smile.stat.distribution

Class BetaDistribution

java.lang.Object

smile.stat.distribution.AbstractDistribution

smile.stat.distribution.BetaDistribution

All Implemented Interfaces:

java.io.Serializable, Distribution, ExponentialFamily

public class BetaDistribution
extends AbstractDistribution
implements ExponentialFamily

The beta distribution is defined on the interval [0, 1] parameterized by two positive shape parameters, typically denoted by α and β. It is the special case of the Dirichlet distribution with only two parameters. The beta distribution is used as a prior distribution for binomial proportions in Bayesian analysis. In Bayesian statistics, it can be seen as the posterior distribution of the parameter α of a binomial distribution after observing α - 1 independent events with probability α and β - 1 with probability 1 - α, if the prior distribution of α was uniform. If α = 1 and β =1, the Beta distribution is the uniform [0, 1] distribution. The probability density function of the beta distribution is f(x;α,β) = x^α-1(1-x)^β-1 / B(α,β) where B(α,β) is the beta function.

See Also:

Serialized Form

Field Summary

Fields

Modifier and Type	Field and Description
double	alpha The shape parameter.
double	beta The shape parameter.

Constructor Summary

Constructors

Constructor and Description
BetaDistribution(double alpha, double beta) Constructor.

Method Summary

Modifier and Type	Method and Description
double	cdf(double x) Cumulative distribution function.
double	entropy() Shannon entropy of the distribution.
static BetaDistribution	fit(double[] data) Estimates the distribution parameters by the moment method.
double	getAlpha() Returns the shape parameter alpha.
double	getBeta() Returns the shape parameter beta.
int	length() The number of parameters of the distribution.
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.
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() Generates a random number following this distribution.
java.lang.String	toString()
double	variance() The variance of distribution.

Methods inherited from class smile.stat.distribution.AbstractDistribution

inverseTransformSampling, quantile, quantile, rejection

Methods inherited from class java.lang.Object

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

Methods inherited from interface smile.stat.distribution.Distribution

likelihood, logLikelihood, rand, sd

Field Detail

alpha

public final double alpha

The shape parameter.

beta

public final double beta

The shape parameter.

Constructor Detail

BetaDistribution

public BetaDistribution(double alpha,
                        double beta)

Constructor.

Parameters:

alpha - shape parameter.

beta - shape parameter.

Method Detail

fit

public static BetaDistribution fit(double[] data)

Estimates the distribution parameters by the moment method.

getAlpha

public double getAlpha()

Returns the shape parameter alpha.

Returns:

the shape parameter alpha

getBeta

public double getBeta()

Returns the shape parameter beta.

Returns:

the shape parameter beta

length

public int length()

Description copied from interface: Distribution

The number of parameters of the distribution. The "length" is in the sense of the minimum description length principle.

Specified by:

length in interface Distribution

mean

public double mean()

Description copied from interface: Distribution

The mean of distribution.

Specified by:

mean in interface Distribution

variance

public double variance()

Description copied from interface: Distribution

The variance of distribution.

Specified by:

variance 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

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.

rand

public double rand()

Description copied from interface: Distribution

Generates a random number following this distribution.

Specified by:

rand in interface Distribution