distance in feature space
between two points given in input space.
Vector x in input space.
Vector y in input space.
Squared distance between x and y in feature (kernel) space.
Computes the squared distance in feature space
between two points given in input space.
Vector x in input space.
Vector y in input space.
Squared distance between x and y in feature (kernel) space.
Computes the squared distance in input space
between two points given in feature space.
Vector x in feature (kernel) space.
Vector y in feature (kernel) space.
Squared distance between x and y in input space.
Computes the distance in input space given
a distance computed in feature space.
Distance in feature space.
Distance in input space.
Constructs a new Gaussian Kernel with a given gamma value. To
construct from a sigma value, use the
constructor instead.
The kernel's gamma parameter.
Estimate appropriate values for sigma given a data set.
This method uses a simple heuristic to obtain appropriate values
for sigma in a radial basis function kernel. The heuristic is shown
by Caputo, Sim, Furesjo and Smola, "Appearance-based object
recognition using SVMs: which kernel should I use?", 2002.
The data set.
A Gaussian kernel initialized with an appropriate sigma value.
Estimate appropriate values for sigma given a data set.
This method uses a simple heuristic to obtain appropriate values
for sigma in a radial basis function kernel. The heuristic is shown
by Caputo, Sim, Furesjo and Smola, "Appearance-based object
recognition using SVMs: which kernel should I use?", 2002.
The data set.
The range of suitable values for sigma.
A Gaussian kernel initialized with an appropriate sigma value.
Estimates appropriate values for sigma given a data set.
This method uses a simple heuristic to obtain appropriate values
for sigma in a radial basis function kernel. The heuristic is shown
by Caputo, Sim, Furesjo and Smola, "Appearance-based object
recognition using SVMs: which kernel should I use?", 2002.
The data set.
The number of random samples to analyze.
A Gaussian kernel initialized with an appropriate sigma value.
Estimates appropriate values for sigma given a data set.
This method uses a simple heuristic to obtain appropriate values
for sigma in a radial basis function kernel. The heuristic is shown
by Caputo, Sim, Furesjo and Smola, "Appearance-based object
recognition using SVMs: which kernel should I use?", 2002.
The data set.
The number of random samples to analyze.
The range of suitable values for sigma.
A Gaussian kernel initialized with an appropriate sigma value.
Estimate appropriate values for sigma given a data set.
This method uses a simple heuristic to obtain appropriate values
for sigma in a radial basis function kernel. The heuristic is shown
by Caputo, Sim, Furesjo and Smola, "Appearance-based object
recognition using SVMs: which kernel should I use?", 2002.
The data set.
A Gaussian kernel initialized with an appropriate sigma value.
Estimate appropriate values for sigma given a data set.
This method uses a simple heuristic to obtain appropriate values
for sigma in a radial basis function kernel. The heuristic is shown
by Caputo, Sim, Furesjo and Smola, "Appearance-based object
recognition using SVMs: which kernel should I use?", 2002.
The data set.
The range of suitable values for sigma.
A Gaussian kernel initialized with an appropriate sigma value.
Estimates appropriate values for sigma given a data set.
This method uses a simple heuristic to obtain appropriate values
for sigma in a radial basis function kernel. The heuristic is shown
by Caputo, Sim, Furesjo and Smola, "Appearance-based object
recognition using SVMs: which kernel should I use?", 2002.
The data set.
The number of random samples to analyze.
A Gaussian kernel initialized with an appropriate sigma value.
Estimates appropriate values for sigma given a data set.
This method uses a simple heuristic to obtain appropriate values
for sigma in a radial basis function kernel. The heuristic is shown
by Caputo, Sim, Furesjo and Smola, "Appearance-based object
recognition using SVMs: which kernel should I use?", 2002.
The data set.
The number of random samples to analyze.
The range of suitable values for sigma.
A Gaussian kernel initialized with an appropriate sigma value.
Estimate appropriate values for sigma given a data set.
This method uses a simple heuristic to obtain appropriate values
for sigma in a radial basis function kernel. The heuristic is shown
by Caputo, Sim, Furesjo and Smola, "Appearance-based object
recognition using SVMs: which kernel should I use?", 2002.
The data set.
The distance function to be used in the Gaussian kernel. Default is .
A Gaussian kernel initialized with an appropriate sigma value.
Estimate appropriate values for sigma given a data set.
This method uses a simple heuristic to obtain appropriate values
for sigma in a radial basis function kernel. The heuristic is shown
by Caputo, Sim, Furesjo and Smola, "Appearance-based object
recognition using SVMs: which kernel should I use?", 2002.
The data set.
The range of suitable values for sigma.
The distance function to be used in the Gaussian kernel. Default is .
A Gaussian kernel initialized with an appropriate sigma value.
Estimates appropriate values for sigma given a data set.
This method uses a simple heuristic to obtain appropriate values
for sigma in a radial basis function kernel. The heuristic is shown
by Caputo, Sim, Furesjo and Smola, "Appearance-based object
recognition using SVMs: which kernel should I use?", 2002.
The data set.
The number of random samples to analyze.
The distance function to be used in the Gaussian kernel. Default is .
A Gaussian kernel initialized with an appropriate sigma value.
Estimates appropriate values for sigma given a data set.
This method uses a simple heuristic to obtain appropriate values
for sigma in a radial basis function kernel. The heuristic is shown
by Caputo, Sim, Furesjo and Smola, "Appearance-based object
recognition using SVMs: which kernel should I use?", 2002.
The data set.
The number of random samples to analyze.
The range of suitable values for sigma.
The distance function to be used in the Gaussian kernel. Default is .
A Gaussian kernel initialized with an appropriate sigma value.
Computes the set of all distances between
all points in a random subset of the data.
The inputs points.
The number of samples.
Computes the set of all distances between
all points in a random subset of the data.
The inputs points.
The number of samples.
Computes the set of all distances between
all points in a random subset of the data.
The inputs points.
The number of samples.
The distance function to be used in the Gaussian kernel. Default is .
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Estimate appropriate values for sigma given a data set.
This method uses a simple heuristic to obtain appropriate values
for sigma in a radial basis function kernel. The heuristic is shown
by Caputo, Sim, Furesjo and Smola, "Appearance-based object
recognition using SVMs: which kernel should I use?", 2002.
The inner kernel.
The data set.
A Gaussian kernel initialized with an appropriate sigma value.
Estimate appropriate values for sigma given a data set.
This method uses a simple heuristic to obtain appropriate values
for sigma in a radial basis function kernel. The heuristic is shown
by Caputo, Sim, Furesjo and Smola, "Appearance-based object
recognition using SVMs: which kernel should I use?", 2002.
The inner kernel.
The data set.
The range of suitable values for sigma.
A Gaussian kernel initialized with an appropriate sigma value.
Estimates appropriate values for sigma given a data set.
This method uses a simple heuristic to obtain appropriate values
for sigma in a radial basis function kernel. The heuristic is shown
by Caputo, Sim, Furesjo and Smola, "Appearance-based object
recognition using SVMs: which kernel should I use?", 2002.
The inner kernel.
The data set.
The number of random samples to analyze.
A Gaussian kernel initialized with an appropriate sigma value.
Estimates appropriate values for sigma given a data set.
This method uses a simple heuristic to obtain appropriate values
for sigma in a radial basis function kernel. The heuristic is shown
by Caputo, Sim, Furesjo and Smola, "Appearance-based object
recognition using SVMs: which kernel should I use?", 2002.
The inner kernel.
The data set.
The number of random samples to analyze.
The range of suitable values for sigma.
A Gaussian kernel initialized with an appropriate sigma value.
Generalized Histogram Intersection Kernel.
The Generalized Histogram Intersection kernel is built based on the
Histogram Intersection Kernel for image classification but applies
in a much larger variety of contexts (Boughorbel, 2005).
Constructs a new Generalized Histogram Intersection Kernel.
Generalized Histogram Intersection Kernel Function
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Hyperbolic Secant Kernel.
References:
-
Chaudhuri et al, A Comparative Study of Kernels for the Multi-class Support Vector
Machine, 2008. Available on: http://www.computer.org/portal/web/csdl/doi/10.1109/ICNC.2008.803
Constructs a new Hyperbolic Secant Kernel
Gets or sets the gamma value for the kernel.
Hyperbolic Secant Kernel function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Hyperbolic Secant Kernel function.
Distance z in input space.
Dot product in feature (kernel) space.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Input space distance interface for kernel functions.
Kernels which implement this interface can be used to solve the pre-image
problem in
Kernel Principal Component Analysis and other methods based in Multi-
Dimensional Scaling.
Computes the squared distance in input space
between two points given in feature space.
Vector x in feature (kernel) space.
Vector y in feature (kernel) space.
Squared distance between x and y in input space.
Kernel function interface.
In Machine Learning and statistics, a Kernel is a function that returns
the value of the dot product between the images of the two arguments.
k(x,y) = ‹S(x),S(y)›
References:
-
http://www.support-vector.net/icml-tutorial.pdf
Laplacian Kernel.
Constructs a new Laplacian Kernel
Constructs a new Laplacian Kernel
The sigma slope value.
Gets or sets the sigma value for the kernel. When setting
sigma, gamma gets updated accordingly (gamma = 0.5*/sigma^2).
Gets or sets the gamma value for the kernel. When setting
gamma, sigma gets updated accordingly (gamma = 0.5*/sigma^2).
Laplacian Kernel function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Laplacian Kernel function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Laplacian Kernel function.
Distance z in input space.
Dot product in feature (kernel) space.
Computes the squared distance in input space
between two points given in feature space.
Vector x in feature (kernel) space.
Vector y in feature (kernel) space.
Squared distance between x and y in input space.
Computes the squared distance in input space
between two points given in feature space.
Vector x in feature (kernel) space.
Vector y in feature (kernel) space.
Squared distance between x and y in input space.
Computes the squared distance in input space
between two points given in feature space.
Vector x in feature (kernel) space.
Vector y in feature (kernel) space.
Squared distance between x and y in input space.
Computes the distance in input space given
a distance computed in feature space.
Distance in feature space.
Distance in input space.
Estimate appropriate values for sigma given a data set.
This method uses a simple heuristic to obtain appropriate values
for sigma in a radial basis function kernel. The heuristic is shown
by Caputo, Sim, Furesjo and Smola, "Appearance-based object
recognition using SVMs: which kernel should I use?", 2002.
The data set.
A Laplacian kernel initialized with an appropriate sigma value.
Estimate appropriate values for sigma given a data set.
This method uses a simple heuristic to obtain appropriate values
for sigma in a radial basis function kernel. The heuristic is shown
by Caputo, Sim, Furesjo and Smola, "Appearance-based object
recognition using SVMs: which kernel should I use?", 2002.
The data set.
The number of random samples to analyze.
The range of suitable values for sigma.
A Laplacian kernel initialized with an appropriate sigma value.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Linear Kernel.
Constructs a new Linear kernel.
A constant intercept term. Default is 0.
Gets or sets the kernel's intercept term. Default is 0.
Linear kernel function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Linear kernel function.
Distance z in input space.
Dot product in feature (kernel) space.
Computes the squared distance in input space
between two points given in feature space.
Vector x in feature (kernel) space.
Vector y in feature (kernel) space.
Squared distance between x and y in input space.
Computes the squared distance in input space
between two points given in feature space.
Vector x in feature (kernel) space.
Vector y in feature (kernel) space.
Squared distance between x and y in input space.
Computes the squared distance in input space
between two points given in feature space.
Vector x in feature (kernel) space.
Vector y in feature (kernel) space.
Squared distance between x and y in input space.
Elementwise addition of a and b, storing in result.
The first vector to add.
The second vector to add.
An array to store the result.
The same vector passed as result.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Elementwise multiplication of scalar a and vector b, storing in result.
The scalar to be multiplied.
The vector to be multiplied.
An array to store the result.
Compress a set of support vectors and weights into a single
parameter vector.
The weights associated with each support vector.
The support vectors.
The constant (bias) value.
A single parameter vector.
Projects an input point into feature space.
The input point to be projected into feature space.
The feature space representation of the given point.
Projects a set of input points into feature space.
The input points to be projected into feature space.
The feature space representation of the given points.
Projects an input point into feature space.
The input point to be projected into feature space.
The parameter of the kernel.
The feature space representation of the given point.
The kernel function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
The kernel function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Elementwise addition of a and b, storing in result.
The first vector to add.
The second vector to add.
An array to store the result.
Elementwise multiplication of scalar a and vector b, storing in result.
The scalar to be multiplied.
The vector to be multiplied.
An array to store the result.
Compress a set of support vectors and weights into a single
parameter vector.
The weights associated with each support vector.
The support vectors.
The constant (bias) value.
A single parameter vector.
Gets the number of parameters in the input vectors.
Gets the number of parameters in the input vectors.
Creates an input vector from the given double values.
Creates an input vector with the given dimensions.
Elementwise multiplication of vector a and vector b, accumulating in result.
The vector to be multiplied.
The vector to be multiplied.
An array to store the result.
Elementwise multiplication of vector a and vector b, accumulating in result.
The vector to be multiplied.
The vector to be multiplied.
An array to store the result.
Converts the input vectors to a double-precision representation.
Converts the input vectors to a double-precision representation.
Logarithm Kernel.
The Log kernel seems to be particularly interesting for
images, but is only conditionally positive definite.
Constructs a new Log Kernel
The kernel's degree.
Constructs a new Log Kernel
The kernel's degree.
Gets or sets the kernel's degree.
Log Kernel function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Log Kernel function.
Distance z in input space.
Dot product in feature (kernel) space.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Multiquadric Kernel.
The multiquadric kernel is only conditionally positive-definite.
Gets or sets the kernel's constant value.
Constructs a new Multiquadric Kernel.
The constant term theta.
Constructs a new Multiquadric Kernel.
Multiquadric Kernel function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Multiquadric Kernel function.
Distance z in input space.
Dot product in feature (kernel) space.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Polynomial Kernel.
Constructs a new Polynomial kernel of a given degree.
The polynomial degree for this kernel.
The polynomial constant for this kernel. Default is 1.
Constructs a new Polynomial kernel of a given degree.
The polynomial degree for this kernel.
Gets or sets the kernel's polynomial degree.
Gets or sets the kernel's polynomial constant term.
Polynomial kernel function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Polynomial kernel function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Polynomial kernel function.
Distance z in input space.
Dot product in feature (kernel) space.
Computes the squared distance in feature space
between two points given in input space.
Vector x in input space.
Vector y in input space.
Squared distance between x and y in feature (kernel) space.
Computes the squared distance in feature space
between two points given in input space.
Vector x in input space.
Vector y in input space.
Squared distance between x and y in feature (kernel) space.
Computes the distance in input space
between two points given in feature space.
Vector x in feature (kernel) space.
Vector y in feature (kernel) space.
Distance between x and y in input space.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Projects an input point into feature space.
The input point to be projected into feature space.
The feature space representation of the given point.
Projects a set of input points into feature space.
The input points to be projected into feature space.
The feature space representation of the given points.
Projects an input point into feature space.
The input point to be projected into feature space.
The parameter of the kernel.
The parameter of the kernel.
The feature space representation of the given point.
Power Kernel, also known as the (Unrectified) Triangular Kernel.
The Power kernel is also known as the (unrectified) triangular kernel.
It is an example of scale-invariant kernel (Sahbi and Fleuret, 2004)
and is also only conditionally positive definite.
Gets or sets the kernel's degree.
Constructs a new Power Kernel.
The kernel's degree.
Constructs a new Power Kernel.
The kernel's degree.
Power Kernel Function
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Power Kernel Function
Distance z in input space.
Dot product in feature (kernel) space.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Precomputed Gram Matrix Kernel.
The following example shows how to learn a multi-class SVM using
a precomputed kernel matrix, obtained from a Polynomial kernel.
-
P. F. Evangelista, M. J. Embrechts, and B. K. Szymanski. Some Properties of the
Gaussian Kernel for One Class Learning. Available on: http://www.cs.rpi.edu/~szymansk/papers/icann07.pdf
Constructs a new Sparse Gaussian Kernel
Constructs a new Sparse Gaussian Kernel
The standard deviation for the Gaussian distribution. Default is 1.
Gets or sets the sigma value for the kernel. When setting
sigma, gamma gets updated accordingly (gamma = 0.5*/sigma^2).
Gets or sets the gamma value for the kernel. When setting
gamma, sigma gets updated accordingly (gamma = 0.5*/sigma^2).
Gaussian Kernel function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Computes the distance in input space
between two points given in feature space.
Vector x in feature (kernel) space.
Vector y in feature (kernel) space.
Distance between x and y in input space.
Computes the squared distance in input space
between two points given in feature space.
Vector x in feature (kernel) space.
Vector y in feature (kernel) space.
Squared distance between x and y in input space.
Estimate appropriate values for sigma given a data set.
This method uses a simple heuristic to obtain appropriate values
for sigma in a radial basis function kernel. The heuristic is shown
by Caputo, Sim, Furesjo and Smola, "Appearance-based object
recognition using SVMs: which kernel should I use?", 2002.
The data set.
The number of random samples to analyze.
The range of suitable values for sigma.
A Gaussian kernel initialized with an appropriate sigma value.
Sparse Laplacian Kernel.
Constructs a new Laplacian Kernel
Constructs a new Laplacian Kernel
The sigma slope value.
Gets or sets the sigma value for the kernel. When setting
sigma, gamma gets updated accordingly (gamma = 0.5*/sigma^2).
Gets or sets the gamma value for the kernel. When setting
gamma, sigma gets updated accordingly (gamma = 0.5*/sigma^2).
Laplacian Kernel function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Computes the distance in input space
between two points given in feature space.
Vector x in feature (kernel) space.
Vector y in feature (kernel) space.
Distance between x and y in input space.
Estimate appropriate values for sigma given a data set.
This method uses a simple heuristic to obtain appropriate values
for sigma in a radial basis function kernel. The heuristic is shown
by Caputo, Sim, Furesjo and Smola, "Appearance-based object
recognition using SVMs: which kernel should I use?", 2002.
The data set.
The number of random samples to analyze.
The range of suitable values for sigma.
A Laplacian kernel initialized with an appropriate sigma value.
Sparse Linear Kernel.
The Sparse Linear kernel accepts inputs in the libsvm sparse format.
The following example shows how to teach a kernel support vector machine using
the linear sparse kernel to perform the AND classification task using sparse
vectors.
// Example AND problem
double[][] inputs =
{
new double[] { }, // 0 and 0: 0 (label -1)
new double[] { 2,1 }, // 0 and 1: 0 (label -1)
new double[] { 1,1 }, // 1 and 0: 0 (label -1)
new double[] { 1,1, 2,1 } // 1 and 1: 1 (label +1)
};
// Dichotomy SVM outputs should be given as [-1;+1]
int[] labels =
{
// 0, 0, 0, 1
-1, -1, -1, 1
};
// Create a Support Vector Machine for the given inputs
// (sparse machines should use 0 as the number of inputs)
var machine = new KernelSupportVectorMachine(new SparseLinear(), inputs: 0);
// Instantiate a new learning algorithm for SVMs
var smo = new SequentialMinimalOptimization(machine, inputs, labels);
// Set up the learning algorithm
smo.Complexity = 100000.0;
// Run
double error = smo.Run(); // should be zero
double[] predicted = inputs.Apply(machine.Compute).Sign();
// Outputs should be -1, -1, -1, +1
Constructs a new Linear kernel.
A constant intercept term. Default is 0.
Constructs a new Linear Kernel.
Gets or sets the kernel's intercept term.
Sparse Linear kernel function.
Sparse vector x in input space.
Sparse vector y in input space.
Dot product in feature (kernel) space.
Computes the squared distance in feature space
between two points given in input space.
Vector x in feature (kernel) space.
Vector y in feature (kernel) space.
Distance between x and y in input space.
Computes the product of two vectors given in sparse representation.
The first vector x.
The second vector y.
The inner product x * y between the given vectors.
Computes the squared Euclidean distance of two vectors given in sparse representation.
The first vector x.
The second vector y.
The squared Euclidean distance d² = |x - y|² between the given vectors.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Sparse Polynomial Kernel.
Constructs a new Sparse Polynomial kernel of a given degree.
The polynomial degree for this kernel.
The polynomial constant for this kernel. Default is 1.
Constructs a new Polynomial kernel of a given degree.
The polynomial degree for this kernel.
Gets or sets the kernel's polynomial degree.
Gets or sets the kernel's polynomial constant term.
Polynomial kernel function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Sparse Sigmoid Kernel.
Sigmoid kernels are not positive definite and therefore do not induce
a reproducing kernel Hilbert space. However, they have been successfully
used in practice (Schölkopf and Smola, 2002).
Constructs a Sparse Sigmoid kernel.
Alpha parameter.
Constant parameter.
Constructs a Sparse Sigmoid kernel.
Gets or sets the kernel's gamma parameter.
In a sigmoid kernel, gamma is a inner product
coefficient for the hyperbolic tangent function.
Gets or sets the kernel's constant term.
Sigmoid kernel function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Spherical Kernel.
The spherical kernel comes from a statistics perspective. It is an example
of an isotropic stationary kernel and is positive definite in R^3.
Gets or sets the kernel's sigma value.
Constructs a new Spherical Kernel.
Value for sigma.
Spherical Kernel Function
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Spherical Kernel Function
Distance z in input space.
Dot product in feature (kernel) space.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Infinite Spline Kernel function.
The Spline kernel is given as a piece-wise cubic
polynomial, as derived in the works by Gunn (1998).
-
Lin, Keng-Pei, and Ming-Syan Chen. "Efficient kernel approximation for large-scale support
vector machine classification." Proceedings of the Eleventh SIAM International Conference on
Data Mining. 2011. Available on: http://epubs.siam.org/doi/pdf/10.1137/1.9781611972818.19
Gets or sets the approximation degree
for this kernel. Default is 1024.
