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/*
* File: StudentTDistribution.java
* Authors: Kevin R. Dixon
* Company: Sandia National Laboratories
* Project: Cognitive Foundry
*
* Copyright August 9, 2007, Sandia Corporation. Under the terms of Contract
* DE-AC04-94AL85000, there is a non-exclusive license for use of this work by
* or on behalf of the U.S. Government. Export of this program may require a
* license from the United States Government. See CopyrightHistory.txt for
* complete details.
*
*/
package gov.sandia.cognition.statistics.distribution;
import gov.sandia.cognition.annotation.PublicationReference;
import gov.sandia.cognition.annotation.PublicationReferences;
import gov.sandia.cognition.annotation.PublicationType;
import gov.sandia.cognition.math.MathUtil;
import gov.sandia.cognition.math.UnivariateStatisticsUtil;
import gov.sandia.cognition.math.matrix.VectorFactory;
import gov.sandia.cognition.math.matrix.Vector;
import gov.sandia.cognition.statistics.AbstractClosedFormSmoothUnivariateDistribution;
import gov.sandia.cognition.statistics.DistributionEstimator;
import gov.sandia.cognition.statistics.DistributionWeightedEstimator;
import gov.sandia.cognition.statistics.EstimableDistribution;
import gov.sandia.cognition.statistics.InvertibleCumulativeDistributionFunction;
import gov.sandia.cognition.statistics.UnivariateProbabilityDensityFunction;
import gov.sandia.cognition.statistics.SmoothCumulativeDistributionFunction;
import gov.sandia.cognition.statistics.method.InverseTransformSampling;
import gov.sandia.cognition.util.AbstractCloneableSerializable;
import gov.sandia.cognition.util.Pair;
import gov.sandia.cognition.util.WeightedValue;
import java.util.ArrayList;
import java.util.Collection;
import java.util.Random;
/**
* Defines a noncentral Student-t Distribution.
*
* @author Kevin R. Dixon
* @since 2.0
*
*/
@PublicationReferences(
references={
@PublicationReference(
author="Christopher Bishop",
title="Pattern Recognition and Machine Learning",
type=PublicationType.Book,
year=2006,
pages={102,105}
)
,
@PublicationReference(
author="Wikipedia",
title="Student's t-distribution",
type=PublicationType.WebPage,
year=2009,
url="http://en.wikipedia.org/wiki/Student_t_distribution"
)
}
)
public class StudentTDistribution
extends AbstractClosedFormSmoothUnivariateDistribution
implements EstimableDistribution
{
/**
* Default degrees of freedom, {@value}.
*/
public static final double DEFAULT_DEGREES_OF_FREEDOM = 3.0;
/**
* Default mean, {@value}.
*/
public static final double DEFAULT_MEAN = 0.0;
/**
* Default precision, {@value}.
*/
public static final double DEFAULT_PRECISION = 1.0;
/**
* Precision, which is proportionate to the inverseRootFinder of variance, of the
* distribution, must be greater than zero.
* Note that we are using "precision" instead of "variance" because the
* variance of the Student-t has another scale factor based on the dofs
* and we want to avoid confusion.
*/
protected double precision;
/**
* Mean, or noncentrality parameter, of the distribution
*/
protected double mean;
/**
* Degrees of freedom in the distribution, usually the number of
* datapoints - 1, DOFs must be greater than zero.
*/
protected double degreesOfFreedom;
/**
* Default degrees of freedom.
*/
public StudentTDistribution()
{
this( DEFAULT_DEGREES_OF_FREEDOM );
}
/**
* Creates a new instance of StudentTDistribution
* @param degreesOfFreedom
* Degrees of freedom in the distribution, usually the number of
* datapoints - 1, DOFs must be greater than zero.
*/
public StudentTDistribution(
final double degreesOfFreedom )
{
this( degreesOfFreedom, DEFAULT_MEAN, DEFAULT_PRECISION );
}
/**
* Creates a new instance of StudentTDistribution
* @param degreesOfFreedom
* Degrees of freedom in the distribution, usually the number of
* datapoints - 1, DOFs must be greater than zero.
* @param mean
* Mean, or noncentrality parameter, of the distribution
* @param precision
* Precision, inverseRootFinder variance, of the distribution, must be greater
* than zero.
