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/*
* File: GammaDistribution.java
* Authors: Kevin R. Dixon
* Company: Sandia National Laboratories
* Project: Cognitive Foundry
*
* Copyright October 2, 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.PublicationType;
import gov.sandia.cognition.math.MathUtil;
import gov.sandia.cognition.math.matrix.VectorFactory;
import gov.sandia.cognition.math.matrix.Vector;
import gov.sandia.cognition.math.UnivariateStatisticsUtil;
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.UnivariateProbabilityDensityFunction;
import gov.sandia.cognition.statistics.SmoothCumulativeDistributionFunction;
import gov.sandia.cognition.util.AbstractCloneableSerializable;
import gov.sandia.cognition.util.ArgumentChecker;
import gov.sandia.cognition.util.Pair;
import gov.sandia.cognition.util.WeightedValue;
import java.util.ArrayList;
import java.util.Collection;
import java.util.Random;
/**
* Class representing the Gamma distribution. This is a two-parameter family
* of continuous distributions. The well-known exponential and chi-square
* distributions are special cases of the Gamma distribution. Please note that
* we use the "shape" and "scale" parameters to describe the PDF/CDF, whereas
* octave/MATLAB use "shape" and "1.0/scale" to parameterize a Gamma
* distribution, so please beware when comparing results to octave/MATLAB.
*
* @author Kevin R. Dixon
* @since 2.0
*
*/
@PublicationReference(
author="Wikipedia",
title="Gamma distribution",
type=PublicationType.WebPage,
year=2009,
url="http://en.wikipedia.org/wiki/Gamma_distribution"
)
public class GammaDistribution
extends AbstractClosedFormSmoothUnivariateDistribution
implements EstimableDistribution
{
/**
* Default shape, {@value}.
*/
public static final double DEFAULT_SHAPE = 1.0;
/**
* Default scale, {@value}.
*/
public static final double DEFAULT_SCALE = 1.0;
/**
* Shape parameter of the Gamma distribution, often written as "k",
* must be greater than zero
*/
private double shape;
/**
* Scale parameters of the Gamma distribution, often written as "theta",
* must be greater than zero.
* Note that this is the INVERSE of what octave uses!!
*/
private double scale;
/**
* Default constructor.
*/
public GammaDistribution()
{
this( DEFAULT_SHAPE, DEFAULT_SCALE );
}
/**
* Creates a new instance of GammaDistribution
* @param shape
* Shape parameter of the Gamma distribution, often written as "k",
* must be greater than zero
* @param scale
* Scale parameters of the Gamma distribution, often written as "theta",
* must be greater than zero.
* Note that this is the INVERSE of what octave uses!!
*/
public GammaDistribution(
final double shape,
final double scale )
{
this.setShape( shape );
this.setScale( scale );
}
/**
* Copy constructor
* @param other
* GammaDistribution to copy
*/
public GammaDistribution(
final GammaDistribution other )
{
this( other.getShape(), other.getScale() );
}
@Override
public GammaDistribution clone()
{
return (GammaDistribution) super.clone();
}
/**
* Getter for shape
* @return
* Shape parameter of the Gamma distribution, often written as "k",
* must be greater than zero
*/
public double getShape()
{
return this.shape;
}
/**
* Setter for shape
* @param shape
* Shape parameter of the Gamma distribution, often written as "k",
* must be greater than zero
*/
public void setShape(
final double shape )
{
if (shape <= 0.0)
{
throw new IllegalArgumentException( "Shape must be > 0.0" );
}
this.shape = shape;
}
/**
* Getter for scale
* @return
* Scale parameters of the Gamma distribution, often written as "theta",
* must be greater than zero.
* Note that this is the INVERSE of what octave uses!!
*/
public double getScale()
{
return this.scale;
}
/**
* Setter for scale
* @param scale
* Scale parameters of the Gamma distribution, often written as "theta",
* must be greater than zero.
* Note that this is the INVERSE of what octave uses!!
*/
public void setScale(
final double scale )
{
ArgumentChecker.assertIsPositive("scale", scale);
this.scale = scale;
}
/**
* Gets the rate parameter, which is just the inverse of the scale parameter.
* It is commonly referred to as beta.
*
* @return
* The rate parameter (1.0 / scale). Must be greater than 0.0.