Gets or sets the Gaussian kernel being
approximated by this Taylor expansion.
Constructs a new kernel with the given sigma.
The kernel's sigma parameter.
The Gaussian approximation degree. Default is 1024.
Constructs a new kernel with the given sigma.
The original Gaussian kernel to be approximated.
The Gaussian approximation degree. Default is 1024.
Gaussian Kernel function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Projects an input point into feature space.
The input point to be projected into feature space.
The feature space representation of the given point.
Projects a set of input points into feature space.
The input points to be projected into feature space.
The feature space representation of the given points.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Elementwise multiplication of scalar a and vector b, storing in result.
The scalar to be multiplied.
The vector to be multiplied.
An array to store the result.
Compress a set of support vectors and weights into a single
parameter vector.
The weights associated with each support vector.
The support vectors.
The constant (bias) value.
A single parameter vector.
Computes the squared distance in input space
between two points given in feature space.
Vector x in feature (kernel) space.
Vector y in feature (kernel) space.
Squared distance between x and y in input space.
Computes the distance d(x,y) between points
and .
The first point x.
The second point y.
A double-precision value representing the distance d(x,y)
between and according
to the distance function implemented by this class.
Elementwise addition of a and b, storing in result.
The first vector to add.
The second vector to add.
An array to store the result.
The same vector passed as result.
Gets the number of parameters in the input vectors.
Creates an input vector from the given double values.
Elementwise multiplication of vector a and vector b, accumulating in result.
The vector to be multiplied.
The vector to be multiplied.
An array to store the result.
Converts the input vectors to a double-precision representation.
Tensor Product combination of Kernels.
Gets or sets the inner kernels used in this tensor kernel.
Constructs a new additive kernel.
Kernels to combine.
Tensor Product Kernel Combination function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Thin Spline Plate Kernel.
Thin plate splines (TPS) are a spline-based technique for data interpolation and smoothing.
Gets or sets the sigma constant for this kernel.
Constructs a new ThinSplinePlate Kernel.
The value for sigma.
Thin Spline Plate Kernel Function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Generalized T-Student Kernel.
The Generalized T-Student Kernel is a Mercer Kernel and thus forms
a positive semi-definite Kernel matrix (Boughorbel, 2004). It has
a similar form to the Inverse Multiquadric Kernel.
Gets or sets the degree of this kernel.
Constructs a new Generalized T-Student Kernel.
The kernel's degree.
Generalized T-Student Kernel function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Generalized T-Student Kernel function.
Distance z in input space.
Dot product in feature (kernel) space.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Wave Kernel.
The Wave kernel is symmetric positive semi-definite (Huang, 2008).
Constructs a new Wave Kernel.
Value for sigma.
Gets or sets the kernel's sigma value.
Wave Kernel Function.
Vector x in input space.
Vector y in input space.
Dot product in feature (kernel) space.
Wave Kernel Function.
Distance z in input space.
Dot product in feature (kernel) space.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Wavelet Kernel.
In Wavelet analysis theory, one of the common goals is to express or
approximate a signal or function using a family of functions generated
by dilations and translations of a function called the mother wavelet.
The Wavelet kernel uses a mother wavelet function together with dilation
and translation constants to produce such representations and build a
inner product which can be used by kernel methods. The default wavelet
used by this class is the mother function h(x) = cos(1.75x)*exp(-x²/2).
References:
-
Li Zhang, Weida Zhou, and Licheng Jiao; Wavelet Support Vector Machine. IEEE
Transactions on Systems, Man, and Cybernetics—Part B: Cybernetics, Vol. 34,
No. 1, February 2004.
-
Wikipedia contributors. "Generalized linear model." Wikipedia, The Free Encyclopedia.
The link function.
An input value.
The transformed input value.
The inverse of the link function.
A transformed value.
The reverse transformed value.
The logarithm of the inverse of the link function.
A transformed value.
The log of the reverse transformed value.
First derivative of the function.
The input value.
The first derivative of the input value.
First derivative of the
function expressed in terms of it's output.
The reverse transformed value.
The first derivative of the input value.
Identity link function.
The identity link function is associated with the
Normal distribution.
Link functions can be used in many models, such as in
and Support
Vector Machines.
Linear scaling coefficient a (intercept).
Linear scaling coefficient b (slope).
Creates a new Identity link function.
The variance value.
The mean value.
Creates a new Identity link function.
The Identity link function.
An input value.
The transformed input value.
The Identity link function is given by f(x) = (x - A) / B.
The mean function.
A transformed value.
The reverse transformed value.
The inverse Identity link function is given by g(x) = B * x + A.
The logarithm of the inverse of the link function.
A transformed value.
The log of the reverse transformed value.
First derivative of the function.
The input value.
The first derivative of the input value.
The first derivative of the identity link
function is given by f'(x) = B.
First derivative of the
function expressed in terms of it's output.
The reverse transformed value.
The first derivative of the input value.
The first derivative of the identity link function
in terms of y = f(x) is given by f'(y) = B.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Logit link function.
The Logit link function is associated with the
Binomial and
Multinomial distributions.
Linear scaling coefficient a (intercept).
Linear scaling coefficient b (slope).
Creates a new Logit link function.
The beta value. Default is 1.
The constant value. Default is 0.
Initializes a new instance of the class.
The Logit link function.
An input value.
The transformed input value.
The inverse Logit link function is given by
f(x) = (Math.Log(x / (1.0 - x)) - A) / B.
The Logit mean (activation) function.
A transformed value.
The reverse transformed value.
The inverse Logit link function is given by
g(x) = 1.0 / (1.0 + Math.Exp(-z) in
which z = B * x + A.
The logarithm of the inverse of the link function.
A transformed value.
The log of the reverse transformed value.
First derivative of the function.
The input value.
The first derivative of the input value.
The first derivative of the identity link
function is given by f'(x) = y * (1.0 - y)
where y = f(x) is the
Logit function.
First derivative of the mean function
expressed in terms of it's output.
The reverse transformed value.
The first derivative of the input value.
The first derivative of the Logit link function
in terms of y = f(x) is given by y * (1.0 - y).
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Inverse link function.
The inverse link function is associated with the
Exponential and
Gamma distributions.
Linear scaling coefficient a (intercept).
Linear scaling coefficient b (slope).
Creates a new Inverse link function.
The alpha value.
The constant value.
Creates a new Inverse link function.
The Inverse link function.
An input value.
The transformed input value.
The Inverse mean (activation) function.
A transformed value.
The reverse transformed value.
The logarithm of the inverse of the link function.
A transformed value.
The log of the reverse transformed value.
First derivative of the function.
The input value.
The first derivative of the input value.
First derivative of the
function expressed in terms of it's output.
The reverse transformed value.
The first derivative of the input value.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Set of statistics functions.
This class represents collection of common functions used in statistics.
Every Matrix function assumes data is organized in a table-like model,
where Columns represents variables and Rows represents a observation of
each variable.
Creates Tukey's box plot inner fence.
Creates Tukey's box plot outer fence.
Gets the rank of a sample, often used with order statistics.
Gets the rank of a sample, often used with order statistics.
Gets the number of ties and distinct elements in a rank vector.
Gets the number of ties and distinct elements in a rank vector.
Generates a random matrix.
The size of the square matrix.
The minimum value for a diagonal element.
The maximum size for a diagonal element.
A square, positive-definite matrix which
can be interpreted as a covariance matrix.
Computes the kernel distance for a kernel function even if it doesn't
implement the interface. Can be used to check
the proper implementation of the distance function.
The kernel function whose distance needs to be evaluated.
An input point x given in input space.
An input point y given in input space.
The distance between and in kernel (feature) space.
Generates the Standard Scores, also known as Z-Scores, from the given data.
A number multi-dimensional array containing the matrix values.
The Z-Scores for the matrix.
Generates the Standard Scores, also known as Z-Scores, from the given data.
A number multi-dimensional array containing the matrix values.
The mean value of the given values, if already known.
The values' standard deviation vector, if already known.
The Z-Scores for the matrix.
Generates the Standard Scores, also known as Z-Scores, from the given data.
A number multi-dimensional array containing the matrix values.
The Z-Scores for the matrix.
Generates the Standard Scores, also known as Z-Scores, from the given data.
A number multi-dimensional array containing the matrix values.
The mean value of the given values, if already known.
The values' standard deviation vector, if already known.
The Z-Scores for the matrix.
Centers an observation, subtracting the empirical
mean from each element in the observation vector.
An array of double precision floating-point numbers.
The destination array where the result of this operation should be stored.
Centers an observation, subtracting the empirical
mean from each element in the observation vector.
An array of double precision floating-point numbers.
The mean of the , if already known.
The destination array where the result of this operation should be stored.
Centers column data, subtracting the empirical mean from each variable.
A matrix where each column represent a variable and each row represent a observation.
True to perform the operation in place, altering the original input matrix.
Centers column data, subtracting the empirical mean from each variable.
A matrix where each column represent a variable and each row represent a observation.
The mean value of the given values, if already known.
True to perform the operation in place, altering the original input matrix.
Centers column data, subtracting the empirical mean from each variable.
A matrix where each column represent a variable and each row represent a observation.
True to perform the operation in place, altering the original input matrix.
Centers column data, subtracting the empirical mean from each variable.
A matrix where each column represent a variable and each row represent a observation.
The mean value of the given values, if already known.
True to perform the operation in place, altering the original input matrix.
Standardizes column data, removing the empirical standard deviation from each variable.
This method does not remove the empirical mean prior to execution.
An array of double precision floating-point numbers.
True to perform the operation in place,
altering the original input matrix.
Standardizes column data, removing the empirical standard deviation from each variable.
This method does not remove the empirical mean prior to execution.
An array of double precision floating-point numbers.
The standard deviation of the given
, if already known.
True to perform the operation in place, altering the original input matrix.
Standardizes column data, removing the empirical standard deviation from each variable.
This method does not remove the empirical mean prior to execution.
A matrix where each column represent a variable and each row represent a observation.
True to perform the operation in place, altering the original input matrix.
Standardizes column data, removing the empirical standard deviation from each variable.
This method does not remove the empirical mean prior to execution.
A matrix where each column represent a variable and each row represent a observation.
The values' standard deviation vector, if already known.
The minimum value that will be taken as the standard deviation in case the deviation is too close to zero.
True to perform the operation in place, altering the original input matrix.
Standardizes column data, removing the empirical standard deviation from each variable.
This method does not remove the empirical mean prior to execution.
A matrix where each column represent a variable and each row represent a observation.
True to perform the operation in place, altering the original input matrix.
Standardizes column data, removing the empirical standard deviation from each variable.
This method does not remove the empirical mean prior to execution.
A matrix where each column represent a variable and each row represent a observation.
The minimum value that will be taken as the standard deviation in case the deviation is too close to zero.
The values' standard deviation vector, if already known.
True to perform the operation in place, altering the original input matrix.
Obsolete. Please use instead.
Obsolete. Please use instead.
Obsolete. Please use instead.
Obsolete. Please use instead.
Obsolete. Please use instead.
Obsolete. Please use instead.
Obsolete. Please use instead.
Obsolete. Please use instead.
Obsolete. Please use instead.
Gets the coefficient of determination, as known as the R-Squared (R²)
The coefficient of determination is used in the context of statistical models
whose main purpose is the prediction of future outcomes on the basis of other
related information. It is the proportion of variability in a data set that
is accounted for by the statistical model. It provides a measure of how well
future outcomes are likely to be predicted by the model.
The R^2 coefficient of determination is a statistical measure of how well the
regression approximates the real data points. An R^2 of 1.0 indicates that the
regression perfectly fits the data.
Obsolete. Please use instead.
Obsolete. Please use instead.
Obsolete. Please use instead.
Obsolete. Please use instead.
Obsolete. Please use instead.
Obsolete. Please use instead.
Obsolete. Please use instead.
Computes the whitening transform for the given data, making
its covariance matrix equals the identity matrix.
A matrix where each column represent a
variable and each row represent a observation.
The base matrix used in the
transformation.
The transformed source data (which now has unit variance).
Computes the whitening transform for the given data, making
its covariance matrix equals the identity matrix.
A matrix where each column represent a
variable and each row represent a observation.
The base matrix used in the
transformation.
The transformed source data (which now has unit variance).
Creates a new distribution that has been fit to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Creates a new distribution that has been fit to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Creates a new distribution that has been fit to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Creates a new distribution that has been fit to a given set of observations.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Creates a new distribution that has been fit to a given set of observations.
The distribution whose parameters should be fitted to the samples.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Creates a new distribution that has been fit to a given set of observations.
The distribution whose parameters should be fitted to the samples.
The array of observations to fit the model against. The array
elements can be either of type double (for univariate data) or
type double[] (for multivariate data).
The weight vector containing the weight for each of the samples.
Optional arguments which may be used during fitting, such
as regularization constants and additional parameters.
Contains statistical models with direct applications in machine learning, such as
Hidden Markov Models,
Conditional Random Fields, Hidden Conditional
Random Fields and linear and
logistic regressions.
The main algorithms and techniques available on this namespaces are certainly
the hidden Markov models.
The Accord.NET Framework contains one of the most popular and well-tested
offerings for creating, training and validating Markov models using either
discrete observations or any arbitrary discrete, continuous or mixed probability distributions to
model the observations.
This namespace also brings
Conditional Random Fields, that alongside the Markov models can be
used to build sequence classifiers,
perform gesture recognition, and can even be combined with neural networks
to create hybrid models. Other
models include regression
and survival models.
The namespace class diagram is shown below.
Please note that class diagrams for each of the inner namespaces are
also available within their own documentation pages.
Contains classes related to Conditional Random
Fields, Hidden Conditional Random
Fields and their learning
algorithms.
The namespace class diagram is shown below.
Please note that class diagrams for each of the inner namespaces are
also available within their own documentation pages.
Utility methods to assist in the creating of s.
Creates a new from the given .
The classifier.
A that implements
exactly the same model as the given .
Creates a new from the given .
The classifier.
A that implements
exactly the same model as the given .
Creates a new from the given .
The classifier.
A that implements
exactly the same model as the given .
Creates a new from the given .
The classifier.
A that implements
exactly the same model as the given .
Creates a new from the given .
The classifier.
A that implements
exactly the same model as the given .
Creates a new from the given .
The classifier.
A that implements
exactly the same model as the given .
Contains learning algorithms for CRFs and
HCRFs, such as
Conjugate Gradient,
L-BFGS and
RProp-based learning.
The namespace class diagram is shown below.
Abstract base class for hidden Conditional Random Fields algorithms.
Gets or sets a cancellation token that can be used to
stop the learning algorithm while it is running.
Gets or sets the potential function to be used if this learning algorithm
needs to create a new .
Gets or sets the model being trained.
Learns a model that can map the given inputs to the given outputs.
The model inputs.
The desired outputs associated with each inputs.
The weight of importance for each input-output pair (if supported by the learning algorithm).
A model that has learned how to produce given .
Creates an instance of the model to be learned. Inheritors of this abstract
class must define this method so new models can be created from the training data.
Inheritors should implement the actual learning algorithm in this method.
Base class for Hidden Conditional Random Fields learning algorithms based
on gradient optimization algorithms.
Gets the optimization algorithm being used.
Gets or sets the amount of the parameter weights
which should be included in the objective function.
Default is 0 (do not include regularization).
Gets or sets the tolerance value used to determine
whether the algorithm has converged.
The tolerance.
Gets or sets the maximum number of iterations
performed by the learning algorithm.
The maximum iterations.
Gets or sets whether the algorithm has converged.
true if this instance has converged; otherwise, false.
Gets or sets the parallelization options for this algorithm.
The parallel options.
Constructs a new L-BFGS learning algorithm.
Inheritors of this class should create the optimization algorithm in this
method, using the current and
settings.
Runs the learning algorithm.
Online learning is not supported.
Runs the learning algorithm with the specified input
training observations and corresponding output labels.
The training observations.
The observation's labels.
Online learning is not supported.
Performs application-defined tasks associated with freeing,
releasing, or resetting unmanaged resources.
Releases unmanaged resources and performs other cleanup operations before
the is reclaimed by garbage
collection.
Releases unmanaged and - optionally - managed resources
true to release both managed
and unmanaged resources; false to release only unmanaged
resources.
Linear Gradient calculator class for
Hidden Conditional Random Fields.
The type of the observations being modeled.
Gets or sets the inputs to be used in the next
call to the Objective or Gradient functions.
Gets or sets the outputs to be used in the next
call to the Objective or Gradient functions.
Gets or sets the current parameter
vector for the model being learned.
Gets the error computed in the last call
to the gradient or objective functions.
Gets or sets the amount of the parameter weights
which should be included in the objective function.
Default is 0 (do not include regularization).
Gets the model being trained.
Initializes a new instance of the class.
Initializes a new instance of the class.
The model to be trained.
Computes the gradient (vector of derivatives) vector for
the cost function, which may be used to guide optimization.
The parameter vector lambda to use in the model.
The inputs to compute the cost function.
The respective outputs to compute the cost function.
The value of the gradient vector for the given parameters.
Computes the gradient (vector of derivatives) vector for
the cost function, which may be used to guide optimization.
The parameter vector lambda to use in the model.
The inputs to compute the cost function.
The respective outputs to compute the cost function.
The value of the gradient vector for the given parameters.
Computes the gradient using the input/outputs stored in this object.
This method is not thread safe.
The parameter vector lambda to use in the model.
The value of the gradient vector for the given parameters.
Computes the gradient using the input/outputs stored in this object.
This method is thread-safe.
The value of the gradient vector for the given parameters.
Computes the objective (cost) function for the Hidden
Conditional Random Field (negative log-likelihood).
The parameter vector lambda to use in the model.
The inputs to compute the cost function.
The respective outputs to compute the cost function.
The value of the objective function for the given parameters.
Computes the objective (cost) function for the Hidden
Conditional Random Field (negative log-likelihood) using
the input/outputs stored in this object.
The parameter vector lambda to use in the model.
Computes the objective (cost) function for the Hidden
Conditional Random Field (negative log-likelihood) using
the input/outputs stored in this object.
Performs application-defined tasks associated with freeing,
releasing, or resetting unmanaged resources.
Releases unmanaged resources and performs other cleanup operations before
the is reclaimed by garbage
collection.
Releases unmanaged and - optionally - managed resources
true to release both managed
and unmanaged resources; false to release only unmanaged
resources.
Common interface for Hidden Conditional Random Fields learning algorithms.
For an example on how to learn Hidden Conditional Random Fields, please see the
Hidden Resilient Gradient Learning
page. All learning algorithms can be utilized in a similar manner.
Runs one iteration of the learning algorithm with the
specified input training observation and corresponding
output label.
The training observations.
The observation labels.
The error in the last iteration.
Runs one iteration of learning algorithm with the specified
input training observations and corresponding output labels.
The training observations.
The observations' labels.
The error in the last iteration.
Runs the learning algorithm with the specified input
training observation and corresponding output label
until convergence.
The training observations.
The observations' labels.
The error in the last iteration.
Common interface for Conditional Random Fields learning algorithms.
Runs the learning algorithm with the specified input
training observations and corresponding output labels.
The training observations.
The observation's labels.
Conjugate Gradient learning algorithm for
Hidden Conditional Hidden Fields.
For an example on how to learn Hidden Conditional Random Fields, please see the
Hidden Resilient Gradient Learning
page. All learning algorithms can be utilized in a similar manner.
Please use HasConverged instead.
Gets the current iteration number.
The current iteration.
Please use MaxIterations instead.
Occurs when the current learning progress has changed.
Constructs a new Conjugate Gradient learning algorithm.
Inheritors of this class should create the optimization algorithm in this
method, using the current and
settings.
ConjugateGradient.
Quasi-Newton (L-BFGS) learning algorithm for
Hidden Conditional Hidden Fields.
The type of the observations.
-
C. Souza, Sequence Classifiers in C# - Part II: Hidden Conditional Random Fields. CodeProject. Available at:
http://www.codeproject.com/Articles/559535/Sequence-Classifiers-in-Csharp-Part-II-Hidden-Cond
-
Chan, Tony F.; Golub, Gene H.; LeVeque, Randall J. (1983). Algorithms for
Computing the Sample Variance: Analysis and Recommendations. The American
Statistician 37, 242-247.
-
H. Lee, J. Kim, An HMM-based threshold model approach for gesture
recognition, IEEE Trans. Pattern Anal. Mach. Intell. 21 (10) (1999)
961–973.
Gets or sets a value governing the rejection given by
a threshold model (if present). Increasing this value
will result in higher rejection rates. Default is 1.
Gets the collection of models specialized in each
class of the sequence classification problem.
Gets the Hidden Markov
Model implementation responsible for recognizing
each of the classes given the desired class label.
The class label of the model to get.
Gets the number of classes which can be recognized by this classifier.
Gets the prior distribution assumed for the classes.
Computes the log-likelihood that the given input vector
belongs to its decided class.
Computes the likelihood that the given input vector
belongs to its decided class.
Computes the log-likelihood that the given input vector
belongs to its decided class.
Computes the probability that the given input vector
belongs to its decided class.
Computes the log-likelihood that the given input vector
belongs to the specified .
The input vector.
The index of the class whose score will be computed.
Computes the probabilities that the given input
vector belongs to each of the possible classes.
The input vector.
The decided class for the input.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Predicts a class label vector for the given input vector, returning the
log-likelihoods of the input vector belonging to each possible class.
A set of input vectors.
The decided class for the input.
An array where the probabilities will be stored,
avoiding unnecessary memory allocations.
Computes a class-label decision for a given .
The input vector that should be classified into
one of the possible classes.
A class-label that best described according
to this classifier.
Returns an enumerator that iterates through the models in the classifier.
A that
can be used to iterate through the collection.
