Module org.bzdev.math

Package org.bzdev.math.stats

Class ChiSquareDistr

java.lang.Object

org.bzdev.math.stats.ProbDistribution

org.bzdev.math.stats.ChiSquareDistr

All Implemented Interfaces:

RealValuedDomainOps

public class ChiSquareDistr
extends ProbDistribution

Class providing methods for the χ² distribution.

The χ² distribution is the distribution of the sum of the squares of k independent random variables, each with a normal distribution and with unit variances. For the default statistic, the means of these variables are zero. For a noncentral statistic, the means are nonzero.

The use of A, P and Q follows the convention in Abramowitz and Stegun, "Handbook of Mathematical Functions" (10th printing [1972], 9th Dover printing), chapter 26. Some of the methods have names that start with an upper-case letter, contrary to the usual Java convention, in order to conform to this text.

For non-central χ² distributions, the the notation in Abramowitz and Stegun (See Equation 26.4.25) is followed. This corresponds to the case where, when there are k degrees of freedom, $\mu_i$ μ_i is the mean value of the random variable $X_i$, and the random variable described by the distribution is $\chi^2 = \sum_{i=1}^k X_i^2$. The parameter λ is the noncentrality parameter and is given by $\lambda = \sum_{i=1}^k \mu_i^2$.

Constructor Summary

Constructors

Constructor	Description
ChiSquareDistr(long nu)	Constructor.
ChiSquareDistr(long nu, double lambda)	Constructor for a non-central χ2 distribution.

Method Summary

Modifier and Type	Method	Description
boolean	domainMinClosed()	Determine if the domain minimum is in the domain.
long	getDegreesOfFreedom()	Get the number of degrees of freedom.
double	getDomainMin()	Get the minimum value in the domain of the function.
double	getLambda()	Get the non-centrality parameter.
boolean	isNonCentral()	Determine if this object is a non-central χ2 distribution.
boolean	isSymmetric(double x)	Determine if this distribution is symmetric about x.
double	P(double x)	Get the probability that a value is no larger than the argument
static double	P(double chi2, long nu)	Get the cumulative probability for a χ2 distribution.
static double	P(double chi2, long nu, double lambda)	Get the cumulative probability distribution for a non-central χ2 distribution.
double	pd(double x)	Get the probability density
static double	pd(double chi2, long nu)	Get the probability density for a χ2 distribution.
static double	pd(double chi2, long nu, double lambda)	Get the probability density for a non-central χ2 distribution.
double	Q(double x)	Get the probability that a value is no smaller than the argument
static double	Q(double chi2, long nu)	Get the complement of the cumulative probability for a χ2 distribution.
static double	Q(double chi2, long nu, double lambda)	Get the complement of the cumulative probability distribution for a non-central χ2 distribution.

Methods inherited from class org.bzdev.math.stats.ProbDistribution

A, cdf, cdfc, domainMaxClosed, getDomainMax, inverseA, inverseP, inverseQ, isInDomain

Methods inherited from class java.lang.Object

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

Constructor Details

ChiSquareDistr

public ChiSquareDistr(long nu)

Constructor.

Parameters:

nu - the number of degrees of freedom

ChiSquareDistr

public ChiSquareDistr(long nu,
 double lambda)

Constructor for a non-central χ² distribution. λ is defined as the sum of the squares of the means of the random variables whose squares are summed.

Parameters:

nu - the number of degrees of freedom

lambda - the non-centrality parameter for the distribution

Method Details

getDegreesOfFreedom

public long getDegreesOfFreedom()

Get the number of degrees of freedom.

Returns:

the number of degrees of freedom

isNonCentral

public boolean isNonCentral()

Determine if this object is a non-central χ² distribution.

Returns:

true if this object is a non-central χ² distribution; false otherwise.

getLambda

public double getLambda()

Get the non-centrality parameter.

Returns:

the non-centrality parameter

Throws:

IllegalStateException - this object is not a non-central χ² distribution

isSymmetric

public boolean isSymmetric(double x)

Description copied from class: ProbDistribution

Determine if this distribution is symmetric about x. A distribution is symmetric if its probability density f satisfies f(x+y) = f(x-y) and that if x+y is in f's domain, so is x-y. Equivalently, it must satisfy Q(x+y) = P(x-y).

Specified by:

isSymmetric in class ProbDistribution

Parameters:

x - the point about which to test if this distribution is symmaetric

Returns:

true if it is symmetric; false otherwise.

pd

public double pd(double x)

Description copied from class: ProbDistribution

Get the probability density

Overrides:

pd in class ProbDistribution

Parameters:

x - the value for which the density is to be computed

Returns:

the probability density

P

public double P(double x)

Description copied from class: ProbDistribution

Get the probability that a value is no larger than the argument

Specified by:

P in class ProbDistribution

Parameters:

x - the argument

Returns:

the probability

Q

public double Q(double x)

Description copied from class: ProbDistribution

Get the probability that a value is no smaller than the argument

Overrides:

Q in class ProbDistribution

Parameters:

x - the argument

Returns:

the probability

getDomainMin

public double getDomainMin()

Description copied from interface: RealValuedDomainOps

Get the minimum value in the domain of the function.

Specified by:

getDomainMin in interface RealValuedDomainOps

Overrides:

getDomainMin in class ProbDistribution

Returns:

the minimum value

domainMinClosed

public boolean domainMinClosed()

Description copied from interface: RealValuedDomainOps

Determine if the domain minimum is in the domain.

Specified by:

domainMinClosed in interface RealValuedDomainOps

Overrides:

domainMinClosed in class ProbDistribution

Returns:

true if the domain minimum is in the domain; false if it is the greatest lower bound for the domain

pd

public static double pd(double chi2,
 long nu)

Get the probability density for a χ² distribution.

Parameters:

chi2 - the χ² value

nu - the number of degrees of freedom

Returns:

the χ² probability density

P

public static double P(double chi2,
 long nu)

Get the cumulative probability for a χ² distribution.

Parameters:

chi2 - the χ² value

nu - the number of degrees of freedom

Returns:

the corresponding value of the cumulative probability distribution

Q

public static double Q(double chi2,
 long nu)

Get the complement of the cumulative probability for a χ² distribution.

Parameters:

chi2 - the χ² value

nu - the number of degrees of freedom

Returns:

a value mathematically equal to 1 - P(chi2, nu)

pd

public static double pd(double chi2,
 long nu,
 double lambda)

Get the probability density for a non-central χ² distribution.

Parameters:

chi2 - the χ² value

nu - the number of degrees of freedom

lambda - the non-centrality parameter for the distribution

Returns:

the value of the noncentral χ² probability density for the specified parameters

P

public static double P(double chi2,
 long nu,
 double lambda)

Get the cumulative probability distribution for a non-central χ² distribution.

Parameters:

chi2 - the χ² value

nu - the number of degrees of freedom

lambda - the non-centrality parameter for the distribution

Returns:

the value of the cummulative distribution function for the specified parameters

Q

public static double Q(double chi2,
 long nu,
 double lambda)

Get the complement of the cumulative probability distribution for a non-central χ² distribution.

Parameters:

chi2 - the χ² value

nu - the number of degrees of freedom

lambda - the non-centrality parameter for the distribution

Returns:

the value of the complement of the cummulative distribution function for the specified parameters