org.apache.commons.math3.distribution
Class GammaDistribution

java.lang.Object
  org.apache.commons.math3.distribution.AbstractRealDistribution
      org.apache.commons.math3.distribution.GammaDistribution

All Implemented Interfaces:

Serializable, RealDistribution

public class GammaDistribution
extends AbstractRealDistribution

Implementation of the Gamma distribution.

Version:

$Id: GammaDistribution.java 1422195 2012-12-15 06:45:18Z psteitz $

See Also:

Gamma distribution (Wikipedia), Gamma distribution (MathWorld), Serialized Form

Field Summary

static double	DEFAULT_INVERSE_ABSOLUTE_ACCURACY Default inverse cumulative probability accuracy.
private double	densityPrefactor1 The constant value of shape / scale * sqrt(e / (2 * pi * (shape + g + 0.5))) / L(shape), where L(shape) is the Lanczos approximation returned by Gamma.lanczos(double).
private double	densityPrefactor2 The constant value of shape * sqrt(e / (2 * pi * (shape + g + 0.5))) / L(shape), where L(shape) is the Lanczos approximation returned by Gamma.lanczos(double).
private double	maxLogY Upper bound on log(y) (y = x / scale) for the selection of the computation method in density(double).
private double	minY Lower bound on y = x / scale for the selection of the computation method in density(double).
private double	scale The scale parameter.
private static long	serialVersionUID Serializable version identifier.
private double	shape The shape parameter.
private double	shiftedShape The constant value of shape + g + 0.5, where g is the Lanczos constant Gamma.LANCZOS_G.
private double	solverAbsoluteAccuracy Inverse cumulative probability accuracy.

Fields inherited from class org.apache.commons.math3.distribution.AbstractRealDistribution

random, randomData, SOLVER_DEFAULT_ABSOLUTE_ACCURACY

Constructor Summary

GammaDistribution(double shape, double scale)
Creates a new gamma distribution with specified values of the shape and scale parameters.

GammaDistribution(double shape, double scale, double inverseCumAccuracy)
Creates a new gamma distribution with specified values of the shape and scale parameters.

GammaDistribution(RandomGenerator rng, double shape, double scale, double inverseCumAccuracy)
Creates a Gamma distribution.

Method Summary

double	cumulativeProbability(double x) For a random variable X whose values are distributed according to this distribution, this method returns P(X <= x).
double	density(double x) Returns the probability density function (PDF) of this distribution evaluated at the specified point x.
double	getAlpha() Deprecated. as of version 3.1, getShape() should be preferred. This method will be removed in version 4.0.
double	getBeta() Deprecated. as of version 3.1, getScale() should be preferred. This method will be removed in version 4.0.
double	getNumericalMean() Use this method to get the numerical value of the mean of this distribution.
double	getNumericalVariance() Use this method to get the numerical value of the variance of this distribution.
double	getScale() Returns the scale parameter of this distribution.
double	getShape() Returns the shape parameter of this distribution.
protected double	getSolverAbsoluteAccuracy() Returns the solver absolute accuracy for inverse cumulative computation.
double	getSupportLowerBound() Access the lower bound of the support.
double	getSupportUpperBound() Access the upper bound of the support.
boolean	isSupportConnected() Use this method to get information about whether the support is connected, i.e.
boolean	isSupportLowerBoundInclusive() Whether or not the lower bound of support is in the domain of the density function.
boolean	isSupportUpperBoundInclusive() Whether or not the upper bound of support is in the domain of the density function.
double	sample() This implementation uses the following algorithms:

Methods inherited from class org.apache.commons.math3.distribution.AbstractRealDistribution

cumulativeProbability, inverseCumulativeProbability, probability, probability, reseedRandomGenerator, sample

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

DEFAULT_INVERSE_ABSOLUTE_ACCURACY

public static final double DEFAULT_INVERSE_ABSOLUTE_ACCURACY

Default inverse cumulative probability accuracy.

