smile.stat.distribution

Class GammaDistribution

java.lang.Object

smile.stat.distribution.AbstractDistribution

smile.stat.distribution.GammaDistribution

All Implemented Interfaces:

Distribution, ExponentialFamily

public class GammaDistribution
extends AbstractDistribution
implements ExponentialFamily

The Gamma distribution is a continuous probability distributions with a scale parameter θ and a shape parameter k. If k is an integer then the distribution represents the sum of k independent exponentially distributed random variables, each of which has a rate parameter of θ). The gamma distribution is frequently a probability model for waiting times; for instance, the waiting time until death in life testing. The probability density function is f(x; k,θ) = x^k-1e^-x/θ / (θ^kΓ(k)) for x > 0 and k, θ > 0.

• If X ∼ Γ(k=1, θ=1/λ), then X has an exponential distribution with rate parameter λ.
• If X ∼ Γ(k=ν/2, θ=2), then X is identical to Χ²(ν), the chi-square distribution with ν degrees of freedom. Conversely, if Q ∼ Χ²(ν), and c is a positive constant, then c ⋅ Q ∼ Γ(k=ν/2, θ=2c).
• If k is an integer, the gamma distribution is an Erlang distribution and is the probability distribution of the waiting time until the k-th "arrival" in a one-dimensional Poisson process with intensity 1/θ.
• If X² ∼ Γ(3/2, 2a²), then X has a Maxwell-Boltzmann distribution with parameter a.

In Bayesian inference, the gamma distribution is the conjugate prior to many likelihood distributions: the Poisson, exponential, normal (with known mean), Pareto, gamma with known shape, and inverse gamma with known shape parameter.

Constructor Summary

Constructors

Constructor and Description
GammaDistribution(double[] data) Constructor.
GammaDistribution(double shape, double scale) Constructor.

Method Summary

Modifier and Type	Method and Description
double	cdf(double x) Cumulative distribution function.
double	entropy() Shannon entropy of the distribution.
double	getScale() Returns the scale parameter.
double	getShape() Returns the shape parameter.
double	logp(double x) The density at x in log scale, which may prevents the underflow problem.
Mixture.Component	M(double[] x, double[] posteriori) The M step in the EM algorithm, which depends the specific distribution.
double	mean() The mean of distribution.
int	npara() The number of parameters of the distribution.
double	p(double x) The probability density function for continuous distribution or probability mass function for discrete distribution at x.
double	quantile(double p) The quantile, the probability to the left of quantile is p.
double	rand() Only support shape parameter k of integer.
double	sd() The standard deviation of distribution.
java.lang.String	toString()
double	var() The variance of distribution.

Methods inherited from class smile.stat.distribution.AbstractDistribution

inverseTransformSampling, likelihood, logLikelihood, quantile, quantile, rejection

Methods inherited from class java.lang.Object

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

Constructor Detail

GammaDistribution

public GammaDistribution(double shape,
                         double scale)

Constructor.

Parameters:

shape - the shape parameter.

scale - the scale parameter.

GammaDistribution

public GammaDistribution(double[] data)

Constructor. Parameter will be estimated from the data by (approximate) MLE.

Method Detail

getScale

public double getScale()

Returns the scale parameter.

getShape

public double getShape()

Returns the shape parameter.

npara

public int npara()

Description copied from interface: Distribution

The number of parameters of the distribution.

Specified by:

npara in interface Distribution

mean

public double mean()

Description copied from interface: Distribution

The mean of distribution.

Specified by:

mean in interface Distribution

var

public double var()

Description copied from interface: Distribution

The variance of distribution.

Specified by:

var in interface Distribution

sd

public double sd()

Description copied from interface: Distribution

The standard deviation of distribution.

Specified by:

sd in interface Distribution

entropy

public double entropy()

Description copied from interface: Distribution

Shannon entropy of the distribution.

Specified by:

entropy in interface Distribution

toString

public java.lang.String toString()

Overrides:

toString in class java.lang.Object

rand

public double rand()

Only support shape parameter k of integer.

Specified by:

rand in interface Distribution

p

public double p(double x)

Description copied from interface: Distribution

The probability density function for continuous distribution or probability mass function for discrete distribution at x.

Specified by:

p in interface Distribution

logp

public double logp(double x)

Description copied from interface: Distribution

The density at x in log scale, which may prevents the underflow problem.

Specified by:

logp in interface Distribution

cdf

public double cdf(double x)

Description copied from interface: Distribution

Cumulative distribution function. That is the probability to the left of x.

Specified by:

cdf in interface Distribution

quantile

public double quantile(double p)

Description copied from interface: Distribution

The quantile, the probability to the left of quantile is p. It is actually the inverse of cdf.

Specified by:

quantile in interface Distribution

M

public Mixture.Component M(double[] x,
                           double[] posteriori)

Description copied from interface: ExponentialFamily

The M step in the EM algorithm, which depends the specific distribution.

Specified by:

M in interface ExponentialFamily

Parameters:

x - the input data for estimation

posteriori - the posteriori probability.

Returns:

the (unnormalized) weight of this distribution in the mixture.