Returns an enumerator that iterates through the models in the classifier.
A that
can be used to iterate through the collection.
Forward-Backward algorithms for Hidden Markov Models.
Forward-Backward algorithms for Hidden Markov Models.
Computes Forward probabilities for a given hidden Markov model and a set of observations.
Computes Forward probabilities for a given hidden Markov model and a set of observations.
Computes Forward probabilities for a given hidden Markov model and a set of observations.
Computes Forward probabilities for a given hidden Markov model and a set of observations.
Computes Backward probabilities for a given hidden Markov model and a set of observations.
Computes Backward probabilities for a given hidden Markov model and a set of observations.
Computes Forward probabilities for a given hidden Markov model and a set of observations.
Computes Forward probabilities for a given hidden Markov model and a set of observations.
Computes Forward probabilities for a given hidden Markov model and a set of observations.
Computes Backward probabilities for a given hidden Markov model and a set of observations.
Computes Backward probabilities for a given hidden Markov model and a set of observations.
Computes Backward probabilities for a given hidden Markov model and a set of observations (no scaling).
Computes Forward probabilities for a given hidden Markov model and a set of observations.
Computes Forward probabilities for a given hidden Markov model and a set of observations (no scaling).
Computes Forward probabilities for a given hidden Markov model and a set of observations.
Computes Forward probabilities for a given hidden Markov model and a set of observations.
Computes Forward probabilities for a given hidden Markov model and a set of observations.
Computes Forward probabilities for a given hidden Markov model and a set of observations.
Computes Forward probabilities for a given hidden Markov model and a set of observations.
Computes Forward probabilities for a given hidden Markov model and a set of observations.
Computes Forward probabilities for a given hidden Markov model and a set of observations.
Computes Backward probabilities for a given hidden Markov model and a set of observations.
Computes Backward probabilities for a given hidden Markov model and a set of observations (no scaling).
Computes Backward probabilities for a given hidden Markov model and a set of observations.
Computes Backward probabilities for a given hidden Markov model and a set of observations (no scaling).
Computes Backward probabilities for a given hidden Markov model and a set of observations.
Computes Backward probabilities for a given hidden Markov model and a set of observations.
Computes Forward probabilities for a given hidden Markov model and a set of observations.
Computes Forward probabilities for a given hidden Markov model and a set of observations.
Computes Forward probabilities for a given hidden Markov model and a set of observations.
Computes Forward probabilities for a given hidden Markov model and a set of observations.
Computes Forward probabilities for a given hidden Markov model and a set of observations.
Computes Forward probabilities for a given hidden Markov model and a set of observations.
Computes Backward probabilities for a given hidden Markov model and a set of observations.
Computes Backward probabilities for a given hidden Markov model and a set of observations.
Computes Backward probabilities for a given hidden Markov model and a set of observations (no scaling).
Computes Backward probabilities for a given hidden Markov model and a set of observations.
Computes Backward probabilities for a given hidden Markov model and a set of observations.
Computes Backward probabilities for a given hidden Markov model and a set of observations (no scaling).
Arbitrary-density Hidden Markov Model Set for Sequence Classification.
This class uses a set of density hidden
Markov models to classify sequences of real (double-precision floating point)
numbers or arrays of those numbers. Each model will try to learn and recognize each
of the different output classes. For examples and details on how to learn such models,
please take a look on the documentation for
.
For the discrete version of this classifier, please see its non-generic counterpart
.
Examples are available at the respective learning algorithm pages. For
example, see .
Creates a new Sequence Classifier with the given number of classes.
The number of classes in the classifier.
Creates a new Sequence Classifier with the given number of classes.
The number of classes in the classifier.
An array specifying the number of hidden states for each
of the classifiers. By default, and Ergodic topology will be used.
The initial probability distributions for the hidden states.
For multivariate continuous density distributions, such as Normal mixtures, the
choice of initial values is crucial for a good performance.
Creates a new Sequence Classifier with the given number of classes.
The number of classes in the classifier.
The topology of the hidden states. A forward-only topology
is indicated to sequence classification problems, such as speech recognition.
The initial probability distributions for the hidden states.
For multivariate continuous density distributions, such as Normal mixtures, the
choice of initial values is crucial for a good performance.
Creates a new Sequence Classifier with the given number of classes.
The number of classes in the classifier.
An array specifying the number of hidden states for each
of the classifiers. By default, and Ergodic topology will be used.
The initial probability distributions for the hidden states.
For multivariate continuous density distributions, such as Normal mixtures, the
choice of initial values is crucial for a good performance.
Creates a new Sequence Classifier with the given number of classes.
The number of classes in the classifier.
The topology of the hidden states. A forward-only topology
is indicated to sequence classification problems, such as speech recognition.
The initial probability distributions for the hidden states.
For multivariate continuous density distributions, such as Normal mixtures, the
choice of initial values is crucial for a good performance.
Creates a new Sequence Classifier with the given number of classes.
The number of classes in the classifier.
The topology of the hidden states. A forward-only topology
is indicated to sequence classification problems, such as speech recognition.
The initial probability distributions for the hidden states.
For multivariate continuous density distributions, such as Normal mixtures, the
choice of initial values is crucial for a good performance.
Creates a new Sequence Classifier with the given number of classes.
The number of classes in the classifier.
The topology of the hidden states. A forward-only topology
is indicated to sequence classification problems, such as speech recognition.
The initial probability distributions for the hidden states.
For multivariate continuous density distributions, such as Normal mixtures, the
choice of initial values is crucial for a good performance.
Creates a new Sequence Classifier with the given number of classes.
The number of classes in the classifier.
The topology of the hidden states. A forward-only topology
is indicated to sequence classification problems, such as speech recognition.
The initial probability distributions for the hidden states.
For multivariate continuous density distributions, such as Normal mixtures, the
choice of initial values is crucial for a good performance.
Creates a new Sequence Classifier with the given number of classes.
The number of classes in the classifier.
The topology of the hidden states. A forward-only topology
is indicated to sequence classification problems, such as speech recognition.
The initial probability distributions for the hidden states.
For multivariate continuous density distributions, such as Normal mixtures, the
choice of initial values is crucial for a good performance.
The class labels for each of the models.
Creates a new Sequence Classifier with the given number of classes.
The models specializing in each of the classes of
the classification problem.
Creates a new Sequence Classifier with the given number of classes.
The number of classes in the classifier.
The topology of the hidden states. A forward-only topology
is indicated to sequence classification problems, such as speech recognition.
The initial probability distributions for the hidden states.
For multivariate continuous density distributions, such as Normal mixtures, the
choice of initial values is crucial for a good performance.
The class labels for each of the models.
Algorithms for solving -related
problems, such as sequence decoding and likelihood evaluation.
Uses the Viterbi algorithm (max-sum) to find the hidden states of a
sequence of observations and to evaluate its likelihood. The likelihood
will be computed along the Viterbi path.
Uses the forward algorithm (sum-prod) to compute the likelihood of a sequence.
The likelihood will be computed considering every possible path in the model (default).
When set, calling LogLikelihoods will give the model's posterior distribution.
Hidden Markov Model for any kind of observations (not only discrete).
Hidden Markov Models (HMM) are stochastic methods to model temporal and sequence
data. They are especially known for their application in temporal pattern recognition
such as speech, handwriting, gesture recognition, part-of-speech tagging, musical
score following, partial discharges and bioinformatics.
This page refers to the arbitrary-density (continuous emission distributions) version
of the model. For discrete distributions, please see .
Dynamical systems of discrete nature assumed to be governed by a Markov chain emits
a sequence of observable outputs. Under the Markov assumption, it is also assumed that
the latest output depends only on the current state of the system. Such states are often
not known from the observer when only the output values are observable.
Hidden Markov Models attempt to model such systems and allow, among other things,
-
To infer the most likely sequence of states that produced a given output sequence,
-
Infer which will be the most likely next state (and thus predicting the next output),
-
Calculate the probability that a given sequence of outputs originated from the system
(allowing the use of hidden Markov models for sequence classification).
The “hidden” in Hidden Markov Models comes from the fact that the observer does not
know in which state the system may be in, but has only a probabilistic insight on where
it should be.
The arbitrary-density Hidden Markov Model uses any probability density function (such
as Gaussian
Mixture Model) for
computing the state probability. In other words, in a continuous HMM the matrix of emission
probabilities B is replaced by an array of either discrete or continuous probability density
functions.
If a general
discrete distribution is used as the underlying probability density function, the
model becomes equivalent to the discrete Hidden Markov Model.
For a more thorough explanation on some fundamentals on how Hidden Markov Models work,
please see the documentation page. To learn a Markov
model, you can find a list of both supervised and
unsupervised learning algorithms in the
namespace.
References:
-
Wikipedia contributors. "Linear regression." Wikipedia, the Free Encyclopedia.
Available at: http://en.wikipedia.org/wiki/Hidden_Markov_model
-
Bishop, Christopher M.; Pattern Recognition and Machine Learning.
Springer; 1st ed. 2006.
The example below reproduces the same example given in the Wikipedia
entry for the Viterbi algorithm (http://en.wikipedia.org/wiki/Viterbi_algorithm).
As an arbitrary density model, one can use it with any available
probability distributions, including with a discrete probability. In the
following example, the generic model is used with a
to reproduce the same example given in .
Below, the model's parameters are initialized manually. However, it is possible to learn
those automatically using .
-
H. Lee, J. Kim, An HMM-based threshold model approach for gesture
recognition, IEEE Trans. Pattern Anal. Mach. Intell. 21 (10) (1999)
961–973.
Gets or sets a value governing the rejection given by
a threshold model (if present). Increasing this value
will result in higher rejection rates. Default is 1.
Gets the collection of models specialized in each
class of the sequence classification problem.
Gets the Hidden Markov
Model implementation responsible for recognizing
each of the classes given the desired class label.
The class label of the model to get.
Gets the number of classes which can be recognized by this classifier.
Gets the prior distribution assumed for the classes.
Computes the most likely class for a given sequence.
The sequence of observations.
Return the label of the given sequence, or -1 if it has
been rejected by the threshold model.
Computes the most likely class for a given sequence.
The sequence of observations.
The probability of the assigned class.
Return the label of the given sequence, or -1 if it has
been rejected by the threshold model.
Computes the most likely class for a given sequence.
The sequence of observations.
The probabilities for each class.
Return the label of the given sequence, or -1 if it has
been rejected by the threshold model.
Computes the log-likelihood that a sequence
belongs to a given class according to this
classifier.
The sequence of observations.
The output class label.
The log-likelihood of the sequence belonging to the given class.
Computes the log-likelihood that a sequence
belongs any of the classes in the classifier.
The sequence of observations.
The log-likelihood of the sequence belonging to the classifier.
Computes the log-likelihood of a set of sequences
belonging to their given respective classes according
to this classifier.
A set of sequences of observations.
The output class label for each sequence.
The log-likelihood of the sequences belonging to the given classes.
Returns an enumerator that iterates through the models in the classifier.
A that
can be used to iterate through the collection.
Returns an enumerator that iterates through the models in the classifier.
A that
can be used to iterate through the collection.
Common interface for sequence classifiers using
hidden Markov models.
Computes the most likely class for a given sequence.
The sequence of observations.
The class responsibilities (or
the probability of the sequence to belong to each class). When
using threshold models, the sum of the probabilities will not
equal one, and the amount left was the threshold probability.
If a threshold model is not being used, the array should sum to
one.
Return the label of the given sequence, or -1 if it has
been rejected.
Gets the number of classes which can be recognized by this classifier.
Obsolete. Please use instead.
Constructs a new Hidden Markov Model with arbitrary-density state probabilities.
A object specifying the initial values of the matrix of transition
probabilities A and initial state probabilities pi to be used by this model.
The initial emission probability distribution to be used by each of the states. This
initial probability distribution will be cloned across all states.
Constructs a new Hidden Markov Model with arbitrary-density state probabilities.
A object specifying the initial values of the matrix of transition
probabilities A and initial state probabilities pi to be used by this model.
The initial emission probability distributions for each state.
Constructs a new Hidden Markov Model with arbitrary-density state probabilities.
The transitions matrix A for this model.
The emissions matrix B for this model.
The initial state probabilities for this model.
Set to true if the matrices are given with logarithms of the
intended probabilities; set to false otherwise. Default is false.
Constructs a new Hidden Markov Model with arbitrary-density state probabilities.
The number of states for the model.
A initial distribution to be copied to all states in the model.
Gets the number of dimensions in the
probability distributions for the states.
Gets the Emission matrix (B) for this model.
Calculates the most likely sequence of hidden states
that produced the given observation sequence.
Decoding problem. Given the HMM M = (A, B, pi) and the observation sequence
O = {o1,o2, ..., oK}, calculate the most likely sequence of hidden states Si
that produced this observation sequence O. This can be computed efficiently
using the Viterbi algorithm.
A sequence of observations.
The sequence of states that most likely produced the sequence.
Calculates the most likely sequence of hidden states
that produced the given observation sequence.
Decoding problem. Given the HMM M = (A, B, pi) and the observation sequence
O = {o1,o2, ..., oK}, calculate the most likely sequence of hidden states Si
that produced this observation sequence O. This can be computed efficiently
using the Viterbi algorithm.
A sequence of observations.
The log-likelihood along the most likely sequence.
The sequence of states that most likely produced the sequence.
Calculates the probability of each hidden state for each
observation in the observation vector.
If there are 3 states in the model, and the
array contains 5 elements, the resulting vector will contain 5 vectors of
size 3 each. Each vector of size 3 will contain probability values that sum
up to one. By following those probabilities in order, we may decode those
probabilities into a sequence of most likely states. However, the sequence
of obtained states may not be valid in the model.
A sequence of observations.
A vector of the same size as the observation vectors, containing
the probabilities for each state in the model for the current observation.
If there are 3 states in the model, and the
array contains 5 elements, the resulting vector will contain 5 vectors of
size 3 each. Each vector of size 3 will contain probability values that sum
up to one.
Calculates the probability of each hidden state for each observation
in the observation vector, and uses those probabilities to decode the
most likely sequence of states for each observation in the sequence
using the posterior decoding method. See remarks for details.
If there are 3 states in the model, and the
array contains 5 elements, the resulting vector will contain 5 vectors of
size 3 each. Each vector of size 3 will contain probability values that sum
up to one. By following those probabilities in order, we may decode those
probabilities into a sequence of most likely states. However, the sequence
of obtained states may not be valid in the model.
A sequence of observations.
The sequence of states most likely associated with each
observation, estimated using the posterior decoding method.
A vector of the same size as the observation vectors, containing
the probabilities for each state in the model for the current observation.
If there are 3 states in the model, and the
array contains 5 elements, the resulting vector will contain 5 vectors of
size 3 each. Each vector of size 3 will contain probability values that sum
up to one.
Calculates the likelihood that this model has generated the given sequence.
Evaluation problem. Given the HMM M = (A, B, pi) and the observation
sequence O = {o1, o2, ..., oK}, calculate the probability that model
M has generated sequence O. This can be computed efficiently using the
either the Viterbi or the Forward algorithms.
A sequence of observations.
The log-likelihood that the given sequence has been generated by this model.
Calculates the log-likelihood that this model has generated the
given observation sequence along the given state path.
A sequence of observations.
A sequence of states.
The log-likelihood that the given sequence of observations has
been generated by this model along the given sequence of states.
Predicts the next observation occurring after a given observation sequence.
A sequence of observations. Predictions will be made regarding
the next observations that should be coming after the last observation in this sequence.
Predicts the next observation occurring after a given observation sequence.
A sequence of observations. Predictions will be made regarding
the next observation that should be coming after the last observation in this sequence.
Predicts the next observation occurring after a given observation sequence.
A sequence of observations. Predictions will be made regarding
the next observations that should be coming after the last observation in this sequence.
The log-likelihood of the given sequence, plus the predicted
next observation. Exponentiate this value (use the System.Math.Exp function) to obtain
a likelihood value.
Predicts the next observation occurring after a given observation sequence.
A sequence of observations. Predictions will be made regarding
the next observations that should be coming after the last observation in this sequence.
The log-likelihood of the given sequence, plus the predicted
next observation. Exponentiate this value (use the System.Math.Exp function) to obtain
a likelihood value.
Predicts the next observation occurring after a given observation sequence.
A sequence of observations. Predictions will be made regarding
the next observations that should be coming after the last observation in this sequence.
The log-likelihood of the given sequence, plus the predicted
next observation. Exponentiate this value (use the System.Math.Exp function) to obtain
a likelihood value.
The continuous probability distribution describing the next observations
that are likely to be generated. Taking the mode of this distribution might give the most likely
next value in the observed sequence.
Predicts the next observation occurring after a given observation sequence.
A sequence of observations. Predictions will be made regarding
the next observations that should be coming after the last observation in this sequence.
The continuous probability distribution describing the next observations
that are likely to be generated. Taking the mode of this distribution might give the most likely
next value in the observed sequence.
Predicts the next observation occurring after a given observation sequence.
A sequence of observations. Predictions will be made regarding
the next observations that should be coming after the last observation in this sequence.
The continuous probability distribution describing the next observations
that are likely to be generated. Taking the mode of this distribution might give the most likely
next value in the observed sequence.
Predicts the next observation occurring after a given observation sequence.
A sequence of observations. Predictions will be made regarding
the next observations that should be coming after the last observation in this sequence.
The log-likelihood of the given sequence, plus the predicted
next observation. Exponentiate this value (use the System.Math.Exp function) to obtain
a likelihood value.
The continuous probability distribution describing the next observations
that are likely to be generated. Taking the mode of this distribution might give the most likely
next value in the observed sequence.
Predicts the next observations occurring after a given observation sequence.
A sequence of observations. Predictions will be made regarding
the next observations that should be coming after the last observation in this sequence.
The number of observations to be predicted. Default is 1.
The log-likelihood of the given sequence, plus the predicted
next observation. Exponentiate this value (use the System.Math.Exp function) to obtain
a likelihood value.
Predicts the next observations occurring after a given observation sequence.
A sequence of observations. Predictions will be made regarding
the next observations that should be coming after the last observation in this sequence.
The number of observations to be predicted. Default is 1.
The log-likelihood of the given sequence, plus the predicted
next observations. Exponentiate this value (use the System.Math.Exp function) to obtain
a likelihood value.
Generates a random vector of observations from the model.
The number of samples to generate.
A random vector of observations drawn from the model.
Generates a random vector of observations from the model.
The number of samples to generate.
The log-likelihood of the generated observation sequence.
The Viterbi path of the generated observation sequence.
A random vector of observations drawn from the model.
Predicts the next observation occurring after a given observation sequence.
Predicts the next observation occurring after a given observation sequence.
Predicts the next observation occurring after a given observation sequence.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Saves the hidden Markov model to a stream.
The stream to which the model is to be serialized.
Saves the hidden Markov model to a stream.
The stream to which the model is to be serialized.
Loads a hidden Markov model from a stream.
The stream from which the model is to be deserialized.
The deserialized model.
Loads a hidden Markov model from a file.
The path to the file from which the model is to be deserialized.
The deserialized model.
Obsolete. Please use instead.
Gets the number of dimensions of the
observations handled by this classifier.
Creates a new Sequence Classifier with the given number of classes.
The number of classes in the classifier.
An array specifying the number of hidden states for each
of the classifiers. By default, and Ergodic topology will be used.
The initial probability distributions for the hidden states.
For multivariate continuous density distributions, such as Normal mixtures, the
choice of initial values is crucial for a good performance.
Creates a new Sequence Classifier with the given number of classes.
The number of classes in the classifier.
The topology of the hidden states. A forward-only topology
is indicated to sequence classification problems, such as speech recognition.
The initial probability distributions for the hidden states.
For multivariate continuous density distributions, such as Normal mixtures, the
choice of initial values is crucial for a good performance.
Creates a new Sequence Classifier with the given number of classes.
The number of classes in the classifier.
The topology of the hidden states. A forward-only topology
is indicated to sequence classification problems, such as speech recognition.
The initial probability distributions for the hidden states.
For multivariate continuous density distributions, such as Normal mixtures, the
choice of initial values is crucial for a good performance.
Creates a new Sequence Classifier with the given number of classes.
The number of classes in the classifier.
The topology of the hidden states. A forward-only topology
is indicated to sequence classification problems, such as speech recognition.
The initial probability distributions for the hidden states.
For multivariate continuous density distributions, such as Normal mixtures, the
choice of initial values is crucial for a good performance.
Creates a new Sequence Classifier with the given number of classes.
The number of classes in the classifier.
The topology of the hidden states. A forward-only topology
is indicated to sequence classification problems, such as speech recognition.
The initial probability distributions for the hidden states.
For multivariate continuous density distributions, such as Normal mixtures, the
choice of initial values is crucial for a good performance.
Creates a new Sequence Classifier with the given number of classes.
The number of classes in the classifier.
The topology of the hidden states. A forward-only topology
is indicated to sequence classification problems, such as speech recognition.
The initial probability distributions for the hidden states.
For multivariate continuous density distributions, such as Normal mixtures, the
choice of initial values is crucial for a good performance.
The class labels for each of the models.
Creates a new Sequence Classifier with the given number of classes.
The models specializing in each of the classes of
the classification problem.
Creates a new Sequence Classifier with the given number of classes.
The number of classes in the classifier.
The topology of the hidden states. A forward-only topology
is indicated to sequence classification problems, such as speech recognition.
The initial probability distributions for the hidden states.
For multivariate continuous density distributions, such as Normal mixtures, the
choice of initial values is crucial for a good performance.
The class labels for each of the models.
Computes the most likely class for a given sequence.
The sequence of observations.
Return the label of the given sequence, or -1 if it has
been rejected by the
threshold model.
Computes the most likely class for a given sequence.
The sequence of observations.
The probability of the assigned class.