*/
public StudentTDistribution(
final double degreesOfFreedom,
final double mean,
final double precision )
{
this.setDegreesOfFreedom(degreesOfFreedom);
this.setPrecision(precision);
this.setMean(mean);
}
/**
* Copy constructor
* @param other
* StudentTDistribution to copy
*/
public StudentTDistribution(
final StudentTDistribution other )
{
this( other.getDegreesOfFreedom(), other.getMean(), other.getPrecision() );
}
@Override
public StudentTDistribution clone()
{
return (StudentTDistribution) super.clone();
}
/**
* Getter for degreesOfFreedom
* @return
* Degrees of freedom in the distribution, usually the number of
* datapoints - 1, DOFs must be greater than zero.
*/
public double getDegreesOfFreedom()
{
return this.degreesOfFreedom;
}
/**
* Setter for degreesOfFreedom
* @param degreesOfFreedom
* Degrees of freedom in the distribution, usually the number of
* datapoints - 1, DOFs must be greater than zero.
*/
public void setDegreesOfFreedom(
final double degreesOfFreedom )
{
if( degreesOfFreedom <= 0.0 )
{
throw new IllegalArgumentException( "DOF must be > 0.0" );
}
this.degreesOfFreedom = degreesOfFreedom;
}
@Override
public double getVariance()
{
final double v = this.getDegreesOfFreedom();
if( v > 2.0 )
{
return v / (v - 2.0) / this.precision;
}
else
{
return Double.POSITIVE_INFINITY;
}
}
/**
* Returns the parameters of this PDF, which is a
* 1-dimensional Vector containing the degrees of freedom
* @return
* 1-dimensional Vector containing the degrees of freedom
*/
@Override
public Vector convertToVector()
{
return VectorFactory.getDefault().copyValues(
this.getDegreesOfFreedom(), this.getMean(), this.getPrecision() );
}
/**
* Sets the parameters of this PDF, which must be a
* 1-dimensional Vector containing the degrees of freedom
* @param parameters
* 1-dimensional Vector containing the degrees of freedom
*/
@Override
public void convertFromVector(
final Vector parameters )
{
parameters.assertDimensionalityEquals(3);
this.setDegreesOfFreedom( parameters.getElement( 0 ) );
this.setMean( parameters.getElement(1) );
this.setPrecision( parameters.getElement(2) );
}
@Override
public void sampleInto(
final Random random,
final double[] output,
final int start,
final int length)
{
final double[] Vs = ChiSquareDistribution.sampleAsDoubles(
this.degreesOfFreedom, random, length);
final double sp = Math.sqrt(this.precision);
final int end = start + length;
for (int i = start; i < end; i++)
{
final double z = random.nextGaussian()/sp;
final double v = Vs[i - start];
output[i] = z * Math.sqrt(this.degreesOfFreedom / v) + this.mean;
}
}
@Override
public StudentTDistribution.CDF getCDF()
{
return new StudentTDistribution.CDF( this );
}
@Override
public StudentTDistribution.PDF getProbabilityFunction()
{
return new StudentTDistribution.PDF( this );
}
/**
* Getter for precision
* @return
* Precision, inverseRootFinder variance, of the distribution, must be greater
* than zero.
*/
public double getPrecision()
{
return this.precision;
}
/**
* Setter for precision
* @param precision
* Precision, inverseRootFinder variance, of the distribution, must be greater
* than zero.
*/
public void setPrecision(
final double precision)
{
if( precision <= 0.0 )
{
throw new IllegalArgumentException(
"Precision must be > 0.0" );
}
this.precision = precision;
}
/**
* Setter for mean
* @param mean
* Mean, or noncentrality parameter, of the distribution
*/
public void setMean(
final double mean)
{
this.mean = mean;
}
@Override
public double getMeanAsDouble()
{
return this.mean;
}
@Override
public Double getMinSupport()
{
return Double.NEGATIVE_INFINITY;
}
@Override
public Double getMaxSupport()
{
return Double.POSITIVE_INFINITY;
}
@Override
public String toString()
{
return "Mean: " + this.getMean() + ", Variance: " + 1.0/this.getPrecision() + ", DOF: " + this.getDegreesOfFreedom();
}
@Override
public StudentTDistribution.MaximumLikelihoodEstimator getEstimator()
{
return new StudentTDistribution.MaximumLikelihoodEstimator();
}
/**
* Evaluator that computes the Probability Density Function (CDF) of
* a Student-t distribution with a fixed number of degrees of freedom
*/
public static class PDF
extends StudentTDistribution
implements UnivariateProbabilityDensityFunction
{
/**
* Default constructor.
*/
public PDF()
{
super();
}
/**
* Creates a new instance of PDF
* @param degreesOfFreedom
* Degrees of freedom in the distribution, usually the number of
* datapoints - 1, DOFs must be greater than zero.