*/
public double getRate()
{
return 1.0 / this.getScale();
}
/**
* Sets the rate parameter, which is just the inverse of the scale parameter.
* It is commonly referred to as beta.
*
* @param rate
* The rate parameter (1.0 / scale). Must be greater than 0.0.
*/
public void setRate(
final double rate)
{
ArgumentChecker.assertIsPositive("rate", rate);
this.setScale(1.0 / rate);
}
@Override
public double getMeanAsDouble()
{
return this.getShape() * this.getScale();
}
@Override
public double getVariance()
{
return this.getShape() * this.getScale() * this.getScale();
}
/**
* Gets the parameters of the distribution
* @return
* 2-dimensional Vector with (shape scale)
*/
@Override
public Vector convertToVector()
{
return VectorFactory.getDefault().copyValues(
this.getShape(), this.getScale() );
}
/**
* Sets the parameters of the distribution
* @param parameters
* 2-dimensional Vector with (shape scale)
*/
@Override
public void convertFromVector(
final Vector parameters )
{
if (parameters.getDimensionality() != 2)
{
throw new IllegalArgumentException(
"Expected a 2-dimensional Vector!" );
}
this.setShape( parameters.getElement( 0 ) );
this.setScale( parameters.getElement( 1 ) );
}
/**
* Efficiently samples from a Gamma distribution given by the
* shape and scale parameters.
* @param shape
* Shape parameter of the Gamma distribution, often written as "k",
* must be greater than zero
* @param scale
* Scale parameters of the Gamma distribution, often written as "theta",
* must be greater than zero.
* Note that this is the INVERSE of what octave uses!!
* @param random
* Random number generator
* @param numSamples
* Number of samples to generate
* @return
* Samples simulated from the Gamma distribution.
*/
public static ArrayList sample(
final double shape,
final double scale,
final Random random,
final int numSamples )
{
final double[] values = sampleAsDoubles(shape, scale, random, numSamples);
final ArrayList result = new ArrayList(numSamples);
for (final double value : values)
{
result.add(value);
}
return result;
}
/**
* Efficiently samples from a Gamma distribution given by the
* shape and scale parameters.
* @param shape
* Shape parameter of the Gamma distribution, often written as "k",
* must be greater than zero
* @param scale
* Scale parameters of the Gamma distribution, often written as "theta",
* must be greater than zero.
* Note that this is the INVERSE of what octave uses!!
* @param random
* Random number generator
* @return
* Samples simulated from the Gamma distribution.
*/
public static double sampleAsDouble(
final double shape,
final double scale,
final Random random)
{
return sample(shape, scale, random);
}
/**
* Efficiently samples from a Gamma distribution given by the
* shape and scale parameters.
* @param shape
* Shape parameter of the Gamma distribution, often written as "k",
* must be greater than zero
* @param scale
* Scale parameters of the Gamma distribution, often written as "theta",
* must be greater than zero.
* Note that this is the INVERSE of what octave uses!!
* @param random
* Random number generator
* @param numSamples
* Number of samples to generate
* @return
* Samples simulated from the Gamma distribution.
*/
public static double[] sampleAsDoubles(
final double shape,
final double scale,
final Random random,
final int numSamples)
{
final double[] result = new double[numSamples];
sampleInto(shape, scale, random, result, 0, numSamples);
return result;
}
/**
* Efficiently samples from a Gamma distribution given by the
* shape and scale parameters.
* @param shape
* Shape parameter of the Gamma distribution, often written as "k",
* must be greater than zero
* @param scale
* Scale parameters of the Gamma distribution, often written as "theta",
* must be greater than zero.
* Note that this is the INVERSE of what octave uses!!
* @param random
* Random number generator
* @param output Array to write samples from Gamma distribution into.
* @param start Starting point in output array to add samples.
* @param length Number of samples to generate
*/
public static void sampleInto(
final double shape,
final double scale,
final Random random,
final double[] output,
final int start,
final int length)
{
final int end = start + length;
for (int i = start; i < end; i++)
{
output[i] = sample(shape, scale, random);
}
}
/**
* Provides a single sample from a Gamma distribution with the given shape
* and scale.
*
* @param shape
* Shape parameter of the Gamma distribution, often written as "k",
* must be greater than zero.
* @param scale
* Scale parameters of the Gamma distribution, often written as "theta",
* must be greater than zero.