Return the label of the given sequence, or -1 if it has
been rejected by the
threshold model.
Computes the most likely class for a given sequence.
The sequence of observations.
The class responsibilities (or
the probability of the sequence to belong to each class). When
using threshold models, the sum of the probabilities will not
equal one, and the amount left was the threshold probability.
If a threshold model is not being used, the array should sum to
one.
Return the label of the given sequence, or -1 if it has
been rejected by the
threshold model.
Computes the log-likelihood of a sequence
belong to a given class according to this
classifier.
The sequence of observations.
The output class label.
The log-likelihood of the sequence belonging to the given class.
Computes the log-likelihood that a sequence
belongs any of the classes in the classifier.
The sequence of observations.
The log-likelihood of the sequence belonging to the classifier.
Computes the log-likelihood of a set of sequences
belonging to their given respective classes according
to this classifier.
A set of sequences of observations.
The output class label for each sequence.
The log-likelihood of the sequences belonging to the given classes.
Saves the classifier to a stream.
The stream to which the classifier is to be serialized.
Saves the classifier to a stream.
The stream to which the classifier is to be serialized.
Loads a classifier from a stream.
The stream from which the classifier is to be deserialized.
The deserialized classifier.
Loads a classifier from a file.
The path to the file from which the classifier is to be deserialized.
The deserialized classifier.
Discrete-density Hidden Markov Model.
Hidden Markov Models (HMM) are stochastic methods to model temporal and sequence
data. They are especially known for their application in temporal pattern recognition
such as speech, handwriting, gesture recognition, part-of-speech tagging, musical
score following, partial discharges and bioinformatics.
This page refers to the discrete-density version of the model. For arbitrary
density (probability distribution) definitions, please see
.
Dynamical systems of discrete nature assumed to be governed by a Markov chain emits
a sequence of observable outputs. Under the Markov assumption, it is also assumed that
the latest output depends only on the current state of the system. Such states are often
not known from the observer when only the output values are observable.
Assuming the Markov probability, the probability of any sequence of observations
occurring when following a given sequence of states can be stated as
in which the probabilities p(yt|yt-1) can be read as the
probability of being currently in state yt given we just were in the
state yt-1 at the previous instant t-1, and the probability
p(xt|yt) can be understood as the probability of observing
xt at instant t given we are currently in the state
yt. To compute those probabilities, we simple use two matrices
A and B.
The matrix A is the matrix of state probabilities:
it gives the probabilities p(yt|yt-1) of jumping from one state
to the other, and the matrix B is the matrix of observation probabilities, which gives the
distribution density p(xt|yt) associated
a given state yt. In the discrete case,
B is really a matrix. In the continuous case,
B is a vector of probability distributions. The overall model definition
can then be stated by the tuple
in which n is an integer representing the total number
of states in the system, A is a matrix
of transition probabilities, B is either
a matrix of observation probabilities (in the discrete case) or a vector of probability
distributions (in the general case) and p is a vector of
initial state probabilities determining the probability of starting in each of the
possible states in the model.
Hidden Markov Models attempt to model such systems and allow, among other things,
-
To infer the most likely sequence of states that produced a given output sequence,
-
Infer which will be the most likely next state (and thus predicting the next output),
-
Calculate the probability that a given sequence of outputs originated from the system
(allowing the use of hidden Markov models for sequence classification).
The “hidden” in Hidden Markov Models comes from the fact that the observer does not
know in which state the system may be in, but has only a probabilistic insight on where
it should be.
To learn a Markov model, you can find a list of both
supervised and unsupervised learning
algorithms in the namespace.
References:
-
Wikipedia contributors. "Linear regression." Wikipedia, the Free Encyclopedia.
Available at: http://en.wikipedia.org/wiki/Hidden_Markov_model
-
Nikolai Shokhirev, Hidden Markov Models. Personal website. Available at:
http://www.shokhirev.com/nikolai/abc/alg/hmm/hmm.html
-
X. Huang, A. Acero, H. Hon. "Spoken Language Processing." pp 396-397.
Prentice Hall, 2001.
-
Dawei Shen. Some mathematics for HMMs, 2008. Available at:
http://courses.media.mit.edu/2010fall/mas622j/ProblemSets/ps4/tutorial.pdf
The example below reproduces the same example given in the Wikipedia
entry for the Viterbi algorithm (http://en.wikipedia.org/wiki/Viterbi_algorithm).
In this example, the model's parameters are initialized manually. However, it is
possible to learn those automatically using .
-
Alexander Schliep, "Learning Hidden Markov Model Topology".
-
Richard Hughey and Anders Krogh, "Hidden Markov models for sequence analysis:
extension and analysis of the basic method", CABIOS 12(2):95-107, 1996. Available in:
http://compbio.soe.ucsc.edu/html_format_papers/hughkrogh96/cabios.html
In a second example, we will create an Ergodic (fully connected)
discrete-density hidden Markov model with uniform probabilities.
// Create a new Ergodic hidden Markov model with three
// fully-connected states and four sequence symbols.
var model = new HiddenMarkovModel(new Ergodic(3), 4);
// After creation, the state transition matrix for the model
// should be given by:
//
// { 0.33, 0.33, 0.33 }
// { 0.33, 0.33, 0.33 }
// { 0.33, 0.33, 0.33 }
//
// in which all state transitions are allowed.
Gets the number of states in this topology.
Gets or sets whether the transition matrix
should be initialized with random probabilities
or not. Default is false.
Creates a new Ergodic topology for a given number of states.
The number of states to be used in the model.
Creates a new Ergodic topology for a given number of states.
The number of states to be used in the model.
Whether to initialize the model with random probabilities
or uniformly with 1 / number of states. Default is false (default is
to use 1/states).
Creates the state transitions matrix and the
initial state probabilities for this topology.
Forward Topology for Hidden Markov Models.
Forward topologies are commonly used to initialize models in which
training sequences can be organized in samples, such as in the recognition
of spoken words. In spoken word recognition, several examples of a single
word can (and should) be used to train a single model, to achieve the most
general model able to generalize over a great number of word samples.
Forward models can typically have a large number of states.
References:
-
Alexander Schliep, "Learning Hidden Markov Model Topology".
-
Richard Hughey and Anders Krogh, "Hidden Markov models for sequence analysis:
extension and analysis of the basic method", CABIOS 12(2):95-107, 1996. Available in:
http://compbio.soe.ucsc.edu/html_format_papers/hughkrogh96/cabios.html
In the following example, we will create a Forward-only
discrete-density hidden Markov model.
// Create a new Forward-only hidden Markov model with
// three forward-only states and four sequence symbols.
var model = new HiddenMarkovModel(new Forward(3), 4);
// After creation, the state transition matrix for the model
// should be given by:
//
// { 0.33, 0.33, 0.33 }
// { 0.00, 0.50, 0.50 }
// { 0.00, 0.00, 1.00 }
//
// in which no backward transitions are allowed (have zero probability).
Gets the number of states in this topology.
Gets or sets the maximum deepness level allowed
for the forward state transition chains.
Gets or sets whether the transition matrix
should be initialized with random probabilities
or not. Default is false.
Gets the initial state probabilities.
Creates a new Forward topology for a given number of states.
The number of states to be used in the model.
Creates a new Forward topology for a given number of states.
The number of states to be used in the model.
The maximum number of forward transitions allowed
for a state. Default is to use the same as the number of states (all forward
connections are allowed).
Creates a new Forward topology for a given number of states.
The number of states to be used in the model.
Whether to initialize the model with random probabilities
or uniformly with 1 / number of states. Default is false (default is
to use 1/states).
Creates a new Forward topology for a given number of states.
The number of states to be used in the model.
The maximum number of forward transitions allowed
for a state. Default is to use the same as the number of states (all forward
connections are allowed).
Whether to initialize the model with random probabilities
or uniformly with 1 / number of states. Default is false (default is
to use 1/states).
Creates the state transitions matrix and the
initial state probabilities for this topology.
Hidden Markov model topology (architecture) specification.
An Hidden Markov Model Topology specifies how many states and which
initial probabilities a Markov model should have. Two common topologies
can be discussed in terms of transition state probabilities and are
available to construction through the and
classes implementing this interface.
Topology specification is important with regard to both learning and
performance: A model with too many states (and thus too many settable
parameters) will require too much training data while an model with an
insufficient number of states will prohibit the HMM from capturing subtle
statistical patterns.
References:
-
Alexander Schliep, "Learning Hidden Markov Model Topology".
-
Richard Hughey and Anders Krogh, "Hidden Markov models for sequence analysis:
extension and analysis of the basic method", CABIOS 12(2):95-107, 1996. Available in:
http://compbio.soe.ucsc.edu/html_format_papers/hughkrogh96/cabios.html
Gets the number of states in this topology.
Creates the state transitions matrix and the
initial state probabilities for this topology.
Polynomial Least-Squares.
In linear regression, the model specification is that the dependent
variable, y is a linear combination of the parameters (but need not
be linear in the independent variables). As the linear regression
has a closed form solution, the regression coefficients can be
efficiently computed using the Regress method of this class.
-
Donghui Chen and Robert J.Plemmons, Nonnegativity Constraints in Numerical Analysis.
Available on: http://users.wfu.edu/plemmons/papers/nonneg.pdf
The following example shows how to fit a multiple linear regression model with the
additional constraint that none of its coefficients should be negative. For this
we can use the learning algorithm instead of
the used above.
-
Bishop, Christopher M.; Pattern Recognition and Machine Learning.
Springer; 1st ed. 2006.
-
Amos Storkey. (2005). Learning from Data: Learning Logistic Regressors. School of Informatics.
Available on: http://www.inf.ed.ac.uk/teaching/courses/lfd/lectures/logisticlearn-print.pdf
-
Cosma Shalizi. (2009). Logistic Regression and Newton's Method. Available on:
http://www.stat.cmu.edu/~cshalizi/350/lectures/26/lecture-26.pdf
-
Edward F. Conor. Logistic Regression. Website. Available on:
http://userwww.sfsu.edu/~efc/classes/biol710/logistic/logisticreg.htm
-
B. Krishnapuram, L. Carin, M.A.T. Figueiredo, A. Hartemink. Sparse Multinomial
Logistic Regression: Fast Algorithms and Generalization Bounds. 2005. Available on:
http://www.lx.it.pt/~mtf/Krishnapuram_Carin_Figueiredo_Hartemink_2005.pdf
-
D. Böhning. Multinomial logistic regression algorithm. Annals of the Institute
of Statistical Mathematics, 44(9):197 ˝U200, 1992. 2. M. Corney.
-
Bishop, Christopher M.; Pattern Recognition and Machine Learning.
Springer; 1st ed. 2006.
-
Bishop, Christopher M.; Pattern Recognition and Machine Learning.
Springer; 1st ed. 2006.
-
Amos Storkey. (2005). Learning from Data: Learning Logistic Regressors. School of Informatics.
Available on: http://www.inf.ed.ac.uk/teaching/courses/lfd/lectures/logisticlearn-print.pdf
-
Cosma Shalizi. (2009). Logistic Regression and Newton's Method. Available on:
http://www.stat.cmu.edu/~cshalizi/350/lectures/26/lecture-26.pdf
-
Edward F. Conor. Logistic Regression. Website. Available on:
http://userwww.sfsu.edu/~efc/classes/biol710/logistic/logisticreg.htm
-
Bishop, Christopher M.; Pattern Recognition and Machine Learning.
Springer; 1st ed. 2006.
-
Amos Storkey. (2005). Learning from Data: Learning Logistic Regressors. School of Informatics.
Available on: http://www.inf.ed.ac.uk/teaching/courses/lfd/lectures/logisticlearn-print.pdf
-
Cosma Shalizi. (2009). Logistic Regression and Newton's Method. Available on:
http://www.stat.cmu.edu/~cshalizi/350/lectures/26/lecture-26.pdf
-
Edward F. Conor. Logistic Regression. Website. Available on:
http://userwww.sfsu.edu/~efc/classes/biol710/logistic/logisticreg.htm
The following example shows how to learn a logistic regression using the
standard algorithm.
-
Student Dave's tutorial on Object Tracking in Images Using 2D Kalman Filters.
Available on: http://studentdavestutorials.weebly.com/object-tracking-2d-kalman-filter.html
-
John D. Cook. Accurately computing running variance. Available on:
http://www.johndcook.com/standard_deviation.html
-
Chan, Tony F.; Golub, Gene H.; LeVeque, Randall J. (1983). Algorithms for
Computing the Sample Variance: Analysis and Recommendations. The American
Statistician 37, 242-247.
-
Ling, Robert F. (1974). Comparison of Several Algorithms for Computing Sample
Means and Variances. Journal of the American Statistical Association, Vol. 69,
No. 348, 859-866.
Gets the sum of the sines of the angles within the window.
Gets the sum of the cosines of the angles within the window.
Gets the current mean of the gathered values.
The mean of the values.
Gets the current variance of the gathered values.
The variance of the values.
Gets the current standard deviation of the gathered values.
The standard deviation of the values.
Gets the current length of the sample mean resultant vector of the gathered values.
Gets the current count of values seen.
The number of samples seen.
Initializes a new instance of the class.
Registers the occurrence of a value.
The value to be registered.
Clears all measures previously computed.
Running (range) statistics.
This class computes the running minimum and maximum values in a stream of values.
Running statistics need only one pass over the data, and do not require all data
to be available prior to computing.
Gets the number of samples seen.
Gets the minimum value seen.
Gets the maximum value seen.
Initializes a new instance of the class.
Registers the occurrence of a value.
The value to be registered.
Clears all measures previously computed.
Running (normal) statistics.
This class computes the running variance using Welford’s method. Running statistics
need only one pass over the data, and do not require all data to be available prior
to computing.
References:
-
John D. Cook. Accurately computing running variance. Available on:
http://www.johndcook.com/standard_deviation.html
-
Chan, Tony F.; Golub, Gene H.; LeVeque, Randall J. (1983). Algorithms for
Computing the Sample Variance: Analysis and Recommendations. The American
Statistician 37, 242-247.
-
Ling, Robert F. (1974). Comparison of Several Algorithms for Computing Sample
Means and Variances. Journal of the American Statistical Association, Vol. 69,
No. 348, 859-866.
Gets the current mean of the gathered values.
The mean of the values.
Gets the current variance of the gathered values.
The variance of the values.
Gets the current standard deviation of the gathered values.
The standard deviation of the values.
Gets the current count of values seen.
The number of samples seen.
Initializes a new instance of the class.
Registers the occurrence of a value.
The value to be registered.
Clears all measures previously computed.
Contains 34+ statistical hypothesis tests, including one way
and two-way ANOVA tests, non-parametric tests such as the
Kolmogorov-Smirnov test and the
Sign Test for the Median, contingency table
tests such as the Kappa test, including variations for
multiple tables, as well as the
Bhapkar and Bowker tests; and the more traditional
Chi-Square, Z, F
, T and Wald tests.
This namespace contains a suite of parametric and non-parametric hypothesis tests. Every
test in this library implements the interface, which defines
a few key methods and properties to assert whether
an statistical hypothesis can be supported or not. Every hypothesis test is associated
with an statistic distribution
which can in turn be queried, inspected and computed as any other distribution in the
namespace.
By default, tests are created using a 0.05 significance level
, which in the framework is referred as the test's size. P-Values are also ready to be
inspected by checking a test's P-Value property.
Furthermore, several tests in this namespace also support
power analysis. The power analysis of a test can be used to suggest an optimal number of samples
which have to be obtained in order to achieve a more interpretable or useful result while doing hypothesis
testing. Power analyses implement the interface, and analyses are available
for the one sample Z, and T tests,
as well as their two sample versions.
Some useful parametric tests are the , ,
, , ,
and . Useful non-parametric tests include the ,
, and the .
Tests are also available for two or more samples. In this case, we can find two sample variants for the
, , ,
, , ,
, as well as the for unpaired samples. For
multiple samples we can find the and , as well as the
and .
Finally, the namespace also includes several tests for contingency tables.
Those tests include Kappa test for inter-rater agreement and its variants, such
as the , and .
Other tests include , , ,
, and the .
The namespace class diagram is shown below.
Please note that class diagrams for each of the inner namespaces are
also available within their own documentation pages.
Hypothesis test for a single ROC curve.
Gets the ROC curve being tested.
Creates a new .
The curve to be tested.
The hypothesized value for the ROC area.
The alternative hypothesis (research hypothesis) to test.
Calculates the standard error of an area calculation for a
curve with the given number of positive and negatives instances
Calculates the standard error of an area calculation for a
curve with the given number of positive and negatives instances
Kappa test for the average of two groups of contingency tables.
The two-matrix Kappa test tries to assert whether the Kappa measure
of two groups of contingency tables, each group created by a different
rater or classification model and measured repeatedly, differs significantly.
This is a two sample t-test kind of test.
References:
- J. L. Fleiss. Statistical methods for rates and proportions.
Wiley-Interscience; 3rd edition (September 5, 2003)
Gets the variance for the first Kappa value.
Gets the variance for the second Kappa value.
Creates a new Two-Table Mean Kappa test.
The average kappa value for the first group of contingency tables.
The average kappa value for the second group of contingency tables.
The kappa's variance in the first group of tables.
The kappa's variance in the first group of tables.
The number of contingency tables averaged in the first group.
The number of contingency tables averaged in the second group.
True to assume equal variances, false otherwise. Default is true.
The alternative hypothesis (research hypothesis) to test.
The hypothesized difference between the two Kappa values.
Creates a new Two-Table Mean Kappa test.
The first group of contingency tables.
The second group of contingency tables.
True to assume equal variances, false otherwise. Default is true.
The hypothesized difference between the two average Kappa values.
The alternative hypothesis (research hypothesis) to test.
Kappa Test for multiple contingency tables.
The multiple-matrix Kappa test tries to assert whether the Kappa measure
of many contingency tables, each of which created by a different rater
or classification model, differs significantly. The computations are
based on the pages 607, 608 of (Fleiss, 2003).
This is a Chi-square kind of test.
References:
- J. L. Fleiss. Statistical methods for rates and proportions.
Wiley-Interscience; 3rd edition (September 5, 2003)
Gets the overall Kappa value
for the many contingency tables.
Gets the overall Kappa variance
for the many contingency tables.
Gets the variance for each kappa value.
Gets the kappa for each contingency table.
Creates a new multiple table Kappa test.
The kappa values.
The variance for each kappa value.
Creates a new multiple table Kappa test.
The contingency tables.
Computes the multiple matrix Kappa test.
Fisher's exact test for contingency tables.
This test statistic distribution is the
Hypergeometric.
-
Wikipedia, The Free Encyclopedia. Lilliefors Test.
Available on: https://en.wikipedia.org/wiki/Lilliefors_test
In this first example, suppose we got a new sample, and we would
like to test whether this sample has been originated from a uniform
continuous distribution. Unlike ,
we can actually use this test whether the data fits a distribution
that has been estimated from the data.
-
Wikipedia, The Free Encyclopedia. Shapiro-Wilk test. Available on:
http://en.wikipedia.org/wiki/Shapiro%E2%80%93Wilk_test
-
Wikipedia, The Free Encyclopedia. Multinomial Test. Available on:
http://en.wikipedia.org/wiki/Multinomial_test
The following example is based on the example available on About.com Statistics,
An Example of Chi-Square Test for a Multinomial Experiment By Courtney Taylor.
In this example, we would like to test if a die is fair. For this, we
will be rolling the die 600 times, annotating the result every time
the die falls. In the end, we got a one 106 times, a two 90 times, a
three 98 times, a four 102 times, a five 100 times and a six 104 times:
int[] sample = { 106, 90, 98, 102, 100, 104 };
// If the die was fair, we should note that we would be expecting the
// probabilities to be all equal to 1 / 6:
double[] hypothesizedProportion =
{
// 1 2 3 4 5 6
1 / 6.0, 1 / 6.0, 1 / 6.0, 1 / 6.0, 1 / 6.0, 1 / 6.0,
};
// Now, we create our test using the samples and the expected proportion
MultinomialTest test = new MultinomialTest(sample, hypothesizedProportion);
double chiSquare = test.Statistic; // 1.6
bool significant = test.Significant; // false
Since the test didn't come up significant, it means that we
don't have enough evidence to to reject the null hypothesis
that the die is fair.
Gets the observed sample proportions.
Gets the hypothesized population proportions.
Creates a new Multinomial test.
The proportions for each category in the sample.
The number of observations in the sample.
Creates a new Multinomial test.
The number of occurrences for each category in the sample.
Creates a new Multinomial test.
The number of occurrences for each category in the sample.
The hypothesized category proportions. Default is
to assume uniformly equal proportions.
Creates a new Multinomial test.
The proportions for each category in the sample.
The number of observations in the sample.
The hypothesized category proportions. Default is
to assume uniformly equal proportions.
Creates a new Multinomial test.
The categories for each observation in the sample.
The number of possible categories.
Creates a new Multinomial test.
The categories for each observation in the sample.
The number of possible categories.
The hypothesized category proportions. Default is
to assume uniformly equal proportions.
Computes the Multinomial test.
Bartlett's test for equality of variances.
In statistics, Bartlett's test is used to test if k samples are from populations
with equal variances. Equal variances across samples is called homoscedasticity
or homogeneity of variances. Some statistical tests, for example the
analysis of variance, assume that variances are equal across groups or samples.
The Bartlett test can be used to verify that assumption.
Bartlett's test is sensitive to departures from normality. That is, if the samples
come from non-normal distributions, then Bartlett's test may simply be testing for
non-normality. Levene's test and the Brown–Forsythe test
are alternatives to the Bartlett test that are less sensitive to departures from
normality.
References:
-
Wikipedia, The Free Encyclopedia. Bartlett's test. Available on:
http://en.wikipedia.org/wiki/Bartlett's_test
Tests the null hypothesis that all group variances are equal.
