Package jsl.utilities.distributions

Class Gamma

java.lang.Object

jsl.utilities.distributions.Distribution

jsl.utilities.distributions.Gamma

All Implemented Interfaces:

ControllableIfc, CDFIfc, ContinuousDistributionIfc, DistributionFunctionIfc, DistributionIfc, DomainIfc, FirstOrderLossFunctionIfc, InverseCDFIfc, LossFunctionDistributionIfc, MeanIfc, PDFIfc, SecondOrderLossFunctionIfc, VarianceIfc, GetNameIfc, IdentityIfc, NewInstanceIfc, ParametersIfc, GetRVariableIfc

public class Gamma
extends Distribution
implements ContinuousDistributionIfc, LossFunctionDistributionIfc, InverseCDFIfc, GetRVariableIfc

Models random variables that have gamma distribution For more information on the gamma distribution and its related functions, see "Object-Oriented Numerical Methods" by D. Besset

Nested Class Summary

Nested classes/interfaces inherited from class jsl.utilities.distributions.Distribution

Distribution.RandomControls

Field Summary

Fields

Modifier and Type	Field	Description
static int	DEFAULT_MAX_ITERATIONS

Fields inherited from class jsl.utilities.distributions.Distribution

myId, myName

Constructor Summary

Constructors

Constructor	Description
Gamma()	Creates new Gamma with shape 1.0, scale 1.0
Gamma(double[] parameters)	Constructs a gamma distribution with shape = parameters[0] and scale = parameters[1]
Gamma(double shape, double scale)	Constructs a gamma distribution with supplied shape and scale
Gamma(double shape, double scale, java.lang.String name)	Constructs a gamma distribution with supplied shape and scale

Method Summary

Modifier and Type	Method	Description
double	cdf(double x)	Returns the F(x) = Pr{X <= x} where F represents the cumulative distribution function
static Gamma	createFromMoments(double... moments)
static double	digamma(double x)	Computes the digamma function Mark Johnson, 2nd September 2007 Computes the Ψ(x) or digamma function, i.e., the derivative of the log gamma function, using a series expansion.
static double	diGammaFunction(double x)	Computes the digamma function using AS 103 Reference: Jose Bernardo, Algorithm AS 103: Psi ( Digamma ) Function, Applied Statistics, Volume 25, Number 3, 1976, pages 315-317.
double	firstOrderLossFunction(double x)	Computes the first order loss function for the function for given value of x, G1(x) = E[max(X-x,0)]
static double	gammaFunction(double x)	Computes the gamma function at x
Interval	getDomain()
double	getKurtosis()	Gets the kurtosis of the distribution
int	getMaxNumIterations()	Gets the maximum number of iterations for the gamma functions
double	getMean()	Returns the mean or expected value of a distribution
double	getMoment(int n)
double	getMoment2()
double	getMoment3()
double	getMoment4()
protected double	getNorm()
double	getNumericalPrecision()	Gets the numerical precision used in computing the gamma functions
double[]	getParameters()	Gets the parameters for the distribution
static double[]	getParametersFromMeanAndVariance(double mean, double var)	Computes the parameters (shape and scale) by matching to mean and variance element[0] = shape element[1] = scale
static double[]	getParametersFromMoments(double... moments)	Computes the parameters (shape and scale) by matching to mean and variance element[0] = shape element[1] = scale
RVariableIfc	getRandomVariable(RNStreamIfc rng)
double	getScale()	Gets the scale parameter
double	getShape()	Gets the shape
double	getSkewness()	Gets the skewness of the distribution
double	getVariance()	Returns the variance of the distribution if defined
static double	incompleteGammaFunction(double x, double alpha, int maxIterations, double eps)	Computes the incomplete gamma function at x
double	invCDF(double p)	Provides the inverse cumulative distribution function for the distribution This is based on a numerical routine that computes the percentage points for the chi-squared distribution
static double	invChiSquareDistribution(double p, double v)	The inverse CDF of the chi-square distribution
static double	invChiSquareDistribution(double p, double v, double g)	Algorithm AS 91 Appl.
static double	invChiSquareDistribution(double p, double v, double g, int maxIterations, double eps)	Algorithm AS 91 Appl.
static double	logGammaFunction(double x)	Computes the natural logarithm of the gamma function at x.
static void	main(java.lang.String[] args)
Gamma	newInstance()	Returns a new instance
double	pdf(double x)	Returns the f(x) where f represents the probability density function for the distribution.
double	secondOrderLossFunction(double x)	Computes the 2nd order loss function for the distribution function for given value of x, G2(x) = (1/2)E[max(X-x,0)*max(X-x-1,0)]
void	setMaxNumIterations(int iterations)	Sets the maximum number of iterations for the gamma functions
protected void	setNorm(double scale, double shape)
void	setNumericalPrecision(double precision)	Sets the numerical precision used in computing the gamma functions
void	setParameters(double[] parameters)	Sets the parameters for the distribution with shape = parameters[0] and scale = parameters[1]
void	setScale(double scale)	Sets the scale parameter
void	setShape(double shape)	Sets the shape parameter

