smile.stat.distribution

Class ChiSquareDistribution

java.lang.Object

smile.stat.distribution.AbstractDistribution

smile.stat.distribution.ChiSquareDistribution

All Implemented Interfaces:

java.io.Serializable, Distribution, ExponentialFamily

public class ChiSquareDistribution
extends AbstractDistribution
implements ExponentialFamily

Chi-square (or chi-squared) distribution with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. It's mean and variance are k and 2k, respectively. The chi-square distribution is a special case of the gamma distribution. It follows from the definition of the chi-square distribution that the sum of independent chi-square variables is also chi-square distributed. Specifically, if X_i are independent chi-square variables with k_i degrees of freedom, respectively, then Y = Σ X_i is chi-square distributed with Σ k_i degrees of freedom.

The chi-square distribution has numerous applications in inferential statistics, for instance in chi-square tests and in estimating variances. Many other statistical tests also lead to a use of this distribution, like Friedman's analysis of variance by ranks.

See Also:

Serialized Form

Field Summary

Fields

Modifier and Type	Field and Description
int	nu The degrees of freedom.

Constructor Summary

Constructors

Constructor and Description
ChiSquareDistribution(int nu) Constructor.

Method Summary

Modifier and Type	Method and Description
double	cdf(double x) Cumulative distribution function.
double	entropy() Shannon entropy of the distribution.
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.
double	sd() The standard deviation of 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

Field Detail

nu

public final int nu

The degrees of freedom.

Constructor Detail

ChiSquareDistribution

public ChiSquareDistribution(int nu)

Constructor.

Parameters:

nu - the degree of freedom.

Method Detail

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

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()

Description copied from interface: Distribution

Generates a random number following this distribution.

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.