* @param random
* Random number generator to use.
* @return
* A value sampled from a Gamma distribution.
*/
public static double sample(
final double shape,
final double scale,
final Random random)
{
// Shape is checked in the next function.
ArgumentChecker.assertIsPositive("scale", scale);
return scale * sampleStandard(shape, random);
}
/**
* Provides a single sample from a Gamma distribution with the given shape
* and a scale of 1.
*
* @param shape
* Shape parameter of the Gamma distribution, often written as "k",
* must be greater than zero.
* @param random
* Random number generator to use.
* @return
* A value sampled from a Gamma distribution.
*/
public static double sampleStandard(
final double shape,
final Random random)
{
ArgumentChecker.assertIsPositive("shape", shape);
// This is based on the gamma distribution algorithm used in numpy:
// https://github.com/numpy/numpy/blob/master/numpy/random/mtrand/distributions.c
if (shape == 1.0)
{
// Sample standard exponential:
return -Math.log(random.nextDouble());
}
else if (shape < 1.0)
{
while (true)
{
final double u = random.nextDouble();
final double v = -Math.log(random.nextDouble());
if (u <= 1.0 - shape)
{
final double x = Math.pow(u, 1.0 / shape);
if (x <= v)
{
return x;
}
}
else
{
final double y = -Math.log((1.0 - u) / shape);
final double x = Math.pow(1.0 - shape + shape * y, 1.0 / shape);
if (x <= (v + y))
{
return x;
}
}
}
}
else
{
// Marsaglia's method.
final double b = shape - 1.0 / 3.0;
final double c = 1.0 / Math.sqrt(9.0 * b);
while (true)
{
double x = 0.0;
double v = 0.0;
do
{
x = random.nextGaussian();
v = 1.0 + c * x;
}
while (v <= 0.0);
v = v * v * v;
final double xx = x * x;
final double u = random.nextDouble();
if (u < (1.0 - 0.0331 * xx * xx)
|| Math.log(u) < ((0.5 * xx) + (b * (1.0 - v + Math.log(v)))))
{
return b * v;
}
}
}
}
@Override
public double sampleAsDouble(
final Random random)
{
return sample(this.getShape(), this.getScale(), random);
}
@Override
public void sampleInto(
final Random random,
final double[] output,
final int start,
final int length)
{
sampleInto(this.getShape(), this.getScale(), random,
output, start, length);
}
@Override
public GammaDistribution.CDF getCDF()
{
return new GammaDistribution.CDF( this );
}
@Override
public GammaDistribution.PDF getProbabilityFunction()
{
return new GammaDistribution.PDF( this );
}
@Override
public String toString()
{
return "Shape = " + this.getShape() + ", Scale = " + this.getScale();
}
@Override
public Double getMinSupport()
{
return 0.0;
}
@Override
public Double getMaxSupport()
{
return Double.POSITIVE_INFINITY;
}
@Override
public GammaDistribution.MomentMatchingEstimator getEstimator()
{
return new GammaDistribution.MomentMatchingEstimator();
}
/**
* Closed-form PDF of the Gamma distribution
*/
public static class PDF
extends GammaDistribution
implements UnivariateProbabilityDensityFunction
{
/**
* Default constructor.
*/
public PDF()
{
super();
}
/**
* Creates a new instance of PDF
* @param shape
* Shape parameter of the Gamma distribution, often written as "k",
* must be greater than zero
* @param scale
* Scale parameters of the Gamma distribution, often written as "theta",
* must be greater than zero.
* Note that this is the INVERSE of what octave uses!!
*/
public PDF(
final double shape,
final double scale )
{
super( shape, scale );
}
/**
* Copy constructor
* @param other
* GammaDistribution to copy
*/
public PDF(
final GammaDistribution 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 evaluate( input, this.getShape(), this.getScale() );
}
/**
* Evaluates the Gamma PDF about the input "x", using the given
* shape and scale
* @param x
* Input to the PDF
* @param shape
* Shape parameter of the Gamma distribution, often written as "k",
* must be greater than zero
* @param scale
* Scale parameters of the Gamma distribution, often written as "theta",
* must be greater than zero.
* Note that this is the INVERSE of what octave uses!!