Since:

2.1

See Also:

Constant Field Values

serialVersionUID

private static final long serialVersionUID

Serializable version identifier.

See Also:

Constant Field Values

shape

private final double shape

The shape parameter.

scale

private final double scale

The scale parameter.

shiftedShape

private final double shiftedShape

The constant value of shape + g + 0.5, where g is the Lanczos constant Gamma.LANCZOS_G.

densityPrefactor1

private final double densityPrefactor1

The constant value of shape / scale * sqrt(e / (2 * pi * (shape + g + 0.5))) / L(shape), where L(shape) is the Lanczos approximation returned by Gamma.lanczos(double). This prefactor is used in density(double), when no overflow occurs with the natural calculation.

densityPrefactor2

private final double densityPrefactor2

The constant value of shape * sqrt(e / (2 * pi * (shape + g + 0.5))) / L(shape), where L(shape) is the Lanczos approximation returned by Gamma.lanczos(double). This prefactor is used in density(double), when overflow occurs with the natural calculation.

minY

private final double minY

Lower bound on y = x / scale for the selection of the computation method in density(double). For y <= minY, the natural calculation overflows.

maxLogY

private final double maxLogY

Upper bound on log(y) (y = x / scale) for the selection of the computation method in density(double). For log(y) >= maxLogY, the natural calculation overflows.

solverAbsoluteAccuracy

private final double solverAbsoluteAccuracy

Inverse cumulative probability accuracy.

Constructor Detail

GammaDistribution

public GammaDistribution(double shape,
                         double scale)
                  throws NotStrictlyPositiveException

Creates a new gamma distribution with specified values of the shape and scale parameters.

Parameters:

shape - the shape parameter

scale - the scale parameter

Throws:

NotStrictlyPositiveException - if shape <= 0 or scale <= 0.

GammaDistribution

public GammaDistribution(double shape,
                         double scale,
                         double inverseCumAccuracy)
                  throws NotStrictlyPositiveException

Creates a new gamma distribution with specified values of the shape and scale parameters.

Parameters:

shape - the shape parameter

scale - the scale parameter

inverseCumAccuracy - the maximum absolute error in inverse cumulative probability estimates (defaults to DEFAULT_INVERSE_ABSOLUTE_ACCURACY).

Throws:

NotStrictlyPositiveException - if shape <= 0 or scale <= 0.

Since:

2.1

GammaDistribution

public GammaDistribution(RandomGenerator rng,
                         double shape,
                         double scale,
                         double inverseCumAccuracy)
                  throws NotStrictlyPositiveException

Creates a Gamma distribution.

Parameters:

rng - Random number generator.

shape - the shape parameter

scale - the scale parameter

inverseCumAccuracy - the maximum absolute error in inverse cumulative probability estimates (defaults to DEFAULT_INVERSE_ABSOLUTE_ACCURACY).

Throws:

NotStrictlyPositiveException - if shape <= 0 or scale <= 0.

Since:

3.1

Method Detail

getAlpha

@Deprecated
public double getAlpha()

Deprecated. as of version 3.1, getShape() should be preferred. This method will be removed in version 4.0.

Returns the shape parameter of this distribution.

Returns:

the shape parameter

getShape

public double getShape()

Returns the shape parameter of this distribution.

Returns:

the shape parameter

Since:

3.1

getBeta

@Deprecated
public double getBeta()

Deprecated. as of version 3.1, getScale() should be preferred. This method will be removed in version 4.0.

Returns the scale parameter of this distribution.

Returns:

the scale parameter

getScale

public double getScale()

Returns the scale parameter of this distribution.

Returns:

the scale parameter

Since:

3.1

density

public double density(double x)

Returns the probability density function (PDF) of this distribution evaluated at the specified point x. In general, the PDF is the derivative of the CDF. If the derivative does not exist at x, then an appropriate replacement should be returned, e.g. Double.POSITIVE_INFINITY, Double.NaN, or the limit inferior or limit superior of the difference quotient.