Methods inherited from class jsl.utilities.distributions.Distribution

getControls, getId, getName, getStandardDeviation, inverseContinuousCDFViaBisection, inverseContinuousCDFViaBisection, inverseDiscreteCDFViaSearchUp, setControls, setId, setName, toString

Methods inherited from class java.lang.Object

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

Methods inherited from interface jsl.utilities.distributions.CDFIfc

cdf, complementaryCDF

Methods inherited from interface jsl.utilities.random.rvariable.GetRVariableIfc

getRandomVariable, getRandomVariable

Methods inherited from interface jsl.utilities.distributions.VarianceIfc

getStandardDeviation

Field Detail

DEFAULT_MAX_ITERATIONS

public static final int DEFAULT_MAX_ITERATIONS

See Also:

Constant Field Values

Constructor Detail

Gamma

public Gamma()

Creates new Gamma with shape 1.0, scale 1.0

Gamma

public Gamma(double[] parameters)

Constructs a gamma distribution with shape = parameters[0] and scale = parameters[1]

Parameters:

parameters - An array with the shape and scale

Gamma

public Gamma(double shape,
             double scale)

Constructs a gamma distribution with supplied shape and scale

Parameters:

shape - The shape parameter of the distribution

scale - The scale parameter of the distribution

Gamma

public Gamma(double shape,
             double scale,
             java.lang.String name)

Constructs a gamma distribution with supplied shape and scale

Parameters:

shape - The shape parameter of the distribution

scale - The scale parameter of the distribution

name - an optional name/label

Method Detail

newInstance

public final Gamma newInstance()

Description copied from interface: NewInstanceIfc

Returns a new instance

Specified by:

newInstance in interface NewInstanceIfc

Specified by:

newInstance in class Distribution

Returns:

the new instance

getDomain

public final Interval getDomain()

Specified by:

getDomain in interface DomainIfc

setShape

public final void setShape(double shape)

Sets the shape parameter

Parameters:

shape - The shape parameter must > 0.0

setScale

public final void setScale(double scale)

Sets the scale parameter

Parameters:

scale - The scale parameter must be > 0.0

getShape

public final double getShape()

Gets the shape

Returns:

The shape parameter as a double

getScale

public final double getScale()

Gets the scale parameter

Returns:

The scale parameter as a double

getMean

public final double getMean()

Description copied from interface: MeanIfc

Returns the mean or expected value of a distribution

Specified by:

getMean in interface MeanIfc

Returns:

double the mean or expected value for the distribution

getMoment2

public final double getMoment2()

Returns:

the 2nd moment

getMoment3

public final double getMoment3()

Returns:

the 3rd moment

getMoment4

public final double getMoment4()

Returns:

the 4th moment

getMoment

public final double getMoment(int n)

Parameters:

n - the desired moment

Returns:

the nth moment

getVariance

public double getVariance()