* @return
* p(x;shape,scale)
*/
public static double evaluate(
final double x,
final double shape,
final double scale )
{
double p;
if (x > 0.0)
{
p = Math.exp( logEvaluate(x, shape, scale) );
}
else
{
p = 0.0;
}
return p;
}
@Override
public double logEvaluate(
final Double input)
{
return this.logEvaluate((double) input);
}
@Override
public double logEvaluate(
final double input)
{
return logEvaluate( input, this.getShape(), this.getScale() );
}
/**
* Evaluates the natural logarithm of the PDF.
* @param input
* Input to consider.
* @param shape
* Shape factor.
* @param scale
* Scale factor.
* @return
* Natural logarithm of the PDF.
*/
public static double logEvaluate(
final double input,
final double shape,
final double scale )
{
if( input <= 0.0 )
{
return Math.log(0.0);
}
else
{
final double n1 = (shape-1.0) * Math.log(input);
final double n2 = -input / scale;
final double d1 = MathUtil.logGammaFunction(shape);
final double d2 = shape * Math.log(scale);
return n1 + n2 - d1 - d2;
}
}
@Override
public GammaDistribution.PDF getProbabilityFunction()
{
return this;
}
}
/**
* CDF of the Gamma distribution
*/
public static class CDF
extends GammaDistribution
implements SmoothCumulativeDistributionFunction
{
/**
* Default constructor.
*/
public CDF()
{
super();
}
/**
* Creates a new instance of CDF
* @param shape
* Shape parameter of the Gamma distribution, often written as "k",
* must be greater than zero
* @param scale
* Scale parameters of the Gamma distribution, often written as "theta",
* must be greater than zero.
* Note that this is the INVERSE of what octave uses!!
*/
public CDF(
final double shape,
final double scale )
{
super( shape, scale );
}
/**
* Copy constructor
* @param other
* GammaDistribution to copy
*/
public CDF(
final GammaDistribution 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 evaluate( input, this.getShape(), this.getScale() );
}
/**
* Evaluates the CDF of the Gamma distribution about x, given
* the shape and scale parameters
* @param x
* Input to the CDF
* @param shape
* Shape parameter of the Gamma distribution, often written as "k",
* must be greater than zero
* @param scale
* Scale parameters of the Gamma distribution, often written as "theta",
* must be greater than zero.
* Note that this is the INVERSE of what octave uses!!
* @return
* Pr(y le x;shape,scale)
*/
public static double evaluate(
final double x,
final double shape,
final double scale )
{
double p;
if (x <= 0.0)
{
p = 0.0;
}
else
{
p = MathUtil.lowerIncompleteGammaFunction( shape, x / scale );
}
return p;
}
@Override
public GammaDistribution.CDF getCDF()
{
return this;
}
@Override
public GammaDistribution.PDF getDerivative()
{
return this.getProbabilityFunction();
}
@Override
public Double differentiate(
final Double input)
{
return this.getDerivative().evaluate(input);
}
}
/**
* Computes the parameters of a Gamma distribution by the
* Method of Moments
*/
public static class MomentMatchingEstimator
extends AbstractCloneableSerializable
implements DistributionEstimator
{
/**
* Default constructor
*/
public MomentMatchingEstimator()
{
}
@Override
public GammaDistribution learn(
final Collection data)
{
Pair pair =
UnivariateStatisticsUtil.computeMeanAndVariance(data);
return learn( pair.getFirst(), pair.getSecond() );
}
/**
* Computes the Gamma distribution describes by the given moments
* @param mean
* Mean of the distribution
* @param variance
* Variance of the distribution
* @return
* Gamma distribution that has the same mean/variance as the
* given parameters.
*/
public static GammaDistribution learn(
final double mean,
final double variance )
{
double scale = variance / mean;
double shape = mean*mean / variance;
return new GammaDistribution(shape, scale);
}
}
/**
* Estimates the parameters of a Gamma distribution using the matching
* of moments, not maximum likelihood.
*/
public static class WeightedMomentMatchingEstimator
extends AbstractCloneableSerializable
implements DistributionWeightedEstimator
{
/**
* Default constructor
*/
public WeightedMomentMatchingEstimator()
{
}
@Override
public GammaDistribution learn(
final Collection> data)
{
Pair pair =
UnivariateStatisticsUtil.computeWeightedMeanAndVariance(data);
return MomentMatchingEstimator.learn(
pair.getFirst(), pair.getSecond());
}
}
}