The grouped samples.
Levene test computation methods.
The test has been computed using the Mean.
The test has been computed using the Median
(which is known as the Brown-Forsythe test).
The test has been computed using the trimmed mean.
Levene's test for equality of variances.
In statistics, Levene's test is an inferential statistic used to assess the
equality of variances for a variable calculated for two or more groups. Some
common statistical procedures assume that variances of the populations from
which different samples are drawn are equal. Levene's test assesses this
assumption. It tests the null hypothesis that the population variances are
equal (called homogeneity of variance or homoscedasticity). If the resulting
P-value of Levene's test is less than some significance level (typically 0.05),
the obtained differences in sample variances are unlikely to have occurred based
on random sampling from a population with equal variances. Thus, the null hypothesis
of equal variances is rejected and it is concluded that there is a difference
between the variances in the population.
Some of the procedures typically assuming homoscedasticity, for which one can use
Levene's tests, include analysis of variance and
t-tests. Levene's test is often used before a comparison of means. When Levene's
test shows significance, one should switch to generalized tests, free from homoscedasticity
assumptions. Levene's test may also be used as a main test for answering a stand-alone
question of whether two sub-samples in a given population have equal or different variances.
References:
-
Wikipedia, The Free Encyclopedia. Levene's test. Available on:
http://en.wikipedia.org/wiki/Levene's_test
Gets the method used to compute the Levene's test.
Tests the null hypothesis that all group variances are equal.
The grouped samples.
True to use the median in the Levene calculation.
False to use the mean. Default is false (use the mean).
Tests the null hypothesis that all group variances are equal.
The grouped samples.
The percentage of observations to discard
from the sample when computing the test with the truncated mean.
Contains methods for power analysis of several related hypothesis tests,
including support for automatic sample size estimation.
The namespace class diagram is shown below.
Please note that class diagrams for each of the inner namespaces are
also available within their own documentation pages.
Common interface for power analysis objects.
The power of a statistical test is the probability that it correctly rejects
the null hypothesis when the null hypothesis is false. That is,
power = P(reject null hypothesis | null hypothesis is false)
It can be equivalently thought of as the probability of correctly accepting the
alternative hypothesis when the alternative hypothesis is true - that is, the ability
of a test to detect an effect, if the effect actually exists. The power is in general
a function of the possible distributions, often determined by a parameter, under the
alternative hypothesis. As the power increases, the chances of a Type II error occurring
decrease. The probability of a Type II error occurring is referred to as the false
negative rate (β) and the power is equal to 1−β. The power is also known as the sensitivity.
Power analysis can be used to calculate the minimum sample size required so that
one can be reasonably likely to detect an effect of a given size. Power analysis
can also be used to calculate the minimum effect size that is likely to be detected
in a study using a given sample size. In addition, the concept of power is used to
make comparisons between different statistical testing procedures: for example,
between a parametric and a nonparametric test of the same hypothesis. There is also
the concept of a power function of a test, which is the probability of rejecting the
null when the null is true.
References:
-
Wikipedia, The Free Encyclopedia. Statistical power. Available on:
http://en.wikipedia.org/wiki/Statistical_power
Gets the test type.
Gets the power of the test, also known as the
(1-Beta error rate) or the test's sensitivity.
Gets the significance level
for the test. Also known as alpha.
Gets the number of samples
considered in the test.
Gets the effect size of the test.
Common interface for two-sample power analysis objects.
Gets the number of observations
contained in the first sample.
Gets the number of observations
contained in the second sample.
Base class for two sample power analysis methods.
This class cannot be instantiated.
Gets the test type.
Gets or sets the power of the test, also
known as the (1-Beta error rate).
Gets or sets the significance level
for the test. Also known as alpha.
Gets or sets the number of observations
in the first sample considered in the test.
Gets or sets the number of observations
in the second sample considered in the test.
Gets the total number of observations
in both samples considered in the test.
Gets the total number of observations
in both samples considered in the test.
Gets or sets the effect size of the test.
Constructs a new power analysis for a two-sample test.
Computes the power for a test with givens values of
effect size and
number of samples under .
The power for the test under the given conditions.
Computes the minimum detectable effect size for the test
considering the power given in , the
number of samples in and the
significance level .
The minimum detectable effect
size for the test under the given conditions.
Computes the minimum significance level for the test
considering the power given in , the
number of samples in and the
effect size .
The minimum detectable effect
size for the test under the given conditions.
Computes the recommended sample size for the test to attain
the power indicated in considering
values of and .
Recommended sample size for attaining the given
for size effect
under the given .
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Converts the numeric power of this test to its equivalent string representation.
Converts the numeric power of this test to its equivalent string representation.
Gets the minimum difference in the experiment units
to which it is possible to detect a difference.
The common standard deviation for the samples.
The minimum difference in means which can be detected by the test.
Gets the minimum difference in the experiment units
to which it is possible to detect a difference.
The variance for the first sample.
The variance for the second sample.
The minimum difference in means which can be detected by the test.
Power analysis for two-sample Z-Tests.
Please take a look at the example section.
Creates a new .
The hypothesis tested.
Creates a new .
The test to create the analysis for.
Computes the power for a test with givens values of
effect size and
number of samples under .
The power for the test under the given conditions.
Gets the recommended sample size for the test to attain
the power indicating in considering
values of and .
Recommended sample size for attaining the given
for size effect
under the given .
Computes the minimum detectable effect size for the test
considering the power given in , the
number of samples in and the significance
level .
The minimum detectable effect
size for the test under the given conditions.
Estimates the number of samples necessary to attain the
required power level for the given effect size.
The minimum detectable difference.
The difference standard deviation.
The desired power level. Default is 0.8.
The desired significance level. Default is 0.05.
The alternative hypothesis (research hypothesis) to be tested.
The proportion of observations in the second group
when compared to the first group. A proportion of 2:1 results in twice more
samples in the second group than in the first. Default is 1.
The required number of samples.
Estimates the number of samples necessary to attain the
required power level for the given effect size.
The number of observations in the first sample.
The number of observations in the second sample.
The desired power level. Default is 0.8.
The desired significance level. Default is 0.05.
The alternative hypothesis (research hypothesis) to be tested.
The required number of samples.
Power analysis for two-sample T-Tests.
There are different ways a power analysis test can be conducted.
// Let's say we have two samples, and we would like to know whether those
// samples have the same mean. For this, we can perform a two sample T-Test:
double[] A = { 5.0, 6.0, 7.9, 6.95, 5.3, 10.0, 7.48, 9.4, 7.6, 8.0, 6.22 };
double[] B = { 5.0, 1.6, 5.75, 5.80, 2.9, 8.88, 4.56, 2.4, 5.0, 10.0 };
// Perform the test, assuming the samples have unequal variances
var test = new TwoSampleTTest(A, B, assumeEqualVariances: false);
double df = test.DegreesOfFreedom; // d.f. = 14.351
double t = test.Statistic; // t = 2.14
double p = test.PValue; // p = 0.04999
bool significant = test.Significant; // true
// The test gave us an indication that the samples may
// indeed have come from different distributions (whose
// mean value is actually distinct from each other).
// Now, we would like to perform an _a posteriori_ analysis of the
// test. When doing an a posteriori analysis, we can not change some
// characteristics of the test (because it has been already done),
// but we can measure some important features that may indicate
// whether the test is trustworthy or not.
// One of the first things would be to check for the test's power.
// A test's power is 1 minus the probability of rejecting the null
// hypothesis when the null hypothesis is actually false. It is
// the other side of the coin when we consider that the P-value
// is the probability of rejecting the null hypothesis when the
// null hypothesis is actually true.
// Ideally, this should be a high value:
double power = test.Analysis.Power; // 0.5376260
// Check how much effect we are trying to detect
double effect = test.Analysis.Effect; // 0.94566
// With this power, that is the minimal difference we can spot?
double sigma = Math.Sqrt(test.Variance);
double thres = test.Analysis.Effect * sigma; // 2.0700909090909
// This means that, using our test, the smallest difference that
// we could detect with some confidence would be something around
// 2 standard deviations. If we would like to say the samples are
// different when they are less than 2 std. dev. apart, we would
// need to do repeat our experiment differently.
Another way to create the power analysis is to pass the
hypothesis test to the t-test power analysis constructor.
// Create an a posteriori analysis of the experiment
var analysis = new TwoSampleTTestPowerAnalysis(test);
// When creating a power analysis, we have three things we can
// change. We can always freely configure two of those things
// and then ask the analysis to give us the third.
// Those are:
double e = analysis.Effect; // the test's minimum detectable effect size (0.94566)
double n = analysis.TotalSamples; // the number of samples in the test (21 or (11 + 10))
double b = analysis.Power; // the probability of committing a type-2 error (0.53)
// Let's say we would like to create a test with 80% power.
analysis.Power = 0.8;
analysis.ComputeEffect(); // what effect could we detect?
double detectableEffect = analysis.Effect; // we would detect a difference of 1.290514
However, to achieve this 80%, we would need to redo our experiment
more carefully. Assuming we are going to redo our experiment, we will
have more freedom about what we can change and what we can not. For
better addressing those points, we will create an a priori analysis
of the experiment:
// We would like to know how many samples we would need to gather in
// order to achieve a 80% power test which can detect an effect size
// of one standard deviation:
//
analysis = TwoSampleTTestPowerAnalysis.GetSampleSize
(
variance1: A.Variance(),
variance2: B.Variance(),
delta: 1.0, // the minimum detectable difference we want
power: 0.8 // the test power that we want
);
// How many samples would we need in order to see the effect we need?
int n1 = (int)Math.Ceiling(analysis.Samples1); // 77
int n2 = (int)Math.Ceiling(analysis.Samples2); // 77
// According to our power analysis, we would need at least 77
// observations in each sample in order to see the effect we
// need with the required 80% power.
Creates a new .
The hypothesis tested.
Creates a new .
The test to create the analysis for.
Computes the power for a test with givens values of
effect size and
number of samples under .
The power for the test under the given conditions.
Estimates the number of samples necessary to attain the
required power level for the given effect size.
The minimum detectable difference.
The difference standard deviation.
The desired power level. Default is 0.8.
The desired significance level. Default is 0.05.
The proportion of observations in the second group
when compared to the first group. A proportion of 2:1 results in twice more
samples in the second group than in the first. Default is 1.
The alternative hypothesis (research hypothesis) to be tested.
The required number of samples.
Estimates the number of samples necessary to attain the
required power level for the given effect size.
The minimum detectable difference.
The first sample variance.
The second sample variance.
The desired power level. Default is 0.8.
The desired significance level. Default is 0.05.
The proportion of observations in the second group
when compared to the first group. A proportion of 2:1 results in twice more
samples in the second group than in the first. Default is 1.
The alternative hypothesis (research hypothesis) to be tested.
The required number of samples.
Base class for one sample power analysis methods.
This class cannot be instantiated.
Gets the test type.
Gets or sets the power of the test, also
known as the (1-Beta error rate).
Gets or sets the significance level
for the test. Also known as alpha.
Gets or sets the number of samples
considered in the test.
Gets or sets the effect size of the test.
Constructs a new power analysis for a one-sample test.
Computes the power for a test with givens values of
effect size and
number of samples under .
The power for the test under the given conditions.
Computes the minimum detectable effect size for the test
considering the power given in , the
number of samples in and the significance
level .
The minimum detectable effect
size for the test under the given conditions.
Computes recommended sample size for the test to attain
the power indicated in considering
values of and .
Recommended sample size for attaining the given
for size effect
under the given .
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Converts the numeric power of this test to its equivalent string representation.
Converts the numeric power of this test to its equivalent string representation.
Gets the minimum difference in the experiment units
to which it is possible to detect a difference.
The standard deviation for the samples.
The minimum difference in means which can be detected by the test.
Power analysis for one-sample T-Tests.
// When creating a power analysis, we have three things we can
// change. We can always freely configure two of those things
// and then ask the analysis to give us the third.
var analysis = new TTestPowerAnalysis(OneSampleHypothesis.ValueIsDifferentFromHypothesis);
// Those are:
double e = analysis.Effect; // the test's minimum detectable effect size
double n = analysis.Samples; // the number of samples in the test
double p = analysis.Power; // the probability of committing a type-2 error
// Let's set the desired effect size and the
// number of samples so we can get the power
analysis.Effect = 0.2; // we would like to detect at least 0.2 std. dev. apart
analysis.Samples = 60; // we would like to use at most 60 samples
analysis.ComputePower(); // what will be the power of this test?
double power = analysis.Power; // The power is going to be 0.33 (or 33%)
// Let's set the desired power and the number
// of samples so we can get the effect size
analysis.Power = 0.8; // we would like to create a test with 80% power
analysis.Samples = 60; // we would like to use at most 60 samples
analysis.ComputeEffect(); // what would be the minimum effect size we can detect?
double effect = analysis.Effect; // The effect will be 0.36 standard deviations.
// Let's set the desired power and the effect
// size so we can get the number of samples
analysis.Power = 0.8; // we would like to create a test with 80% power
analysis.Effect = 0.2; // we would like to detect at least 0.2 std. dev. apart
analysis.ComputeSamples();
double samples = analysis.Samples; // We would need around 199 samples.
Creates a new .
The hypothesis tested.
Creates a new .
The test to create the analysis for.
Computes the power for a test with givens values of
effect size and
number of samples under .
The power for the test under the given conditions.
Estimates the number of samples necessary to attain the
required power level for the given effect size.
The minimum detectable difference.
The difference standard deviation.
The desired power level. Default is 0.8.
The desired significance level. Default is 0.05.
The alternative hypothesis (research hypothesis) to be tested.
The required number of samples.
Estimates the number of samples necessary to attain the
required power level for the given effect size.
The number of observations in the sample.
The desired power level. Default is 0.8.
The desired significance level. Default is 0.05.
The alternative hypothesis (research hypothesis) to be tested.
The required number of samples.
Power analysis for one-sample Z-Tests.
// When creating a power analysis, we have three things we can
// change. We can always freely configure two of those things
// and then ask the analysis to give us the third.
var analysis = new ZTestPowerAnalysis(OneSampleHypothesis.ValueIsDifferentFromHypothesis);
// Those are:
double e = analysis.Effect; // the test's minimum detectable effect size
double n = analysis.Samples; // the number of samples in the test
double p = analysis.Power; // the probability of committing a type-2 error
// Let's set the desired effect size and the
// number of samples so we can get the power
analysis.Effect = 0.2; // we would like to detect at least 0.2 std. dev. apart
analysis.Samples = 60; // we would like to use at most 60 samples
analysis.ComputePower(); // what will be the power of this test?
double power = analysis.Power; // The power is going to be 0.34 (or 34%)
// Let's set the desired power and the number
// of samples so we can get the effect size
analysis.Power = 0.8; // we would like to create a test with 80% power
analysis.Samples = 60; // we would like to use at most 60 samples
analysis.ComputeEffect(); // what would be the minimum effect size we can detect?
double effect = analysis.Effect; // The effect will be 0.36 standard deviations.
// Let's set the desired power and the effect
// size so we can get the number of samples
analysis.Power = 0.8; // we would like to create a test with 80% power
analysis.Effect = 0.2; // we would like to detect at least 0.2 std. dev. apart
analysis.ComputeSamples();
double samples = analysis.Samples; // We would need around 197 samples.
Creates a new .
The hypothesis tested.
Creates a new .
The test to create the analysis for.
Computes the power for a test with givens values of
effect size and
number of samples under .
The power for the test under the given conditions.
Gets the recommended sample size for the test to attain
the power indicating in considering
values of and .
Recommended sample size for attaining the given
for size effect
under the given .
Computes the minimum detectable effect size for the test
considering the power given in , the
number of samples in and the significance
level .
The minimum detectable effect
size for the test under the given conditions.
Estimates the number of samples necessary to attain the
required power level for the given effect size.
The minimum detectable difference.
The difference standard deviation.
The desired power level. Default is 0.8.
The desired significance level. Default is 0.05.
The alternative hypothesis (research hypothesis) to be tested.
The required number of samples.
Estimates the number of samples necessary to attain the
required power level for the given effect size.
The number of observations in the sample.
The desired power level. Default is 0.8.
The desired significance level. Default is 0.05.
The alternative hypothesis (research hypothesis) to be tested.
The required number of samples.
T-Test for two paired samples.
The Paired T-test can be used when the samples are dependent; that is, when there
is only one sample that has been tested twice (repeated measures) or when there are
two samples that have been matched or "paired". This is an example of a paired difference
test.
References:
-
Wikipedia, The Free Encyclopedia. Student's t-test.
Available from: http://en.wikipedia.org/wiki/Student%27s_t-test#Dependent_t-test_for_paired_samples
Suppose we would like to know the effect of a treatment (such
as a new drug) in improving the well-being of 9 patients. The
well-being is measured in a discrete scale, going from 0 to 10.
// To do so, we need to register the initial state of each patient
// and then register their state after a given time under treatment.
double[,] patients =
{
// before after
// treatment treatment
/* Patient 1.*/ { 0, 1 },
/* Patient 2.*/ { 6, 5 },
/* Patient 3.*/ { 4, 9 },
/* Patient 4.*/ { 8, 6 },
/* Patient 5.*/ { 1, 6 },
/* Patient 6.*/ { 6, 7 },
/* Patient 7.*/ { 3, 4 },
/* Patient 8.*/ { 8, 7 },
/* Patient 9.*/ { 6, 5 },
};
// Extract the before and after columns
double[] before = patients.GetColumn(0);
double[] after = patients.GetColumn(1);
// Create the paired-sample T-test. Our research hypothesis is
// that the treatment does improve the patient's well-being. So
// we will be testing the hypothesis that the well-being of the
// "before" sample, the first sample, is "smaller" in comparison
// to the "after" treatment group.
PairedTTest test = new PairedTTest(before, after,
TwoSampleHypothesis.FirstValueIsSmallerThanSecond);
bool significant = test.Significant; // not significant
double pvalue = test.PValue; // p-value = 0.1650
double tstat = test.Statistic; // t-stat = -1.0371
Gets the alternative hypothesis under test. If the test is
, the null hypothesis can be rejected
in favor of this alternative hypothesis.
Gets the first sample's mean.
Gets the second sample's mean.
Gets the observed mean difference between the two samples.
Gets the standard error of the difference.
Gets the size of a sample.
Both samples have equal size.
Gets the 95% confidence interval for the
statistic.
Gets a confidence interval for the
statistic within the given confidence level percentage.
The confidence level. Default is 0.95.
A confidence interval for the estimated value.
Creates a new paired t-test.
The observations in the first sample.
The observations in the second sample.
The alternative hypothesis (research hypothesis) to test.
Update event.
Converts a given p-value to a test statistic.
The p-value.
The test statistic which would generate the given p-value.
Converts a given test statistic to a p-value.
The value of the test statistic.
The p-value for the given statistic.
Z-Test for two sample proportions.
Creates a new Z-Test for two sample proportions.
The proportion of success observations in the first sample.
The total number of observations in the first sample.
The proportion of success observations in the second sample.
The total number of observations in the second sample.
The alternative hypothesis (research hypothesis) to test.
Creates a new Z-Test for two sample proportions.
The number of successes in the first sample.
The total number of trials (observations) in the first sample.
The number of successes in the second sample.
The total number of trials (observations) in the second sample.
The alternative hypothesis (research hypothesis) to test.
Computes the Z-test for two sample proportions.
Hypothesis test for two Receiver-Operating
Characteristic (ROC) curve areas (ROC-AUC).
First Receiver-Operating Characteristic curve.
First Receiver-Operating Characteristic curve.
Gets the summed Kappa variance
for the two contingency tables.
Gets the variance for the first Kappa value.
Gets the variance for the second Kappa value.
Creates a new test for two ROC curves.
The first ROC curve.
The second ROC curve.
The hypothesized difference between the two areas.
The alternative hypothesis (research hypothesis) to test.
Grubb's Test for Outliers (for approximately Normal distributions).
Grubbs' test (named after Frank E. Grubbs, who published the test in 1950),
also known as the maximum normed residual test or extreme studentized deviate
test, is a statistical test used to detect outliers in a univariate data set
assumed to come from a normally distributed population.
References:
-
Wikipedia, The Free Encyclopedia. Grubb's test for outliers. Available on:
https://en.wikipedia.org/wiki/Grubbs%27_test_for_outliers
Gets the number of observations in the sample.
The number of observations in the sample.
Gets the sample's mean.
Gets the sample's standard deviation.
Gets the difference between the minimum value and the mean.
Gets the difference between the maximum value and the mean.
Gets the maximum absolute difference between
an observation in the sample and the mean.
Gets the being tested.
Constructs a Grubb's test.
Converts a given test statistic to a p-value.
The value of the test statistic.
The p-value for the given statistic.
Converts a given p-value to a test statistic.
The p-value.
The test statistic which would generate the given p-value.
Base class for Wilcoxon's W tests.
This is a base class which doesn't need to be used directly.
Instead, you may wish to call
and .
Gets the number of samples being tested.
Gets the signs for each of the differences.
Gets the differences between the samples.
Gets the rank statistics for the differences.
Gets wether the samples to be ranked contain zeros.
Gets wether the samples to be ranked contain ties.
Gets whether we are using a exact test.
Creates a new Wilcoxon's W+ test.
The signs for the sample differences.
The differences between samples.
The distribution tail to test.
Creates a new Wilcoxon's W+ test.
Computes the Wilcoxon Signed-Rank test.
Computes the Wilcoxon Signed-Rank test.
Converts a given test statistic to a p-value.
The value of the test statistic.
The p-value for the given statistic.
Converts a given p-value to a test statistic.
The p-value.
The test statistic which would generate the given p-value.
Mann-Whitney-Wilcoxon test for unpaired samples.