Description copied from interface: VarianceIfc

Returns the variance of the distribution if defined

Specified by:

getVariance in interface VarianceIfc

Returns:

double the variance of the random variable

cdf

public double cdf(double x)

Description copied from interface: CDFIfc

Returns the F(x) = Pr{X <= x} where F represents the cumulative distribution function

Specified by:

cdf in interface CDFIfc

Parameters:

x - a double representing the upper limit

Returns:

a double representing the probability

invCDF

public double invCDF(double p)

Provides the inverse cumulative distribution function for the distribution This is based on a numerical routine that computes the percentage points for the chi-squared distribution

Specified by:

invCDF in interface InverseCDFIfc

Parameters:

p - The probability to be evaluated for the inverse, p must be [0,1] or an IllegalArgumentException is thrown p = 0.0 returns 0.0 p = 1.0 returns Double.POSITIVE_INFINITY

Returns:

The inverse cdf evaluated at p

pdf

public double pdf(double x)

Description copied from interface: PDFIfc

Returns the f(x) where f represents the probability density function for the distribution. Note this is not a probability.

Specified by:

pdf in interface PDFIfc

Parameters:

x - a double representing the value to be evaluated

Returns:

f(x)

setNorm

protected final void setNorm(double scale,
                             double shape)

getNorm

protected final double getNorm()

getKurtosis

public final double getKurtosis()

Gets the kurtosis of the distribution

Returns:

the kurtosis

getSkewness

public final double getSkewness()

Gets the skewness of the distribution

Returns:

the skewness

invChiSquareDistribution

public static double invChiSquareDistribution(double p,
                                              double v)

The inverse CDF of the chi-square distribution

Parameters:

p - the probability to be evaluated

v - degrees of freedom

Returns:

the inverse value

invChiSquareDistribution

public static double invChiSquareDistribution(double p,
                                              double v,
                                              double g)

Algorithm AS 91 Appl. Statist. (1975) Vol.24, P.35 To evaluate the percentage points of the chi-squared probability distribution function. The default maximum number of iterations is DEFAULT_MAX_ITERATIONS The default numerical precision is JSLMath.getDefaultNumericalPrecision() logGammaFunction(v/2.0)) Incorporates the suggested changes in AS R85 (vol.40(1), pp.233-5, 1991) Auxiliary routines required: PPND = AS 111 (or AS 241) and GAMMAD = AS 239.

Parameters:

p - must lie in the range [0.0,1.0]

v - must be positive, degrees of freedom

g - must be supplied and should be equal to natural logarithm of the gamma function evaluated at half of the degrees of freedom

Returns:

The percentage point

invChiSquareDistribution

public static double invChiSquareDistribution(double p,
                                              double v,
                                              double g,
                                              int maxIterations,
                                              double eps)

Algorithm AS 91 Appl. Statist. (1975) Vol.24, P.35 To evaluate the percentage points of the chi-squared probability distribution function. logGammaFunction(v/2.0)) Incorporates the suggested changes in AS R85 (vol.40(1), pp.233-5, 1991) Auxiliary routines required: PPND = AS 111 (or AS 241) and GAMMAD = AS 239.

Parameters:

p - must lie in the range [0.0,1.0]

v - must be positive, degrees of freedom

g - must be supplied and should be equal to natural logarithm of the gamma function evaluated at half of the degrees of freedom

maxIterations - maximum number of iterations for series/continued fraction evaluation

eps - the numerical precision for convergence of series/continued fraction evaluation

Returns:

The percentage point

gammaFunction

public static double gammaFunction(double x)

Computes the gamma function at x

Parameters:

x - The value to be evaluated, x must be > 0

Returns:

The value of the gamma function evaluated at x

logGammaFunction

public static double logGammaFunction(double x)

Computes the natural logarithm of the gamma function at x. Useful when x gets large.

