umontreal.iro.lecuyer.probdistmulti

Class MultinomialDist

java.lang.Object

umontreal.iro.lecuyer.probdistmulti.DiscreteDistributionIntMulti

umontreal.iro.lecuyer.probdistmulti.MultinomialDist

public class MultinomialDist
extends DiscreteDistributionIntMulti

Implements the abstract class DiscreteDistributionIntMulti for the multinomial distribution with parameters n and (p₁, ...,p_d). The probability mass function is

P[X = (x₁,..., x_d)] = n!∏_i=1^dp_i^x_i/(∏_i=1^dx_i!),

where ∑_i=1^dx_i = n and ∑_i=1^dp_i = 1.

Constructor Summary

Constructors

Constructor and Description
MultinomialDist(int n, double[] p) Creates a MultinomialDist object with parameters n and (p1,...,pd) such that ∑i=1dpi = 1.

Method Summary

Modifier and Type	Method and Description
double	cdf(int[] x) Computes the cumulative probability function F of the distribution evaluated at x, assuming the lowest values start at 0, i.e.
static double	cdf(int n, double[] p, int[] x) Computes the function F of the multinomial distribution with parameters n and (p1,...,pd) evaluated at x.
double[][]	getCorrelation() Returns the correlation matrix of the distribution, defined as ρij = σij/(σ_iiσ_jj)1/2.
static double[][]	getCorrelation(int n, double[] p) Computes the correlation matrix of the multinomial distribution with parameters n and (p1,...,pd).
double[][]	getCovariance() Returns the variance-covariance matrix of the distribution, defined as σij = E[(Xi - μi)(Xj - μj)].
static double[][]	getCovariance(int n, double[] p) Computes the covariance matrix of the multinomial distribution with parameters n and (p1,...,pd).
static double[]	getMaximumLikelihoodEstimate(int[][] x, int m, int d, int n) Deprecated.
double[]	getMean() Returns the mean vector of the distribution, defined as μi = E[Xi].
static double[]	getMean(int n, double[] p) Computes the mean E[Xi] = npi of the multinomial distribution with parameters n and (p1,...,pd).
static double[]	getMLE(int[][] x, int m, int d, int n) Estimates and returns the parameters [hat(p_i),...,hat(p_d)] of the multinomial distribution using the maximum likelihood method.
int	getN() Returns the parameter n of this object.
double[]	getP() Returns the parameters (p1,...,pd) of this object.
double	prob(int[] x) Returns the probability mass function p(x1, x2,…, xd), which should be a real number in [0, 1].
static double	prob(int n, double[] p, int[] x) Computes the probability mass function of the multinomial distribution with parameters n and (p1,...,pd) evaluated at x.
void	setParams(int n, double[] p) Sets the parameters n and (p1,...,pd) of this object.

Methods inherited from class umontreal.iro.lecuyer.probdistmulti.DiscreteDistributionIntMulti

getDimension

Methods inherited from class java.lang.Object

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

Constructor Detail

MultinomialDist

public MultinomialDist(int n,
                       double[] p)

Creates a MultinomialDist object with parameters n and (p₁,...,p_d) such that ∑_i=1^dp_i = 1. We have p_i = p[i-1].

Method Detail

prob

public double prob(int[] x)

Description copied from class: DiscreteDistributionIntMulti

Returns the probability mass function p(x₁, x₂,…, x_d), which should be a real number in [0, 1].

Specified by:

prob in class DiscreteDistributionIntMulti

Parameters:

x - value at which the mass function must be evaluated

Returns:

the mass function evaluated at x

cdf

public double cdf(int[] x)

Description copied from class: DiscreteDistributionIntMulti

Computes the cumulative probability function F of the distribution evaluated at x, assuming the lowest values start at 0, i.e. computes

F(x₁, x₂,…, x_d) = ∑_s1=0^x₁∑_s2=0^{x₂ ...} ∑_sd=0^x_dp(s₁, s₂,…, s_d).

Uses the naive implementation, is very inefficient and may underflows.

Overrides:

cdf in class DiscreteDistributionIntMulti

getMean

public double[] getMean()

Description copied from class: DiscreteDistributionIntMulti

Returns the mean vector of the distribution, defined as μ_i = E[X_i].

Specified by:

getMean in class DiscreteDistributionIntMulti

getCovariance

public double[][] getCovariance()

Description copied from class: DiscreteDistributionIntMulti

Returns the variance-covariance matrix of the distribution, defined as
σ_ij = E[(X_i - μ_i)(X_j - μ_j)].

Specified by:

getCovariance in class DiscreteDistributionIntMulti

getCorrelation

public double[][] getCorrelation()

Description copied from class: DiscreteDistributionIntMulti

Returns the correlation matrix of the distribution, defined as ρ_ij = σ_ij/(σ_iiσ_jj)^1/2.

Specified by:

getCorrelation in class DiscreteDistributionIntMulti

prob

public static double prob(int n,
                          double[] p,
                          int[] x)

Computes the probability mass function of the multinomial distribution with parameters n and (p₁,...,p_d) evaluated at x.

cdf

public static double cdf(int n,
                         double[] p,
                         int[] x)

Computes the function F of the multinomial distribution with parameters n and (p₁,...,p_d) evaluated at x.

getMean

public static double[] getMean(int n,
                               double[] p)

Computes the mean E[X_i] = np_i of the multinomial distribution with parameters n and (p₁,...,p_d).

getCovariance

public static double[][] getCovariance(int n,
                                       double[] p)

Computes the covariance matrix of the multinomial distribution with parameters n and (p₁,...,p_d).

getCorrelation

public static double[][] getCorrelation(int n,
                                        double[] p)

Computes the correlation matrix of the multinomial distribution with parameters n and (p₁,...,p_d).

getMaximumLikelihoodEstimate

@Deprecated
public static double[] getMaximumLikelihoodEstimate(int[][] x,
                                                                int m,
                                                                int d,
                                                                int n)

Deprecated.

It is now called getMLE.

getMLE

public static double[] getMLE(int[][] x,
                              int m,
                              int d,
                              int n)

Estimates and returns the parameters [hat(p_i),...,hat(p_d)] of the multinomial distribution using the maximum likelihood method. It uses the m observations of d components in table x[i][j], i = 0, 1,…, m - 1 and j = 0, 1,…, d - 1.

Parameters:

x - the list of observations used to evaluate parameters

m - the number of observations used to evaluate parameters

d - the dimension of each observation

n - the number of independant trials for each series

Returns:

returns the parameters [hat(p_i),...,hat(p_d)]

getN

public int getN()

Returns the parameter n of this object.

getP

public double[] getP()

Returns the parameters (p₁,...,p_d) of this object.

setParams

public void setParams(int n,
                      double[] p)

Sets the parameters n and (p₁,...,p_d) of this object.