The Mann–Whitney U test (also called the Mann–Whitney–Wilcoxon (MWW),
Wilcoxon rank-sum test, or Wilcoxon–Mann–Whitney test) is a non-parametric
test of the null hypothesis that two populations are the same against
an alternative hypothesis, especially that a particular population tends
to have larger values than the other.
It has greater efficiency than the t-test on
non-normal distributions, such as a mixture
of normal distributions, and it is
nearly as efficient as the t-test on normal
distributions.
The following example comes from Richard Lowry's page at
http://vassarstats.net/textbook/ch11a.html. As stated by
Richard, this example deals with persons seeking treatment
by claustrophobia. Those persons are randomly divided into
two groups, and each group receive a different treatment
for the disorder.
The hypothesis would be that treatment A would more effective
than B. To check this hypothesis, we can use Mann-Whitney's Test
to compare the medians of both groups.
// Claustrophobia test scores for people treated with treatment A
double[] sample1 = { 4.6, 4.7, 4.9, 5.1, 5.2, 5.5, 5.8, 6.1, 6.5, 6.5, 7.2 };
// Claustrophobia test scores for people treated with treatment B
double[] sample2 = { 5.2, 5.3, 5.4, 5.6, 6.2, 6.3, 6.8, 7.7, 8.0, 8.1 };
// Create a new Mann-Whitney-Wilcoxon's test to compare the two samples
MannWhitneyWilcoxonTest test = new MannWhitneyWilcoxonTest(sample1, sample2,
TwoSampleHypothesis.FirstValueIsSmallerThanSecond);
double sum1 = test.RankSum1; // 96.5
double sum2 = test.RankSum2; // 134.5
double statistic1 = test.Statistic1; // 79.5
double statistic2 = test.Statistic2; // 30.5
double pvalue = test.PValue; // 0.043834132843420748
// Check if the test was significant
bool significant = test.Significant; // true
Gets the alternative hypothesis under test. If the test is
, the null hypothesis can be rejected
in favor of this alternative hypothesis.
Gets the number of samples in the first sample.
Gets the number of samples in the second sample.
Gets the rank statistics for the first sample.
Gets the rank statistics for the second sample.
Gets the sum of ranks for the first sample. Often known as Ta.
Gets the sum of ranks for the second sample. Often known as Tb.
Gets the difference between the expected value for
the observed value of and its
expected value under the null hypothesis. Often known as U_a.
Gets the difference between the expected value for
the observed value of and its
expected value under the null hypothesis. Often known as U_b.
Gets a value indicating whether the provided samples have tied ranks.
Gets whether we are using a exact test.
Tests whether two samples comes from the
same distribution without assuming normality.
The first sample.
The second sample.
The alternative hypothesis (research hypothesis) to test.
True to compute the exact distribution. May require a significant
amount of processing power for large samples (n > 12). If left at null, whether to
compute the exact or approximate distribution will depend on the number of samples.
Default is null.
Whether to account for ties when computing the
rank statistics or not. Default is true.
Converts a given test statistic to a p-value.
The value of the test statistic.
The p-value for the given statistic.
Converts a given p-value to a test statistic.
The p-value.
The test statistic which would generate the given p-value.
Base class for Hypothesis Tests.
A statistical hypothesis test is a method of making decisions using data, whether from
a controlled experiment or an observational study (not controlled). In statistics, a
result is called statistically significant if it is unlikely to have occurred by chance
alone, according to a pre-determined threshold probability, the significance level.
References:
-
Wikipedia, The Free Encyclopedia. Statistical Hypothesis Testing.
Initializes a new instance of the class.
Gets the distribution associated
with the test statistic.
Gets the P-value associated with this test.
In statistical hypothesis testing, the p-value is the probability of
obtaining a test statistic at least as extreme as the one that was
actually observed, assuming that the null hypothesis is true.
The lower the p-value, the less likely the result can be explained
by chance alone, assuming the null hypothesis is true.
Gets the test statistic.
Gets the test type.
Gets the significance level for the
test. Default value is 0.05 (5%).
Gets whether the null hypothesis should be rejected.
A test result is said to be statistically significant when the
result would be very unlikely to have occurred by chance alone.
Gets the critical value for the current significance level.
Converts a given test statistic to a p-value.
The value of the test statistic.
The p-value for the given statistic.
Converts a given p-value to a test statistic.
The p-value.
The test statistic which would generate the given p-value.
Called whenever the test significance level changes.
Converts the numeric P-Value of this test to its equivalent string representation.
Converts the numeric P-Value of this test to its equivalent string representation.
Common interface for Hypothesis tests depending on a statistical distribution.
The test statistic distribution.
Gets the distribution associated
with the test statistic.
Common interface for Hypothesis tests depending on a statistical distribution.
Gets the test type.
Gets whether the null hypothesis should be rejected.
A test result is said to be statistically significant when the
result would be very unlikely to have occurred by chance alone.
Converts a given test statistic to a p-value.
The value of the test statistic.
The p-value for the given statistic.
Converts a given p-value to a test statistic.
The p-value.
The test statistic which would generate the given p-value.
Bhapkar test of homogeneity for contingency tables.
The Bhapkar test is a more powerful alternative to the
Stuart-Maxwell test.
This is a Chi-square kind of test.
References:
-
Bhapkar, V.P. (1966). A note on the equivalence of two test criteria
for hypotheses in categorical data. Journal of the American Statistical
Association, 61, 228-235.
Gets the delta vector d used
in the test calculations.
Gets the covariance matrix S
used in the test calculations.
Gets the inverse covariance matrix
S^-1 used in the calculations.
Creates a new Bhapkar test.
The contingency table to test.
Bowker test of symmetry for contingency tables.
This is a Chi-square kind of test.
Creates a new Bowker test.
The contingency table to test.
Two-Sample (Goodness-of-fit) Chi-Square Test (Upper Tail)
A chi-square test (also chi-squared or χ² test) is any statistical
hypothesis test in which the sampling distribution of the test statistic
is a chi-square distribution when
the null hypothesis is true, or any in which this is asymptotically true,
meaning that the sampling distribution (if the null hypothesis is true)
can be made to approximate a chi-square distribution as closely as desired
by making the sample size large enough.
The chi-square test is used whenever one would like to test whether the
actual data differs from a random distribution.
References:
-
Wikipedia, The Free Encyclopedia. Chi-Square Test. Available on:
http://en.wikipedia.org/wiki/Chi-square_test
-
J. S. McLaughlin. Chi-Square Test. Available on:
http://www2.lv.psu.edu/jxm57/irp/chisquar.html
The following example has been based on the example section
of the
Pearson's chi-squared test article on Wikipedia.
// Suppose we would like to test the hypothesis that a random sample of
// 100 people has been drawn from a population in which men and women are
// equal in frequency.
// Under this hypothesis, the observed number of men and women would be
// compared to the theoretical frequencies of 50 men and 50 women. So,
// after drawing our sample, we found out that there were 44 men and 56
// women in the sample:
// man woman
double[] observed = { 44, 56 };
double[] expected = { 50, 50 };
// If the null hypothesis is true (i.e., men and women are chosen with
// equal probability), the test statistic will be drawn from a chi-squared
// distribution with one degree of freedom. If the male frequency is known,
// then the female frequency is determined.
//
int degreesOfFreedom = 1;
// So now we have:
//
var chi = new ChiSquareTest(expected, observed, degreesOfFreedom);
// The chi-squared distribution for 1 degree of freedom shows that the
// probability of observing this difference (or a more extreme difference
// than this) if men and women are equally numerous in the population is
// approximately 0.23.
double pvalue = chi.PValue; // 0.23
// This probability is higher than conventional criteria for statistical
// significance (0.001 or 0.05), so normally we would not reject the null
// hypothesis that the number of men in the population is the same as the
// number of women.
bool significant = chi.Significant; // false
Gets the degrees of freedom for the Chi-Square distribution.
Constructs a Chi-Square Test.
The test statistic.
The chi-square distribution degrees of freedom.
Constructs a Chi-Square Test.
The expected variable values.
The observed variable values.
The chi-square distribution degrees of freedom.
Constructs a Chi-Square Test.
Constructs a Chi-Square Test.
Constructs a Chi-Square Test.
Constructs a Chi-Square Test.
Computes the Chi-Square Test.
Converts a given test statistic to a p-value.
The value of the test statistic.
The p-value for the given statistic.
Converts a given p-value to a test statistic.
The p-value.
The test statistic which would generate the given p-value.
Kappa Test for two contingency tables.
The two-matrix Kappa test tries to assert whether the Kappa measure
of two contingency tables, each of which created by a different rater
or classification model, differs significantly.
This is a two sample z-test kind of test.
References:
- J. L. Fleiss. Statistical methods for rates and proportions.
Wiley-Interscience; 3rd edition (September 5, 2003)
-
Ientilucci, Emmett (2006). "On Using and Computing the Kappa Statistic".
Available on: http://www.cis.rit.edu/~ejipci/Reports/On_Using_and_Computing_the_Kappa_Statistic.pdf
Gets the summed Kappa variance
for the two contingency tables.
Gets the variance for the first Kappa value.
Gets the variance for the second Kappa value.
Creates a new Two-Table Kappa test.
The kappa value for the first contingency table to test.
The kappa value for the second contingency table to test.
The variance of the kappa value for the first contingency table to test.
The variance of the kappa value for the second contingency table to test.
The alternative hypothesis (research hypothesis) to test.
The hypothesized difference between the two Kappa values.
Creates a new Two-Table Kappa test.
The first contingency table to test.
The second contingency table to test.
The hypothesized difference between the two Kappa values.
The alternative hypothesis (research hypothesis) to test.
Kappa Test for agreement in contingency tables.
The Kappa test tries to assert whether the Kappa measure of a
a contingency table, is significantly different from another
hypothesized value.
The computations used by the test are the same found in the 1969 paper by
J. L. Fleiss, J. Cohen, B. S. Everitt, in which they presented the finally
corrected version of the Kappa's variance formulae. This is contrast to the
computations traditionally found in the remote sensing literature. For those
variance computations, see the method.
This is a z-test kind of test.
References:
- J. L. Fleiss. Statistical methods for rates and proportions.
Wiley-Interscience; 3rd edition (September 5, 2003)
- J. L. Fleiss, J. Cohen, B. S. Everitt. Large sample standard errors of
kappa and weighted kappa. Psychological Bulletin, Volume: 72, Issue: 5. Washington,
DC: American Psychological Association, Pages: 323-327, 1969.
Gets the variance of the Kappa statistic.
Creates a new Kappa test.
The estimated Kappa statistic.
The standard error of the kappa statistic. If the test is
being used to assert independency between two raters (i.e. testing the null hypothesis
that the underlying Kappa is zero), then the
standard error should be computed with the null hypothesis parameter set to true.
The alternative hypothesis (research hypothesis) to test. If the
hypothesized kappa is left unspecified, a one-tailed test will be used. Otherwise, the
default is to use a two-sided test.
Creates a new Kappa test.
The estimated Kappa statistic.
The standard error of the kappa statistic. If the test is
being used to assert independency between two raters (i.e. testing the null hypothesis
that the underlying Kappa is zero), then the
standard error should be computed with the null hypothesis parameter set to true.
The hypothesized value for the Kappa statistic.
The alternative hypothesis (research hypothesis) to test. If the
hypothesized kappa is left unspecified, a one-tailed test will be used. Otherwise, the
default is to use a two-sided test.
Creates a new Kappa test.
The contingency table to test.
The alternative hypothesis (research hypothesis) to test. If the
hypothesized kappa is left unspecified, a one-tailed test will be used. Otherwise, the
default is to use a two-sided test.
Creates a new Kappa test.
The contingency table to test.
The alternative hypothesis (research hypothesis) to test. If the
hypothesized kappa is left unspecified, a one-tailed test will be used. Otherwise, the
default is to use a two-sided test.
The hypothesized value for the Kappa statistic. If the test
is being used to assert independency between two raters (i.e. testing the null hypothesis
that the underlying Kappa is zero), then the
standard error will be computed with the null hypothesis parameter set to true.
Creates a new Kappa test.
The contingency table to test.
The alternative hypothesis (research hypothesis) to test. If the
hypothesized kappa is left unspecified, a one-tailed test will be used. Otherwise, the
default is to use a two-sided test.
The hypothesized value for the Kappa statistic. If the test
is being used to assert independency between two raters (i.e. testing the null hypothesis
that the underlying Kappa is zero), then the
standard error will be computed with the null hypothesis parameter set to true.
Compute Cohen's Kappa variance using the large sample approximation
given by Congalton, which is common in the remote sensing literature.
A representing the ratings.
Kappa's variance.
Compute Cohen's Kappa variance using the large sample approximation
given by Congalton, which is common in the remote sensing literature.
A representing the ratings.
Kappa's standard deviation.
Kappa's variance.
Computes the asymptotic variance for Fleiss's Kappa variance using the formulae
by (Fleiss et al, 1969) when the underlying Kappa is assumed different from zero.
A representing the ratings.
Kappa's variance.
Computes the asymptotic variance for Fleiss's Kappa variance using the formulae
by (Fleiss et al, 1969). If is set to true, the
method will return the variance under the null hypothesis.
A representing the ratings.
Kappa's standard deviation.
True to compute Kappa's variance when the null hypothesis
is true (i.e. that the underlying kappa is zer). False otherwise. Default is false.
Kappa's variance.
Computes the asymptotic variance for Fleiss's Kappa variance using the formulae
by (Fleiss et al, 1969). If is set to true, the
method will return the variance under the null hypothesis.
A representing the ratings.
Kappa's standard deviation.
True to compute Kappa's variance when the null hypothesis
is true (i.e. that the underlying kappa is zer). False otherwise. Default is false.
Kappa's variance.
Snedecor's F-Test.
A F-test is any statistical test in which the test statistic has an
F-distribution under the null hypothesis.
It is most often used when comparing statistical models that have been fit
to a data set, in order to identify the model that best fits the population
from which the data were sampled.
References:
-
Wikipedia, The Free Encyclopedia. F-Test. Available on:
http://en.wikipedia.org/wiki/F-test
// The following example has been based on the page "F-Test for Equality
// of Two Variances", from NIST/SEMATECH e-Handbook of Statistical Methods:
//
// http://www.itl.nist.gov/div898/handbook/eda/section3/eda359.htm
//
// Consider a data set containing 480 ceramic strength
// measurements for two batches of material. The summary
// statistics for each batch are shown below:
// Batch 1:
int numberOfObservations1 = 240;
// double mean1 = 688.9987;
double stdDev1 = 65.54909;
double var1 = stdDev1 * stdDev1;
// Batch 2:
int numberOfObservations2 = 240;
// double mean2 = 611.1559;
double stdDev2 = 61.85425;
double var2 = stdDev2 * stdDev2;
// Here, we will be testing the null hypothesis that
// the variances for the two batches are equal.
int degreesOfFreedom1 = numberOfObservations1 - 1;
int degreesOfFreedom2 = numberOfObservations2 - 1;
// Now we can create a F-Test to test the difference between variances
var ftest = new FTest(var1, var2, degreesOfFreedom1, degreesOfFreedom2);
double statistic = ftest.Statistic; // 1.123037
double pvalue = ftest.PValue; // 0.185191
bool significant = ftest.Significant; // false
// The F test indicates that there is not enough evidence
// to reject the null hypothesis that the two batch variances
// are equal at the 0.05 significance level.
Gets the alternative hypothesis under test. If the test is
, the null hypothesis can be rejected
in favor of this alternative hypothesis.
Gets the degrees of freedom for the
numerator in the test distribution.
Gets the degrees of freedom for the
denominator in the test distribution.
Creates a new F-Test for a given statistic with given degrees of freedom.
The variance of the first sample.
The variance of the second sample.
The degrees of freedom for the first sample.
The degrees of freedom for the second sample.
The alternative hypothesis (research hypothesis) to test.
Creates a new F-Test for a given statistic with given degrees of freedom.
The test statistic.
The degrees of freedom for the numerator.
The degrees of freedom for the denominator.
The alternative hypothesis (research hypothesis) to test.
Computes the F-test.
Creates a new F-Test.
Converts a given test statistic to a p-value.
The value of the test statistic.
The p-value for the given statistic.
Converts a given p-value to a test statistic.
The p-value.
The test statistic which would generate the given p-value.
McNemar test of homogeneity for 2 x 2 contingency tables.
McNemar's test is a non-parametric method used on nominal data. It is applied to
2 × 2 contingency tables with a dichotomous trait, with matched pairs of subjects,
to determine whether the row and column marginal frequencies are equal, i.e. if
the contingency table presents marginal homogeneity.
This is a Chi-square kind of test.
References:
-
Wikipedia contributors, "McNemar's test," Wikipedia, The Free Encyclopedia,
Available on: http://http://en.wikipedia.org/wiki/McNemar's_test.
Creates a new McNemar test.
The contingency table to test.
True to use Yate's correction of
continuity, falser otherwise. Default is false.
One-sample Kolmogorov-Smirnov (KS) test.
The Kolmogorov-Smirnov test tries to determine if a sample differs significantly
from an hypothesized theoretical probability distribution. The Kolmogorov-Smirnov
test has an interesting advantage in which it does not requires any assumptions
about the data. The distribution of the K-S test statistic does not depend on
which distribution is being tested.
The K-S test has also the advantage of being an exact test (other tests, such as the
chi-square goodness-of-fit test depends on an adequate sample size). One disadvantage
is that it requires a fully defined distribution which should not have been estimated
from the data. If the parameters of the theoretical distribution have been estimated
from the data, the critical region of the K-S test will be no longer valid.
This class uses an efficient and high-accuracy algorithm based on work by Richard
Simard (2010). Please see for more details.
References:
-
Wikipedia, The Free Encyclopedia. Kolmogorov-Smirnov Test.
Available on: http://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test
-
NIST/SEMATECH e-Handbook of Statistical Methods. Kolmogorov-Smirnov Goodness-of-Fit Test.
Available on: http://www.itl.nist.gov/div898/handbook/eda/section3/eda35g.htm
-
Richard Simard, Pierre L’Ecuyer. Computing the Two-Sided Kolmogorov-Smirnov Distribution.
Journal of Statistical Software. Volume VV, Issue II. Available on:
http://www.iro.umontreal.ca/~lecuyer/myftp/papers/ksdist.pdf
In this first example, suppose we got a new sample, and we would
like to test whether this sample has been originated from a uniform
continuous distribution.
double[] sample =
{
0.621, 0.503, 0.203, 0.477, 0.710, 0.581, 0.329, 0.480, 0.554, 0.382
};
// First, we create the distribution we would like to test against:
//
var distribution = UniformContinuousDistribution.Standard;
// Now we can define our hypothesis. The null hypothesis is that the sample
// comes from a standard uniform distribution, while the alternate is that
// the sample is not from a standard uniform distribution.
//
var kstest = new KolmogorovSmirnovTest(sample, distribution);
double statistic = kstest.Statistic; // 0.29
double pvalue = kstest.PValue; // 0.3067
bool significant = kstest.Significant; // false
Since the null hypothesis could not be rejected, then the sample
can perhaps be from a uniform distribution. However, please note
that this doesn't means that the sample *is* from the uniform, it
only means that we could not rule out the possibility.
Before we could not rule out the possibility that the sample came from
a uniform distribution, which means the sample was not very far from
uniform. This would be an indicative that it would be far from what
would be expected from a Normal distribution:
// First, we create the distribution we would like to test against:
//
NormalDistribution distribution = NormalDistribution.Standard;
// Now we can define our hypothesis. The null hypothesis is that the sample
// comes from a standard Normal distribution, while the alternate is that
// the sample is not from a standard Normal distribution.
//
var kstest = new KolmogorovSmirnovTest(sample, distribution);
double statistic = kstest.Statistic; // 0.580432
double pvalue = kstest.PValue; // 0.000999
bool significant = kstest.Significant; // true
Since the test says that the null hypothesis should be rejected, then
this can be regarded as a strong indicative that the sample does not
comes from a Normal distribution, just as we expected.
Gets the alternative hypothesis under test. If the test is
, the null hypothesis can be rejected
in favor of this alternative hypothesis.
Gets the theoretical, hypothesized distribution for the samples,
which should have been stated before any measurements.
Gets the empirical distribution measured from the sample.
Creates a new One-Sample Kolmogorov test.
The sample we would like to test as belonging to the .
A fully specified distribution (which must NOT have been estimated from the data).
Creates a new One-Sample Kolmogorov test.
The sample we would like to test as belonging to the .
A fully specified distribution (which must NOT have been estimated from the data).
The alternative hypothesis (research hypothesis) to test.
Gets the appropriate Kolmogorov-Sminorv D statistic for the samples and target distribution.
The sorted samples.
The target distribution.
The alternate hypothesis for the KS test. For , this
is the two-sided Dn statistic; for this is the one sided Dn+ statistic;
and for this is the one sided Dn- statistic.
Gets the one-sided "Dn-" Kolmogorov-Sminorv statistic for the samples and target distribution.
The sorted samples.
The target distribution.
Gets the one-sided "Dn+" Kolmogorov-Sminorv statistic for the samples and target distribution.
The sorted samples.
The target distribution.
Gets the two-sided "Dn" Kolmogorov-Sminorv statistic for the samples and target distribution.
The sorted samples.
The target distribution.
Converts a given p-value to a test statistic.
The p-value.
The test statistic which would generate the given p-value.
Converts a given test statistic to a p-value.
The value of the test statistic.
The p-value for the given statistic.
ANOVA's result table.
This class represents the results obtained from an ANOVA experiment.
Source of variation in an ANOVA experiment.
Creates a new object representation of a variation source in an ANOVA experiment.
The associated ANOVA analysis.
The name of the variation source.
Creates a new object representation of a variation source in an ANOVA experiment.
The associated ANOVA analysis.
The name of the variation source.
The degrees of freedom for the source.