Parameters:

x - The value to be evaluated, x must be > 0

Returns:

The natural logarithm of the gamma function

incompleteGammaFunction

public static double incompleteGammaFunction(double x,
                                             double alpha,
                                             int maxIterations,
                                             double eps)

Computes the incomplete gamma function at x

Parameters:

x - The value to be evaluated, must be > 0

alpha - must be > 0

maxIterations - maximum number of iterations for series/continued fraction evaluation

eps - the numerical precision for convergence of series/continued fraction evaluation

Returns:

the value of the incomplete gamma function evaluated at x

diGammaFunction

public static double diGammaFunction(double x)

Computes the digamma function using AS 103 Reference: Jose Bernardo, Algorithm AS 103: Psi ( Digamma ) Function, Applied Statistics, Volume 25, Number 3, 1976, pages 315-317.

Parameters:

x - the value to evaluate

Returns:

the function value

digamma

public static double digamma(double x)

Computes the digamma function Mark Johnson, 2nd September 2007 Computes the Ψ(x) or digamma function, i.e., the derivative of the log gamma function, using a series expansion. http://www.cog.brown.edu/~mj/code/digamma.c

Parameters:

x - the value to evaluate

Returns:

the function value

getMaxNumIterations

public final int getMaxNumIterations()

Gets the maximum number of iterations for the gamma functions

Returns:

the maximum number of iterations

setMaxNumIterations

public final void setMaxNumIterations(int iterations)

Sets the maximum number of iterations for the gamma functions

Parameters:

iterations - the maximum number of iterations

getNumericalPrecision

public final double getNumericalPrecision()

Gets the numerical precision used in computing the gamma functions

Returns:

the numerical precision

setNumericalPrecision

public final void setNumericalPrecision(double precision)

Sets the numerical precision used in computing the gamma functions

Parameters:

precision - the numerical precision

setParameters

public void setParameters(double[] parameters)

Sets the parameters for the distribution with shape = parameters[0] and scale = parameters[1]

Specified by:

setParameters in interface ParametersIfc

Parameters:

parameters - an array of doubles representing the parameters for the distribution

getParameters

public double[] getParameters()

Gets the parameters for the distribution

Specified by:

getParameters in interface ParametersIfc

Returns:

Returns an array of the parameters for the distribution

getParametersFromMeanAndVariance

public static double[] getParametersFromMeanAndVariance(double mean,
                                                        double var)

Computes the parameters (shape and scale) by matching to mean and variance element[0] = shape element[1] = scale

Parameters:

mean - must be > 0

var - must be > 0

Returns:

the parameter array

getParametersFromMoments

public static double[] getParametersFromMoments(double... moments)

Computes the parameters (shape and scale) by matching to mean and variance element[0] = shape element[1] = scale

createFromMoments

public static Gamma createFromMoments(double... moments)

firstOrderLossFunction

public double firstOrderLossFunction(double x)

Description copied from interface: FirstOrderLossFunctionIfc

Computes the first order loss function for the function for given value of x, G1(x) = E[max(X-x,0)]

Specified by:

firstOrderLossFunction in interface FirstOrderLossFunctionIfc

Parameters:

x - The value to be evaluated

Returns:

The loss function value, E[max(X-x,0)]

secondOrderLossFunction

public double secondOrderLossFunction(double x)

Description copied from interface: SecondOrderLossFunctionIfc

Computes the 2nd order loss function for the distribution function for given value of x, G2(x) = (1/2)E[max(X-x,0)*max(X-x-1,0)]

Specified by:

secondOrderLossFunction in interface SecondOrderLossFunctionIfc

Parameters:

x - The value to be evaluated

Returns:

The loss function value, (1/2)E[max(X-x,0)*max(X-x-1,0)]

getRandomVariable

public final RVariableIfc getRandomVariable(RNStreamIfc rng)

Specified by:

getRandomVariable in interface GetRVariableIfc

Overrides:

getRandomVariable in class Distribution

Parameters:

rng - the stream to use

Returns:

a random variable

main

public static void main(java.lang.String[] args)