umontreal.iro.lecuyer.probdistmulti
Class DirichletDist

java.lang.Object
  umontreal.iro.lecuyer.probdistmulti.ContinuousDistributionMulti
      umontreal.iro.lecuyer.probdistmulti.DirichletDist

public class DirichletDist
extends ContinuousDistributionMulti

Implements the abstract class ContinuousDistributionMulti for the Dirichlet distribution with parameters (α₁,...,α_d), α_i > 0. The probability density is

f (x₁,…, x_d) = Γ(α₀)∏_i=1^dx_i^α_i-1/(∏_i=1^dΓ(α_i))

where x_i >= 0, ∑_i=1^dx_i = 1, α₀ = ∑_i=1^dα_i, and Γ is the Gamma function.

Constructor Summary

DirichletDist(double[] alpha)

Method Summary

double	density(double[] x) Returns f (x1, x2,…, xd), the probability density of X evaluated at the point x, where x = {x1, x2,…, xd}.
static double	density(double[] alpha, double[] x) Computes the density of the Dirichlet distribution with parameters (α1, ..., αd).
double[]	getAlpha() Returns the parameters (α1, ..., αd) of this object.
double	getAlpha(int i) Returns the ith component of the alpha vector.
double[][]	getCorrelation() Returns the correlation matrix of the distribution, defined as ρij = σij/(σ_iiσ_jj)1/2.
static double[][]	getCorrelation(double[] alpha) Computes the correlation matrix of the Dirichlet distribution with parameters (α1, ..., αd).
double[][]	getCovariance() Returns the variance-covariance matrix of the distribution, defined as σij = E[(Xi - μi)(Xj - μj)].
static double[][]	getCovariance(double[] alpha) Computes the covariance matrix of the Dirichlet distribution with parameters (α1, ..., αd).
double[]	getMean() Returns the mean vector of the distribution, defined as μi = E[Xi].
static double[]	getMean(double[] alpha) Computes the mean E[X] = αi/α0 of the Dirichlet distribution with parameters (α1, ..., αd), where α0 = ∑i=1dαi.
static double[]	getMLE(double[][] x, int n, int d) Estimates the parameters [ hat(α_1),…, hat(α_d)] of the Dirichlet distribution using the maximum likelihood method.
void	setParams(double[] alpha) Sets the parameters (α1, ..., αd) of this object.

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

getDimension

Methods inherited from class java.lang.Object

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

Constructor Detail

DirichletDist

public DirichletDist(double[] alpha)

Method Detail

density

public double density(double[] x)

Description copied from class: ContinuousDistributionMulti

Returns f (x₁, x₂,…, x_d), the probability density of X evaluated at the point x, where x = {x₁, x₂,…, x_d}. The convention is that x[i - 1] = x_i.

Specified by:

density in class ContinuousDistributionMulti

Parameters:

x - value at which the density is evaluated

Returns:

density function evaluated at x

getMean

public double[] getMean()

Description copied from class: ContinuousDistributionMulti

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

Specified by:

getMean in class ContinuousDistributionMulti

getCovariance

public double[][] getCovariance()

Description copied from class: ContinuousDistributionMulti

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 ContinuousDistributionMulti

getCorrelation

public double[][] getCorrelation()

Description copied from class: ContinuousDistributionMulti

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

Specified by:

getCorrelation in class ContinuousDistributionMulti

density

public static double density(double[] alpha,
                             double[] x)

Computes the density of the Dirichlet distribution with parameters (α₁, ..., α_d).

getCovariance

public static double[][] getCovariance(double[] alpha)

Computes the covariance matrix of the Dirichlet distribution with parameters (α₁, ..., α_d).

getCorrelation

public static double[][] getCorrelation(double[] alpha)

Computes the correlation matrix of the Dirichlet distribution with parameters (α₁, ..., α_d).

getMLE

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

Estimates the parameters [ hat(α_1),…, hat(α_d)] of the Dirichlet distribution using the maximum likelihood method. It uses the n observations of d components in table x[i][j], i = 0, 1,…, n - 1 and j = 0, 1,…, d - 1.

Parameters:

x - the list of observations to use to evaluate parameters

n - the number of observations to use to evaluate parameters

d - the dimension of each vector

Returns:

returns the parameter [ hat(α_1),…, hat(α_d)]

getMean

public static double[] getMean(double[] alpha)

Computes the mean E[X] = α_i/α₀ of the Dirichlet distribution with parameters (α₁, ..., α_d), where α₀ = ∑_i=1^dα_i.

getAlpha

public double[] getAlpha()

Returns the parameters (α₁, ..., α_d) of this object.

getAlpha

public double getAlpha(int i)

Returns the ith component of the alpha vector.

setParams

public void setParams(double[] alpha)

Sets the parameters (α₁, ..., α_d) of this object.