The sum of squares of the source.
Creates a new object representation of a variation source in an ANOVA experiment.
The associated ANOVA analysis.
The name of the variation source.
The degrees of freedom for the source.
The mean sum of squares of the source.
The sum of squares of the source.
Creates a new object representation of a variation source in an ANOVA experiment.
The associated ANOVA analysis.
The name of the variation source.
The degrees of freedom for the source.
The sum of squares of the source.
The F-Test containing the F-Statistic for the source.
Creates a new object representation of a variation source in an ANOVA experiment.
The associated ANOVA analysis.
The name of the variation source.
The degrees of freedom for the source.
The sum of squares of the source.
The mean sum of squares of the source.
The F-Test containing the F-Statistic for the source.
Gets the ANOVA associated with this source.
Gets the name of the variation source.
Gets the sum of squares associated with the variation source.
Gets the degrees of freedom associated with the variation source.
Get the mean squares, or the variance, associated with the source.
Gets the significance of the source.
Gets the F-Statistic associated with the source's significance.
Common interface for analyses of variance.
Gets the ANOVA results in the form of a table.
One-way Analysis of Variance (ANOVA).
The one-way ANOVA is a way to test for the equality of three or more means at the same
time by using variances. In its simplest form ANOVA provides a statistical test of whether
or not the means of several groups are all equal, and therefore generalizes t-test to more
than two groups.
References:
-
Wikipedia, The Free Encyclopedia. Analysis of variance.
-
Wikipedia, The Free Encyclopedia. F-Test.
-
Wikipedia, The Free Encyclopedia. One-way ANOVA.
The following is the same example given in Wikipedia's page for the
F-Test [1]. Suppose one would like to test the effect of three levels
of a fertilizer on plant growth.
To achieve this goal, an experimenter has divided a set of 18 plants on
three groups, 6 plants each. Each group has received different levels of
the fertilizer under question.
After some months, the experimenter registers the growth for each plant:
double[][] samples =
{
new double[] { 6, 8, 4, 5, 3, 4 }, // records for the first group
new double[] { 8, 12, 9, 11, 6, 8 }, // records for the second group
new double[] { 13, 9, 11, 8, 7, 12 }, // records for the third group
};
Now, he would like to test whether the different fertilizer levels has
indeed caused any effect in plant growth. In other words, he would like
to test if the three groups are indeed significantly different.
// To do it, he runs an ANOVA test:
OneWayAnova anova = new OneWayAnova(samples);
After the Anova object has been created, one can display its findings
in the form of a standard ANOVA table by binding anova.Table to a
DataGridView or any other display object supporting data binding. To
illustrate, we could use Accord.NET's DataGridBox to inspect the
table's contents.
DataGridBox.Show(anova.Table);
Result in:
The p-level for the analysis is about 0.002, meaning the test is
significant at the 5% significance level. The experimenter would
thus reject the null hypothesis, concluding there is a strong
evidence that the three groups are indeed different. Assuming the
experiment was correctly controlled, this would be an indication
that the fertilizer does indeed affect plant growth.
[1] http://en.wikipedia.org/wiki/F_test
Gets the F-Test produced by this one-way ANOVA.
Gets the ANOVA results in the form of a table.
Creates a new one-way ANOVA test.
The sampled values.
The independent, nominal variables.
Creates a new one-way ANOVA test.
The grouped sampled values.
Two-way ANOVA model types.
References:
-
Wikipedia, The Free Encyclopedia. Analysis of variance.
Fixed-effects model (Model 1).
The fixed-effects model of analysis of variance, as known as model 1, applies
to situations in which the experimenter applies one or more treatments to the
subjects of the experiment to see if the response variable values change.
This allows the experimenter to estimate the ranges of response variable values
that the treatment would generate in the population as a whole.
References:
-
Wikipedia, The Free Encyclopedia. Analysis of variance.
Random-effects model (Model 2).
Random effects models are used when the treatments are not fixed. This occurs when
the various factor levels are sampled from a larger population. Because the levels
themselves are random variables, some assumptions and the method of contrasting the
treatments differ from ANOVA model 1.
References:
-
Wikipedia, The Free Encyclopedia. Analysis of variance.
Mixed-effects models (Model 3).
A mixed-effects model contains experimental factors of both fixed and random-effects
types, with appropriately different interpretations and analysis for the two types.
References:
-
Wikipedia, The Free Encyclopedia. Analysis of variance.
Two-way Analysis of Variance.
The two-way ANOVA is an extension of the one-way ANOVA for two independent
variables. There are three classes of models which can also be used in the
analysis, each of which determining the interpretation of the independent
variables in the analysis.
References:
-
Wikipedia, The Free Encyclopedia. Analysis of variance.
-
Carsten Dahl Mørch, ANOVA. Aalborg Universitet. Available on:
http://www.smi.hst.aau.dk/~cdahl/BiostatPhD/ANOVA.pdf
-
Uebersax, John (2006). "McNemar Tests of Marginal Homogeneity".
Available on: http://www.john-uebersax.com/stat/mcnemar.htm
-
Sun, Xuezheng; Yang, Zhao (2008). "Generalized McNemar's Test for Homogeneity of the Marginal
Distributions". Available on: http://www2.sas.com/proceedings/forum2008/382-2008.pdf
Gets the delta vector d used
in the test calculations.
Gets the covariance matrix S
used in the test calculations.
Gets the inverse covariance matrix
S^-1 used in the calculations.
Creates a new Stuart-Maxwell test.
The contingency table to test.
Sign test for the median.
In statistics, the sign test can be used to test the hypothesis that the difference
median is zero between the continuous distributions of two random variables X and Y,
in the situation when we can draw paired samples from X and Y. It is a non-parametric
test which makes very few assumptions about the nature of the distributions under test
- this means that it has very general applicability but may lack the
statistical power of other tests such as the paired-samples
t-test or the Wilcoxon signed-rank test.
References:
-
Wikipedia, The Free Encyclopedia. Sign test. Available on:
http://en.wikipedia.org/wiki/Sign_test
// This example has been adapted from the Wikipedia's page about
// the Z-Test, available from: http://en.wikipedia.org/wiki/Z-test
// We would like to check whether a sample of 20
// students with a median score of 96 points ...
double[] sample =
{
106, 115, 96, 88, 91, 88, 81, 104, 99, 68,
104, 100, 77, 98, 96, 104, 82, 94, 72, 96
};
// ... could have happened just by chance inside a
// population with an hypothesized median of 100 points.
double hypothesizedMedian = 100;
// So we start by creating the test:
SignTest test = new SignTest(sample, hypothesizedMedian,
OneSampleHypothesis.ValueIsSmallerThanHypothesis);
// Now, we can check whether this result would be
// unlikely under a standard significance level:
bool significant = test.Significant; // false (so the event was likely)
// We can also check the test statistic and its P-Value
double statistic = test.Statistic; // 5
double pvalue = test.PValue; // 0.99039
Gets the alternative hypothesis under test. If the test is
, the null hypothesis
can be rejected in favor of this alternative hypothesis.
Tests the null hypothesis that the sample median is equal to a hypothesized value.
The number of positive samples.
The total number of samples.
The alternative hypothesis (research hypothesis) to test.
Tests the null hypothesis that the sample median is equal to a hypothesized value.
The data samples from which the test will be performed.
The constant to be compared with the samples.
The alternative hypothesis (research hypothesis) to test.
Computes the one sample sign test.
Wilcoxon signed-rank test for the median.
The Wilcoxon signed-rank test is a non-parametric statistical hypothesis test
used when comparing two related samples, matched samples, or repeated measurements
on a single sample to assess whether their population mean ranks differ (i.e. it is
a paired difference test). It can be used as an alternative to the paired
Student's t-test, t-test for matched pairs, or the t-test
for dependent samples when the population cannot be assumed to be normally distributed.
The Wilcoxon signed-rank test is not the same as the Wilcoxon rank-sum
test, although both are nonparametric and involve summation of ranks.
This test uses the positive W statistic, as explained in
https://onlinecourses.science.psu.edu/stat414/node/319
References:
-
Wikipedia, The Free Encyclopedia. Wilcoxon signed-rank test. Available on:
http://en.wikipedia.org/wiki/Wilcoxon_signed-rank_test
// This example has been adapted from the Wikipedia's page about
// the Z-Test, available from: http://en.wikipedia.org/wiki/Z-test
// We would like to check whether a sample of 20
// students with a median score of 96 points ...
double[] sample =
{
106, 115, 96, 88, 91, 88, 81, 104, 99, 68,
104, 100, 77, 98, 96, 104, 82, 94, 72, 96
};
// ... could have happened just by chance inside a
// population with an hypothesized median of 100 points.
double hypothesizedMedian = 100;
// So we start by creating the test:
WilcoxonSignedRankTest test = new WilcoxonSignedRankTest(sample,
hypothesizedMedian, OneSampleHypothesis.ValueIsSmallerThanHypothesis);
// Now, we can check whether this result would be
// unlikely under a standard significance level:
bool significant = test.Significant; // false (so the event was likely)
// We can also check the test statistic and its P-Value
double statistic = test.Statistic; // 40.0
double pvalue = test.PValue; // 0.98585347446367344
Gets the alternative hypothesis under test. If the test is
, the null hypothesis can be rejected
in favor of this alternative hypothesis.
Tests the null hypothesis that the sample median is equal to a hypothesized value.
The data samples from which the test will be performed.
The constant to be compared with the samples.
The alternative hypothesis (research hypothesis) to test.
True to compute the exact distribution. May require a significant
amount of processing power for large samples (n > 12). If left at null, whether to
compute the exact or approximate distribution will depend on the number of samples.
Default is null.
Whether to account for ties when computing the
rank statistics or not. Default is true.
Sign test for two paired samples.
This is a Binomial kind of test.
Gets the alternative hypothesis under test. If the test is
, the null hypothesis
can be rejected in favor of this alternative hypothesis.
Creates a new sign test for two samples.
The number of positive samples (successes).
The total number of samples (trials).
The alternative hypothesis (research hypothesis) to test.
Creates a new sign test for two samples.
The first sample of observations.
The second sample of observations.
The alternative hypothesis (research hypothesis) to test.
Computes the two sample sign test.
Binomial test.
In statistics, the binomial test is an exact test of the statistical significance
of deviations from a theoretically expected distribution of observations into two
categories. The most common use of the binomial test is in the case where the null
hypothesis is that two categories are equally likely to occur (such as a coin toss).
When there are more than two categories, and an exact test is required, the
multinomial test, based on the multinomial
distribution, must be used instead of the binomial test.
References:
-
Wikipedia, The Free Encyclopedia. Binomial-Test. Available from:
http://en.wikipedia.org/wiki/Binomial_test
This is the second example from Wikipedia's page on hypothesis testing. In this example,
a person is tested for clairvoyance (ability of gaining information about something through
extra sensory perception; detecting something without using the known human senses.
// A person is shown the reverse of a playing card 25 times and is
// asked which of the four suits the card belongs to. Every time
// the person correctly guesses the suit of the card, we count this
// result as a correct answer. Let's suppose the person obtained 13
// correctly answers out of the 25 cards.
// Since each suit appears 1/4 of the time in the card deck, we
// would assume the probability of producing a correct answer by
// chance alone would be of 1/4.
// And finally, we must consider we are interested in which the
// subject performs better than what would expected by chance.
// In other words, that the person's probability of predicting
// a card is higher than the chance hypothesized value of 1/4.
BinomialTest test = new BinomialTest(
successes: 13, trials: 25,
hypothesizedProbability: 1.0 / 4.0,
alternate: OneSampleHypothesis.ValueIsGreaterThanHypothesis);
Console.WriteLine("Test p-Value: " + test.PValue); // ~ 0.003
Console.WriteLine("Significant? " + test.Significant); // True.
Gets the alternative hypothesis under test. If the test is
, the null hypothesis can be rejected
in favor of this alternative hypothesis.
Tests the probability of two outcomes in a series of experiments.
The experimental trials.
The hypothesized occurrence probability.
The alternative hypothesis (research hypothesis) to test.
Tests the probability of two outcomes in a series of experiments.
The number of successes in the trials.
The total number of experimental trials.
The hypothesized occurrence probability.
The alternative hypothesis (research hypothesis) to test.
Creates a Binomial test.
Computes the Binomial test.
Converts a given test statistic to a p-value.
The value of the test statistic.
The p-value for the given statistic.
Converts a given p-value to a test statistic.
The p-value.
The test statistic which would generate the given p-value.
Computes the two-tail probability using the Wilson-Sterne rule,
which defines the tail of the distribution based on a ordering
of the null probabilities of X. (Smirnoff, 2003)
References: Jeffrey S. Simonoff, Analyzing
Categorical Data, Springer, 2003 (pg 64).
Wilcoxon signed-rank test for paired samples.
Gets the alternative hypothesis under test. If the test is
, the null hypothesis can be rejected
in favor of this alternative hypothesis.
Tests whether the medians of two paired samples are different.
The first sample.
The second sample.
The alternative hypothesis (research hypothesis) to test.
True to compute the exact distribution. May require a significant
amount of processing power for large samples (n > 12). If left at null, whether to
compute the exact or approximate distribution will depend on the number of samples.
Default is null.
Whether to account for ties when computing the
rank statistics or not. Default is true.
One-sample Student's T test.
The one-sample t-test assesses whether the mean of a sample is
statistically different from a hypothesized value.
This test supports creating power analyses
through its property.
References:
-
Wikipedia, The Free Encyclopedia. Student's T-Test.
-
William M.K. Trochim. The T-Test. Research methods Knowledge Base, 2009.
Available on: http://www.le.ac.uk/bl/gat/virtualfc/Stats/ttest.html
-
Graeme D. Ruxton. The unequal variance t-test is an underused alternative to Student's
t-test and the Mann–Whitney U test. Oxford Journals, Behavioral Ecology Volume 17, Issue 4, pp.
688-690. 2006. Available on: http://beheco.oxfordjournals.org/content/17/4/688.full
// Consider a sample generated from a Gaussian
// distribution with mean 0.5 and unit variance.
double[] sample =
{
-0.849886940156521, 3.53492346633185, 1.22540422494611, 0.436945126810344, 1.21474290382610,
0.295033941700225, 0.375855651783688, 1.98969760778547, 1.90903448980048, 1.91719241342961
};
// One may rise the hypothesis that the mean of the sample is not
// significantly different from zero. In other words, the fact that
// this particular sample has mean 0.5 may be attributed to chance.
double hypothesizedMean = 0;
// Create a T-Test to check this hypothesis
TTest test = new TTest(sample, hypothesizedMean,
OneSampleHypothesis.ValueIsDifferentFromHypothesis);
// Check if the mean is significantly different
test.Significant should be true
// Now, we would like to test if the sample mean is
// significantly greater than the hypothesized zero.
// Create a T-Test to check this hypothesis
TTest greater = new TTest(sample, hypothesizedMean,
OneSampleHypothesis.ValueIsGreaterThanHypothesis);
// Check if the mean is significantly larger
greater.Significant should be true
// Now, we would like to test if the sample mean is
// significantly smaller than the hypothesized zero.
// Create a T-Test to check this hypothesis
TTest smaller = new TTest(sample, hypothesizedMean,
OneSampleHypothesis.ValueIsSmallerThanHypothesis);
// Check if the mean is significantly smaller
smaller.Significant should be false
Gets the power analysis for the test, if available.
Gets the standard error of the estimated value.
Gets the estimated parameter value, such as the sample's mean value.
Gets the hypothesized parameter value.
Gets the 95% confidence interval for the .
Gets the alternative hypothesis under test. If the test is
, the null hypothesis can be rejected
in favor of this alternative hypothesis.
Gets a confidence interval for the estimated value
within the given confidence level percentage.
The confidence level. Default is 0.95.
A confidence interval for the estimated value.
Tests the null hypothesis that the population mean is equal to a specified value.
The test statistic.
The degrees of freedom for the test distribution.
The alternative hypothesis (research hypothesis) to test.
Tests the null hypothesis that the population mean is equal to a specified value.
The estimated value (θ).
The standard error of the estimation (SE).
The hypothesized value (θ').
The degrees of freedom for the test distribution.
The alternative hypothesis (research hypothesis) to test.
Tests the null hypothesis that the population mean is equal to a specified value.
The data samples from which the test will be performed.
The constant to be compared with the samples.
The alternative hypothesis (research hypothesis) to test.
Creates a T-Test.
Tests the null hypothesis that the population mean is equal to a specified value.
The sample's mean value.
The standard deviation for the samples.
The number of observations in the sample.
The constant to be compared with the samples.
The alternative hypothesis (research hypothesis) to test.
Computes the T-Test.
Computes the T-test.
Computes the T-test.
Update event.
Converts a given p-value to a test statistic.
The p-value.
The test statistic which would generate the given p-value.
Converts a given test statistic to a p-value.
The value of the test statistic.
The p-value for the given statistic.
Converts a given test statistic to a p-value.
The value of the test statistic.
The tail of the test distribution.
The test distribution.
The p-value for the given statistic.
Converts a given p-value to a test statistic.
The p-value.
The tail of the test distribution.
The test distribution.
The test statistic which would generate the given p-value.
Two-sample Kolmogorov-Smirnov (KS) test.
The Kolmogorov-Smirnov test tries to determine if two samples have been
drawn from the same probability distribution. The Kolmogorov-Smirnov test
has an interesting advantage in which it does not requires any assumptions
about the data. The distribution of the K-S test statistic does not depend
on which distribution is being tested.
The K-S test has also the advantage of being an exact test (other tests, such as the
chi-square goodness-of-fit test depends on an adequate sample size). One disadvantage
is that it requires a fully defined distribution which should not have been estimated
from the data. If the parameters of the theoretical distribution have been estimated
from the data, the critical region of the K-S test will be no longer valid.
The two-sample KS test is one of the most useful and general nonparametric methods for
comparing two samples, as it is sensitive to differences in both location and shape of
the empirical cumulative distribution functions of the two samples.
This class uses an efficient and high-accuracy algorithm based on work by Richard
Simard (2010). Please see for more details.
References:
-
Wikipedia, The Free Encyclopedia. Kolmogorov-Smirnov Test.
Available at: http://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test
-
NIST/SEMATECH e-Handbook of Statistical Methods. Kolmogorov-Smirnov Goodness-of-Fit Test.
Available at: http://www.itl.nist.gov/div898/handbook/eda/section3/eda35g.htm
-
Richard Simard, Pierre L’Ecuyer. Computing the Two-Sided Kolmogorov-Smirnov Distribution.
Journal of Statistical Software. Volume VV, Issue II. Available at:
http://www.iro.umontreal.ca/~lecuyer/myftp/papers/ksdist.pdf
-
Kirkman, T.W. (1996) Statistics to Use. Available at:
http://www.physics.csbsju.edu/stats/
In the following example, we will be creating a K-S test to verify
if two samples have been drawn from different populations. In this
example, we will first generate a number of samples from two different
distributions and then check if the K-S can indeed see the difference
between them:
// Generate 15 points from a Normal distribution with mean 5 and sigma 2
double[] sample1 = new NormalDistribution(mean: 5, stdDev: 1).Generate(25);
// Generate 15 points from an uniform distribution from 0 to 10
double[] sample2 = new UniformContinuousDistribution(a: 0, b: 10).Generate(25);
// Now we can create a K-S test and test the unequal hypothesis:
var test = new TwoSampleKolmogorovSmirnovTest(sample1, sample2,
TwoSampleKolmogorovSmirnovTestHypothesis.SamplesDistributionsAreUnequal);
bool significant = test.Significant; // outputs true
The following example comes from the stats page of the College of Saint Benedict and Saint John's
University (Kirkman, 1996). It is a very interesting example as it shows a case in which a t-test
fails to see a difference between the samples because of the non-normality of the sample's
distributions. The Kolmogorov-Smirnov nonparametric test, on the other hand, succeeds.
The example deals with the preference of bees between two nearby blooming trees in an empty field.
The experimenter has collected data measuring how much time does a bee spent near a particular
tree. The time starts to be measured when a bee first touches the tree, and is stopped when the bee
moves more than 1 meter far from it. The samples below represents the measured time, in seconds, of
the observed bees for each of the trees.
double[] redwell =
{
23.4, 30.9, 18.8, 23.0, 21.4, 1, 24.6, 23.8, 24.1, 18.7, 16.3, 20.3,
14.9, 35.4, 21.6, 21.2, 21.0, 15.0, 15.6, 24.0, 34.6, 40.9, 30.7,
24.5, 16.6, 1, 21.7, 1, 23.6, 1, 25.7, 19.3, 46.9, 23.3, 21.8, 33.3,
24.9, 24.4, 1, 19.8, 17.2, 21.5, 25.5, 23.3, 18.6, 22.0, 29.8, 33.3,
1, 21.3, 18.6, 26.8, 19.4, 21.1, 21.2, 20.5, 19.8, 26.3, 39.3, 21.4,
22.6, 1, 35.3, 7.0, 19.3, 21.3, 10.1, 20.2, 1, 36.2, 16.7, 21.1, 39.1,
19.9, 32.1, 23.1, 21.8, 30.4, 19.62, 15.5
};
double[] whitney =
{
16.5, 1, 22.6, 25.3, 23.7, 1, 23.3, 23.9, 16.2, 23.0, 21.6, 10.8, 12.2,
23.6, 10.1, 24.4, 16.4, 11.7, 17.7, 34.3, 24.3, 18.7, 27.5, 25.8, 22.5,
14.2, 21.7, 1, 31.2, 13.8, 29.7, 23.1, 26.1, 25.1, 23.4, 21.7, 24.4, 13.2,
22.1, 26.7, 22.7, 1, 18.2, 28.7, 29.1, 27.4, 22.3, 13.2, 22.5, 25.0, 1,
6.6, 23.7, 23.5, 17.3, 24.6, 27.8, 29.7, 25.3, 19.9, 18.2, 26.2, 20.4,
23.3, 26.7, 26.0, 1, 25.1, 33.1, 35.0, 25.3, 23.6, 23.2, 20.2, 24.7, 22.6,
39.1, 26.5, 22.7
};
// Create a t-test as a first attempt.
var t = new TwoSampleTTest(redwell, whitney);
Console.WriteLine("T-Test");
Console.WriteLine("Test p-value: " + t.PValue); // ~0.837
Console.WriteLine("Significant? " + t.Significant); // false
// Create a non-parametric Kolmogorov-Smirnov test
var ks = new TwoSampleKolmogorovSmirnovTest(redwell, whitney);
Console.WriteLine("KS-Test");
Console.WriteLine("Test p-value: " + ks.PValue); // ~0.038
Console.WriteLine("Significant? " + ks.Significant); // true
Gets the alternative hypothesis under test. If the test is
, the null hypothesis can be rejected
in favor of this alternative hypothesis.