*/
public PDF(
final double degreesOfFreedom )
{
super( degreesOfFreedom );
}
/**
* Creates a new instance of PDF
* @param degreesOfFreedom
* Degrees of freedom in the distribution, usually the number of
* datapoints - 1, DOFs must be greater than zero.
* @param mean
* Mean, or noncentrality parameter, of the distribution
* @param precision
* Precision, inverseRootFinder variance, of the distribution, must be greater
* than zero.
*/
public PDF(
final double degreesOfFreedom,
final double mean,
final double precision )
{
super( degreesOfFreedom, mean, precision );
}
/**
* Creates a new instance of PDF
* @param other
* The underlying Student t-distribution
*/
public PDF(
final StudentTDistribution other )
{
super( other );
}
@Override
public Double evaluate(
final Double input )
{
return this.evaluate( input.doubleValue() );
}
@Override
public double evaluateAsDouble(
final Double input)
{
return this.evaluate(input.doubleValue());
}
@Override
public double evaluate(
final double input )
{
return Math.exp( this.logEvaluate(input) );
}
@Override
public double logEvaluate(
final Double input)
{
return this.logEvaluate((double) input);
}
@Override
public double logEvaluate(
final double input)
{
double logSum = 0.0;
final double v2 = this.degreesOfFreedom / 2.0;
final double delta = input-this.mean;
logSum += MathUtil.logGammaFunction( v2 + 0.5 );
logSum -= MathUtil.logGammaFunction( v2 );
logSum += 0.5 * Math.log( this.precision / (Math.PI*this.degreesOfFreedom) );
logSum -= (v2 + 0.5) * Math.log( 1.0 + this.precision * delta*delta / this.degreesOfFreedom );
return logSum;
}
@Override
public StudentTDistribution.PDF getProbabilityFunction()
{
return this;
}
}
/**
* Evaluator that computes the Cumulative Distribution Function (CDF) of
* a Student-t distribution with a fixed number of degrees of freedom
*/
public static class CDF
extends StudentTDistribution
implements SmoothCumulativeDistributionFunction,
InvertibleCumulativeDistributionFunction
{
/**
* Default constructor.
*/
public CDF()
{
super();
}
/**
* Creates a new instance of CDF
* @param degreesOfFreedom
* Degrees of freedom in the distribution, usually the number of
* datapoints - 1, DOFs must be greater than zero.
*/
public CDF(
final double degreesOfFreedom )
{
super( degreesOfFreedom );
}
/**
* Creates a new instance of PDF
* @param degreesOfFreedom
* Degrees of freedom in the distribution, usually the number of
* datapoints - 1, DOFs must be greater than zero.
* @param mean
* Mean, or noncentrality parameter, of the distribution
* @param precision
* Precision, inverseRootFinder variance, of the distribution, must be greater
* than zero.
*/
public CDF(
final double degreesOfFreedom,
final double mean,
final double precision )
{
super( degreesOfFreedom, mean, precision );
}
/**
* Creates a new instance of CDF
* @param other
* The underlying Student t-distribution
*/
public CDF(
final StudentTDistribution other )
{
super( other );
}
@Override
public Double evaluate(
final Double input )
{
return this.evaluate( input.doubleValue() );
}
@Override
public double evaluateAsDouble(
final Double input)
{
return this.evaluate(input.doubleValue());
}
@Override
public double evaluate(
final double input )
{
double delta = input-this.mean;
double a = 0.5 * BetaDistribution.CDF.evaluate(
degreesOfFreedom / (degreesOfFreedom + this.precision*(delta*delta)),
0.5 * degreesOfFreedom, 0.5 );
// Take advantage of the symmetry to compute the CDF for values
// less than the mean
if (delta < 0.0)
{
a = 1.0 - a;
}
double p = 1.0 - a;
if (p < 0.0)
{
p = 0.0;
}
if (p > 1.0)
{
p = 1.0;
}
return p;
}
/**
* Evaluates the Inverse Student-t CDF for the given probability
* and degrees of freedom
* @param p
* Value at which to solve for x such that x=CDF(p)
* @return
* Value of x such that x=CDF(p)
*/
@Override
public Double inverse(
final double p )
{
return InverseTransformSampling.inverse( this, p ).getInput();
}
@Override
public StudentTDistribution.CDF getCDF()
{
return this;
}
@Override
public StudentTDistribution.PDF getDerivative()
{
return this.getProbabilityFunction();
}
@Override
public Double differentiate(
final Double input)
{
return this.getDerivative().evaluate(input);
}
}
/**
* Estimates the parameters of the Student-t distribution from the given
* data, where the degrees of freedom are estimated from the Kurtosis
* of the sample data.