Parameters:

x - the point at which the PDF is evaluated

Returns:

the value of the probability density function at point x

cumulativeProbability

public double cumulativeProbability(double x)

For a random variable X whose values are distributed according to this distribution, this method returns P(X <= x). In other words, this method represents the (cumulative) distribution function (CDF) for this distribution. The implementation of this method is based on:

• Chi-Squared Distribution, equation (9).
• Casella, G., & Berger, R. (1990). Statistical Inference. Belmont, CA: Duxbury Press.

Parameters:

x - the point at which the CDF is evaluated

Returns:

the probability that a random variable with this distribution takes a value less than or equal to x

getSolverAbsoluteAccuracy

protected double getSolverAbsoluteAccuracy()

Returns the solver absolute accuracy for inverse cumulative computation. You can override this method in order to use a Brent solver with an absolute accuracy different from the default.

Overrides:

getSolverAbsoluteAccuracy in class AbstractRealDistribution

Returns:

the maximum absolute error in inverse cumulative probability estimates

getNumericalMean

public double getNumericalMean()

Use this method to get the numerical value of the mean of this distribution. For shape parameter alpha and scale parameter beta, the mean is alpha * beta.

Returns:

the mean or Double.NaN if it is not defined

getNumericalVariance

public double getNumericalVariance()

Use this method to get the numerical value of the variance of this distribution. For shape parameter alpha and scale parameter beta, the variance is alpha * beta^2.

Returns:

the variance (possibly Double.POSITIVE_INFINITY as for certain cases in TDistribution) or Double.NaN if it is not defined

getSupportLowerBound

public double getSupportLowerBound()

Access the lower bound of the support. This method must return the same value as inverseCumulativeProbability(0). In other words, this method must return

inf {x in R | P(X <= x) > 0}.

The lower bound of the support is always 0 no matter the parameters.

Returns:

lower bound of the support (always 0)

getSupportUpperBound

public double getSupportUpperBound()

Access the upper bound of the support. This method must return the same value as inverseCumulativeProbability(1). In other words, this method must return

inf {x in R | P(X <= x) = 1}.

The upper bound of the support is always positive infinity no matter the parameters.

Returns:

upper bound of the support (always Double.POSITIVE_INFINITY)

isSupportLowerBoundInclusive

public boolean isSupportLowerBoundInclusive()

Whether or not the lower bound of support is in the domain of the density function. Returns true iff getSupporLowerBound() is finite and density(getSupportLowerBound()) returns a non-NaN, non-infinite value.

Returns:

true if the lower bound of support is finite and the density function returns a non-NaN, non-infinite value there

isSupportUpperBoundInclusive

public boolean isSupportUpperBoundInclusive()

Whether or not the upper bound of support is in the domain of the density function. Returns true iff getSupportUpperBound() is finite and density(getSupportUpperBound()) returns a non-NaN, non-infinite value.

Returns:

true if the upper bound of support is finite and the density function returns a non-NaN, non-infinite value there

isSupportConnected

public boolean isSupportConnected()

Use this method to get information about whether the support is connected, i.e. whether all values between the lower and upper bound of the support are included in the support. The support of this distribution is connected.

Returns:

true

sample

public double sample()

This implementation uses the following algorithms:

For 0 < shape < 1:
Ahrens, J. H. and Dieter, U., Computer methods for sampling from gamma, beta, Poisson and binomial distributions. Computing, 12, 223-246, 1974.

For shape >= 1:
Marsaglia and Tsang, A Simple Method for Generating Gamma Variables. ACM Transactions on Mathematical Software, Volume 26 Issue 3, September, 2000.

Specified by:

sample in interface RealDistribution

Overrides:

sample in class AbstractRealDistribution

Returns:

random value sampled from the Gamma(shape, scale) distribution