Gets the first empirical distribution being tested.
Gets the second empirical distribution being tested.
Creates a new Two-Sample Kolmogorov test.
The first sample.
The second sample.
Creates a new Two-Sample Kolmogorov test.
The first sample.
The second sample.
The alternative hypothesis (research hypothesis) to test.
Converts a given p-value to a test statistic.
The p-value.
The test statistic which would generate the given p-value.
Converts a given test statistic to a p-value.
The value of the test statistic.
The p-value for the given statistic.
Two-sample Student's T test.
The two-sample t-test assesses whether the means of two groups are statistically
different from each other.
References:
-
Wikipedia, The Free Encyclopedia. Student's T-Test.
-
William M.K. Trochim. The T-Test. Research methods Knowledge Base, 2009.
Available on: http://www.le.ac.uk/bl/gat/virtualfc/Stats/ttest.html
-
Graeme D. Ruxton. The unequal variance t-test is an underused alternative to Student's
t-test and the Mann–Whitney U test. Oxford Journals, Behavioral Ecology Volume 17, Issue 4, pp.
688-690. 2006. Available on: http://beheco.oxfordjournals.org/content/17/4/688.full
Gets the power analysis for the test, if available.
Gets the alternative hypothesis under test. If the test is
, the null hypothesis can be rejected
in favor of this alternative hypothesis.
Gets whether the test assumes equal sample variance.
Gets the standard error for the difference.
Gets the combined sample variance.
Gets the estimated value for the first sample.
Gets the estimated value for the second sample.
Gets the hypothesized difference between the two estimated values.
Gets the actual difference between the two estimated values.
Gets the degrees of freedom for the test statistic.
Gets the 95% confidence interval for the
statistic.
Gets a confidence interval for the
statistic within the given confidence level percentage.
The confidence level. Default is 0.95.
A confidence interval for the estimated value.
Tests whether the means of two samples are different.
The first sample.
The second sample.
The hypothesized sample difference.
True to assume equal variances, false otherwise. Default is true.
The alternative hypothesis (research hypothesis) to test.
Tests whether the means of two samples are different.
The first sample's mean.
The second sample's mean.
The first sample's variance.
The second sample's variance.
The number of observations in the first sample.
The number of observations in the second sample.
True assume equal variances, false otherwise. Default is true.
The hypothesized sample difference.
The alternative hypothesis (research hypothesis) to test.
Creates a new two-sample T-Test.
Computes the T Test.
Update event.
Converts a given p-value to a test statistic.
The p-value.
The test statistic which would generate the given p-value.
Converts a given test statistic to a p-value.
The value of the test statistic.
The p-value for the given statistic.
Two sample Z-Test.
References:
-
Wikipedia, The Free Encyclopedia. Z-Test. Available on:
http://en.wikipedia.org/wiki/Z-test
Gets the power analysis for the test, if available.
Gets the alternative hypothesis under test. If the test is
, the null hypothesis can be rejected
in favor of this alternative hypothesis.
Gets the standard error for the difference.
Gets the estimated value for the first sample.
Gets the estimated value for the second sample.
Gets the hypothesized difference between the two estimated values.
Gets the actual difference between the two estimated values.
Gets the 95% confidence interval for the
statistic.
Gets a confidence interval for the
statistic within the given confidence level percentage.
The confidence level. Default is 0.95.
A confidence interval for the estimated value.
Constructs a Z test.
The first data sample.
The second data sample.
The hypothesized sample difference.
The alternative hypothesis (research hypothesis) to test.
Constructs a Z test.
The first sample's mean.
The second sample's mean.
The first sample's variance.
The second sample's variance.
The number of observations in the first sample.
The number of observations in the second sample.
The hypothesized sample difference.
The alternative hypothesis (research hypothesis) to test.
Constructs a Z test.
Computes the Z test.
Computes the Z test.
Computes the Z test.
Update event.
Converts a given p-value to a test statistic.
The p-value.
The test statistic which would generate the given p-value.
Converts a given test statistic to a p-value.
The value of the test statistic.
The p-value for the given statistic.
Wald's Test using the Normal distribution.
The Wald test is a parametric statistical test named after Abraham Wald
with a great variety of uses. Whenever a relationship within or between
data items can be expressed as a statistical model with parameters to be
estimated from a sample, the Wald test can be used to test the true value
of the parameter based on the sample estimate.
Under the Wald statistical test, the maximum likelihood estimate of the
parameter(s) of interest θ is compared with the proposed value θ', with
the assumption that the difference between the two will be approximately
normal.
References:
-
Wikipedia, The Free Encyclopedia. Wald Test. Available on:
http://en.wikipedia.org/wiki/Wald_test
Constructs a Wald's test.
The test statistic, as given by (θ-θ')/SE.
Constructs a Wald's test.
The estimated value (θ).
The hypothesized value (θ').
The standard error of the estimation (SE).
One-sample Z-Test (location test).
The term Z-test is often used to refer specifically to the one-sample
location test comparing the mean of a set of measurements to a given
constant. Due to the central limit theorem, many test statistics are
approximately normally distributed for large samples. Therefore, many
statistical tests can be performed as approximate Z-tests if the sample
size is large.
If the test is , the null hypothesis
can be rejected in favor of the alternate hypothesis
specified at the creation of the test.
This test supports creating power analyses
through its property.
References:
-
Wikipedia, The Free Encyclopedia. Z-Test. Available on:
http://en.wikipedia.org/wiki/Z-test
This example has been gathered from the Wikipedia's page about
the Z-Test, available from: http://en.wikipedia.org/wiki/Z-test
Suppose there is a text comprehension test being run across
a given demographic region. The mean score of the population
from this entire region are around 100 points, with a standard
deviation of 12 points.
There is a local school, however, whose 55 students attained
an average score in the test of only about 96 points. Would
their scores be surprisingly that low, or could this event
have happened due to chance?
// So we would like to check that a sample of
// 55 students with a mean score of 96 points:
int sampleSize = 55;
double sampleMean = 96;
// Was expected to have happened by chance in a population with
// an hypothesized mean of 100 points and standard deviation of
// about 12 points:
double standardDeviation = 12;
double hypothesizedMean = 100;
// So we start by creating the test:
ZTest test = new ZTest(sampleMean, standardDeviation, sampleSize,
hypothesizedMean, OneSampleHypothesis.ValueIsSmallerThanHypothesis);
// Now, we can check whether this result would be
// unlikely under a standard significance level:
bool significant = test.Significant;
// We can also check the test statistic and its P-Value
double statistic = test.Statistic;
double pvalue = test.PValue;
Gets the power analysis for the test, if available.
Gets the standard error of the estimated value.
Gets the estimated value, such as the mean estimated from a sample.
Gets the hypothesized value.
Gets the 95% confidence interval for the .
Gets the alternative hypothesis under test. If the test is
, the null hypothesis can be rejected
in favor of this alternative hypothesis.
Gets a confidence interval for the
statistic within the given confidence level percentage.
The confidence level. Default is 0.95.
A confidence interval for the estimated value.
Constructs a Z test.
The data samples from which the test will be performed.
The constant to be compared with the samples.
The alternative hypothesis (research hypothesis) to test.
Constructs a Z test.
The sample's mean.
The sample's standard error.
The hypothesized value for the distribution's mean.
The alternative hypothesis (research hypothesis) to test.
Constructs a Z test.
The sample's mean.
The sample's standard deviation.
The hypothesized value for the distribution's mean.
The sample's size.
The alternative hypothesis (research hypothesis) to test.
Constructs a Z test.
The test statistic, as given by (x-μ)/SE.
The alternate hypothesis to test.
Computes the Z test.
Computes the Z test.
Constructs a T-Test.
Update event.
Converts a given p-value to a test statistic.
The p-value.
The test statistic which would generate the given p-value.
Converts a given test statistic to a p-value.
The value of the test statistic.
The p-value for the given statistic.
Converts a given test statistic to a p-value.
The value of the test statistic.
The tail of the test distribution.
The p-value for the given statistic.
Converts a given p-value to a test statistic.
The p-value.
The tail of the test distribution.
The test statistic which would generate the given p-value.
Hypothesis type
The type of the hypothesis being made expresses the way in
which a value of a parameter may deviate from that assumed
in the null hypothesis. It can either state that a value is
higher, lower or simply different than the one assumed under
the null hypothesis.
The test considers the two tails from a probability distribution.
The two-tailed test is a statistical test in which a given statistical
hypothesis, H0 (the null hypothesis), will be rejected when the value of
the test statistic is either sufficiently small or sufficiently large.
The test considers the upper (right) tail from a probability distribution.
The one-tailed, upper tail test is a statistical test in which a given
statistical hypothesis, H0 (the null hypothesis), will be rejected when
the value of the test statistic is sufficiently large.
The test considers the lower (left) tail from a probability distribution.
The one-tailed, lower tail test is a statistical test in which a given
statistical hypothesis, H0 (the null hypothesis), will be rejected when
the value of the test statistic is sufficiently small.
Common test Hypothesis for one sample tests, such
as and .
Tests if the mean (or the parameter under test)
is significantly different from the hypothesized
value, without considering the direction for this
difference.
Tests if the mean (or the parameter under test)
is significantly greater (larger, bigger) than
the hypothesized value.
Tests if the mean (or the parameter under test)
is significantly smaller (lesser) than the
hypothesized value.
Common test Hypothesis for two sample tests, such as
and .
Tests if the mean (or the parameter under test) of
the first sample is different from the mean of the
second sample, without considering any particular
direction for the difference.
Tests if the mean (or the parameter under test) of
the first sample is greater (larger, bigger) than
the mean of the second sample.
Tests if the mean (or the parameter under test) of
the first sample is smaller (lesser) than the mean
of the second sample.
Hypothesis for the one-sample Kolmogorov-Smirnov test.
Tests whether the sample's distribution is
different from the reference distribution.
Tests whether the distribution of one sample is greater
than the reference distribution, in a statistical sense.
Tests whether the distribution of one sample is smaller
than the reference distribution, in a statistical sense.
Test hypothesis for the two-sample Kolmogorov-Smirnov tests.
Tests whether samples have been drawn
from significantly unequal distributions.
Tests whether the distribution of one sample is
greater than the other, in a statistical sense.
Tests whether the distribution of one sample is
smaller than the other, in a statistical sense.
Hypothesis for the one-sample Grubb's test.
Tests whether there is at least one outlier in the data.
Tests whether the maximum value in the data is actually an outlier.
Tests whether the minimum value in the data is actually an outlier.
Scatter Plot class.
Gets the integer label associated with this class.
Gets the indices of all points of this class.
Gets all X values of this class.
Gets all Y values of this class.
Gets or sets the class' text label.
Returns an enumerator that iterates through a collection.
An object that can be used to iterate through the collection.
Returns an enumerator that iterates through a collection.
An object that can be used to iterate through the collection.
Collection of Histogram bins. This class cannot be instantiated.
Searches for a bin containing the specified value.
The value to search for.
The histogram bin containing the searched value.
Attempts to find the index of the bin containing the specified value.
The value to search for.
The index of the bin containing the specified value.
Histogram Bin
A "bin" is a container, where each element stores the total number of observations of a sample
whose values lie within a given range. A histogram of a sample consists of a list of such bins
whose range does not overlap with each other; or in other words, bins that are mutually exclusive.
Unless is true, the ranges of all bins i are
defined as Edge[i] <= x < Edge[i+1]. Otherwise, the last bin will have an inclusive upper
bound (i.e. will be defined as Edge[i] <= x <= Edge[i+1].
Gets the actual range of data this bin represents.
Gets the Width (range) for this histogram bin.
Gets the Value (number of occurrences of a variable in a range)
for this histogram bin.
Gets whether the Histogram Bin contains the given value.
Optimum histogram bin size adjustment rule.
Does not attempts to automatically calculate
an optimum bin width and preserves the current
histogram organization.
Calculates the optimum bin width as 3.49σN, where σ
is the sample standard deviation and N is the number
of samples.
Scott, D. 1979. On optimal and data-based histograms. Biometrika, 66:605-610.
Calculates the optimum bin width as ceiling( log2(N) + 1 )m
where N is the number of samples. The rule implicitly bases
the bin sizes on the range of the data, and can perform poorly
if n < 30.
Calculates the optimum bin width as the square root of the
number of samples. This is the same rule used by Microsoft (c)
Excel and many others.
Histogram.
In a more general mathematical sense, a histogram is a mapping Mi
that counts the number of observations that fall into various
disjoint categories (known as bins).
This class represents a Histogram mapping of Discrete or Continuous
data. To use it as a discrete mapping, pass a bin size (length) of 1.
To use it as a continuous mapping, pass any real number instead.
Currently, only a constant bin width is supported.
Constructs an empty histogram
Constructs an empty histogram
The values to be binned in the histogram.
Constructs an empty histogram
The values to be binned in the histogram.
Initializes the histogram's bins.
Sets the histogram's bin ranges (edges).
Update statistical value of the histogram.
The method recalculates statistical values of the histogram, like mean,
standard deviation, etc., in the case if histogram's values were changed directly.
The method should be called only in the case if histogram's values were retrieved
through property and updated after that.
Gets the Bin values of this Histogram.
Bin index.
The number of hits of the selected bin.
Gets the Bin values for this Histogram.
Gets the Range of the values in this Histogram.
Gets the edges of each bin in this Histogram.
Gets the collection of bins of this Histogram.
Gets or sets whether this histogram represents a cumulative distribution.
Gets or sets the bin size auto adjustment rule
to be used when computing this histogram from
new data. Default is .
The bin size auto adjustment rule.
Mean value.
The property allows to retrieve mean value of the histogram.
Standard deviation.
The property allows to retrieve standard deviation value of the histogram.
Median value.
The property allows to retrieve median value of the histogram.
Minimum value.
The property allows to retrieve minimum value of the histogram with non zero
hits count.
Maximum value.
The property allows to retrieve maximum value of the histogram with non zero
hits count.
Total count of values.
The property represents total count of values contributed to the histogram, which is
essentially sum of the array.
Gets or sets a value indicating whether the last bin
should have an inclusive upper bound. Default is true.
If set to false, the last bin's range will be defined
as Edge[i] <= x < Edge[i+1]. If set to true, the
last bin will have an inclusive upper bound and be defined as
Edge[i] <= x <= Edge[i+1] instead.
true if the last bin should have an inclusive upper bound;
false otherwise.
Get range around median containing specified percentage of values.
Values percentage around median.
Returns the range which containes specifies percentage of values.
The method calculates range of stochastic variable, which summary probability
comprises the specified percentage of histogram's hits.
Sample usage:
// create histogram
Histogram histogram = new Histogram( new int[10] { 0, 0, 1, 3, 6, 8, 11, 0, 0, 0 } );
// get 50% range
IntRange range = histogram.GetRange( 0.5 );
// show the range ([4, 6])
Console.WriteLine( "50% range = [" + range.Min + ", " + range.Max + "]" );
Computes (populates) an Histogram mapping with values from a sample.
The values to be binned in the histogram.
The desired width for the histogram's bins.
Computes (populates) an Histogram mapping with values from a sample.
The values to be binned in the histogram.
The desired number of histogram's bins.
Computes (populates) an Histogram mapping with values from a sample.
The values to be binned in the histogram.
The desired number of histogram's bins.
Whether to include an extra upper bin going to infinity.
Computes (populates) an Histogram mapping with values from a sample.
The values to be binned in the histogram.
The desired number of histogram's bins.
The desired width for the histogram's bins.
Computes (populates) an Histogram mapping with values from a sample.
The values to be binned in the histogram.
Actually computes the histogram.
Computes the optimum number of bins based on a .
Integer array implicit conversion.
Converts this histogram into an integer array representation.
Creates a histogram of values from a sample.
The values to be binned in the histogram.
A histogram reflecting the distribution of values in the sample.
Subtracts one histogram from the other, storing
results in a new histogram, without changing the
current instance.
The histogram whose bin values will be subtracted.
A new containing the result of this operation.
Subtracts one histogram from the other, storing
results in a new histogram, without changing the
current instance.
The histogram whose bin values will be subtracted.
A new containing the result of this operation.
Adds one histogram from the other, storing
results in a new histogram, without changing the
current instance.
The histogram whose bin values will be added.
A new containing the result of this operation.
Adds one histogram from the other, storing
results in a new histogram, without changing the
current instance.
The histogram whose bin values will be added.
A new containing the result of this operation.
Multiplies one histogram from the other, storing
results in a new histogram, without changing the
current instance.
The histogram whose bin values will be multiplied.
A new containing the result of this operation.
Multiplies one histogram from the other, storing
results in a new histogram, without changing the
current instance.
The value to be multiplied.
A new containing the result of this operation.
Adds a value to each histogram bin.
The value to be added.
A new containing the result of this operation.
Subtracts a value to each histogram bin.
The value to be subtracted.
A new containing the result of this operation.
Creates a new object that is a copy of the current instance.
A new object that is a copy of this instance.
Scatter Plot.
Gets the title of the scatter plot.
Gets the name of the X-axis.
Gets the name of the Y-axis.
Gets the name of the label axis.
Gets the values associated with the X-axis.
Gets the corresponding Y values associated with each X.
Gets the label of each (x,y) pair.
Gets an integer array containing the integer labels
associated with each of the classes in the scatter plot.
Gets the class labels for each of the classes in the plot.
Gets a collection containing information about
each of the classes presented in the scatter plot.
Constructs an empty Scatter plot.
Constructs an empty Scatter plot with given title.
Scatter plot title.
Constructs an empty scatter plot with
given title and axis names.
Scatter Plot title.
Title for the x-axis.
Title for the y-axis.
Constructs an empty Scatter Plot with
given title and axis names.
Scatter Plot title.
Title for the x-axis.
Title for the y-axis.
Title for the labels.
Computes the scatter plot.
Array of values.
Computes the scatter plot.
Array of X values.
Array of corresponding Y values.
Computes the scatter plot.
Array of X values.
Array of corresponding Y values.
Array of integer labels defining a class for each (x,y) pair.
Computes the scatter plot.
Array of { x,y } values.
Computes the scatter plot.
Array of { x,y } values.
Array of integer labels defining a class for each (x,y) pair.
Computes the scatter plot.
Array of { x,y } values.
Computes the scatter plot.
Array of { x,y } values.
Array of integer labels defining a class for each (x,y) pair.
Bhattacharyya distance.
Initializes a new instance of the class.
Bhattacharyya distance between two histograms.
The first histogram.
The second histogram.
The Bhattacharyya between the two histograms.
Bhattacharyya distance between two histograms.
The first histogram.
The second histogram.
The Bhattacharyya between the two histograms.
Bhattacharyya distance between two histograms.
The first histogram.
The second histogram.
The Bhattacharyya between the two histograms.
Bhattacharyya distance between two datasets, assuming
their contents can be modelled by multivariate Gaussians.
The first dataset.
The second dataset.
The Bhattacharyya between the two datasets.
Bhattacharyya distance between two datasets, assuming
their contents can be modelled by multivariate Gaussians.
The first dataset.
The second dataset.
The Bhattacharyya between the two datasets.
Bhattacharyya distance between two Gaussian distributions.
Mean for the first distribution.
Covariance matrix for the first distribution.
Mean for the second distribution.
Covariance matrix for the second distribution.
The Bhattacharyya distance between the two distributions.
Bhattacharyya distance between two Gaussian distributions.
Mean for the first distribution.
Covariance matrix for the first distribution.
Mean for the second distribution.
Covariance matrix for the second distribution.
The Bhattacharyya distance between the two distributions.
Bhattacharyya distance between two Gaussian distributions.
Mean for the first distribution.
Covariance matrix for the first distribution.
Mean for the second distribution.
Covariance matrix for the second distribution.
The logarithm of the determinant for
the covariance matrix of the first distribution.
The logarithm of the determinant for
the covariance matrix of the second distribution.
The Bhattacharyya distance between the two distributions.
Bhattacharyya distance between two Gaussian distributions.
Mean for the first distribution.
Covariance matrix for the first distribution.
Mean for the second distribution.
Covariance matrix for the second distribution.
The logarithm of the determinant for
the covariance matrix of the first distribution.
The logarithm of the determinant for
the covariance matrix of the second distribution.
The Bhattacharyya distance between the two distributions.
Bhattacharyya distance between two Gaussian distributions.
The first Normal distribution.
The second Normal distribution.
The Bhattacharyya distance between the two distributions.
Log-likelihood distance between a sample and a statistical distribution.
The type of the distribution.
Initializes a new instance of the class.
Computes the distance d(x,y) between points
and .
The first point x.
The second point y.
A double-precision value representing the distance d(x,y)
between and according
to the distance function implemented by this class.
Computes the distance d(x,y) between points
and .
The first point x.
The second point y.
A double-precision value representing the distance d(x,y)
between and according
to the distance function implemented by this class.