*/
public static class MaximumLikelihoodEstimator
extends AbstractCloneableSerializable
implements DistributionEstimator
{
/**
* Amount to add to the variance to keep it from being 0.0
*/
private double defaultVariance;
/**
* Typical value of a defaultVariance, {@value}
*/
public static final double DEFAULT_VARIANCE = 1e-5;
/**
* Default constructor
*/
public MaximumLikelihoodEstimator()
{
this( DEFAULT_VARIANCE );
}
/**
* Creates a new instance of MaximumLikelihoodEstimator
* @param defaultVariance
* Amount to add to the variance to keep it from being 0.0
*/
public MaximumLikelihoodEstimator(
final double defaultVariance )
{
this.defaultVariance = defaultVariance;
}
/**
* Creates a new instance of UnivariateGaussian from the given data
* @param data
* Data to fit a UnivariateGaussian against
* @return
* Maximum likelihood estimate of the UnivariateGaussian that generated
* the data
*/
@Override
public StudentTDistribution learn(
final Collection data )
{
return MaximumLikelihoodEstimator.learn( data, this.defaultVariance );
}
/**
* Creates a new instance of UnivariateGaussian from the given data
* @param data
* Data to fit a UnivariateGaussian against
* @return
* Maximum likelihood estimate of the UnivariateGaussian that generated
* the data
* @param defaultVariance
* Amount to add to the variance to keep it from being 0.0
*/
public static StudentTDistribution.PDF learn(
final Collection data,
final double defaultVariance )
{
Pair mle =
UnivariateStatisticsUtil.computeMeanAndVariance(data);
double kurtosis = UnivariateStatisticsUtil.computeKurtosis(data);
double mean = mle.getFirst();
double variance = mle.getSecond();
variance += defaultVariance;
double dofs = (6.0/(kurtosis+defaultVariance))+4.0;
double precision = dofs / (variance*(dofs-2.0));
return new StudentTDistribution.PDF( dofs, mean, precision );
}
}
/**
* Creates a UnivariateGaussian from weighted data
*/
public static class WeightedMaximumLikelihoodEstimator
extends AbstractCloneableSerializable
implements DistributionWeightedEstimator
{
/**
* Amount to add to the variance to keep it from being 0.0
*/
private double defaultVariance;
/**
* Default constructor
*/
public WeightedMaximumLikelihoodEstimator()
{
this( MaximumLikelihoodEstimator.DEFAULT_VARIANCE );
}
/**
* Creates a new instance of WeightedMaximumLikelihoodEstimator
* @param defaultVariance
* Amount to add to the variance to keep it from being 0.0
*/
public WeightedMaximumLikelihoodEstimator(
final double defaultVariance )
{
this.defaultVariance = defaultVariance;
}
/**
* Creates a new instance of UnivariateGaussian using a weighted
* Maximum Likelihood estimate based on the given data
* @param data
* Weighed pairs of data (first is data, second is weight) that was
* generated by some unknown UnivariateGaussian distribution
* @return
* Maximum Likelihood UnivariateGaussian that generated the data
*/
@Override
public StudentTDistribution.PDF learn(
final Collection> data )
{
return WeightedMaximumLikelihoodEstimator.learn(
data, this.defaultVariance );
}
/**
* Creates a new instance of UnivariateGaussian using a weighted
* Maximum Likelihood estimate based on the given data
* @param data
* Weighed pairs of data (first is data, second is weight) that was
* generated by some unknown UnivariateGaussian distribution
* @return
* Maximum Likelihood UnivariateGaussian that generated the data
* @param defaultVariance
* Amount to add to the variance to keep it from being 0.0
*/
public static StudentTDistribution.PDF learn(
final Collection> data,
final double defaultVariance )
{
Pair mle =
UnivariateStatisticsUtil.computeWeightedMeanAndVariance(data);
double kurtosis = UnivariateStatisticsUtil.computeWeightedKurtosis(data);
double mean = mle.getFirst();
double variance = mle.getSecond();
variance += defaultVariance;
double dofs = (6.0/(Math.abs(kurtosis)+defaultVariance))+4.0;
double precision = dofs / (variance*(dofs-2.0));
return new StudentTDistribution.PDF( dofs, mean, precision );
}
}
}