dev.nm.stat.distribution.multivariate

Class MultivariateNormalDistribution

java.lang.Object

dev.nm.stat.distribution.multivariate.MultivariateNormalDistribution

All Implemented Interfaces:

MultivariateProbabilityDistribution

public class MultivariateNormalDistribution
extends Object
implements MultivariateProbabilityDistribution

The multivariate Normal distribution or multivariate Gaussian distribution, is a generalization of the one-dimensional (univariate) Normal distribution to higher dimensions.

An equivalent function in R is dmvnorm from the package mvtnorm.

See Also:

Wikipedia: Multivariate normal distribution

Constructor Summary

Constructors

Constructor and Description
MultivariateNormalDistribution(int dim) Constructs an instance of the standard Normal distribution.
MultivariateNormalDistribution(Vector mu, Matrix Sigma) Constructs an instance with the given mean and covariance matrix.

Method Summary

Modifier and Type	Method and Description
double	cdf(Vector x) Gets the cumulative probability F(x) = Pr(X ≤ x).
Matrix	covariance() Gets the covariance matrix of this distribution.
double	density(Vector x) The density function, which, if exists, is the derivative of F.
double	entropy() Gets the entropy of this distribution.
Vector	mean() Gets the mean of this distribution.
Vector	mode() Gets the mode of this distribution.
double	moment(Vector t) The moment generating function is the expected value of \(e^{tX}\).

Methods inherited from class java.lang.Object

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

Constructor Detail

MultivariateNormalDistribution

public MultivariateNormalDistribution(Vector mu,
                                      Matrix Sigma)

Constructs an instance with the given mean and covariance matrix.

Parameters:

mu - the mean

Sigma - the covariance matrix which must be positive definite

MultivariateNormalDistribution

public MultivariateNormalDistribution(int dim)

Constructs an instance of the standard Normal distribution.

Parameters:

dim - the dimensionality of the distribution

Method Detail

cdf

public double cdf(Vector x)

Description copied from interface: MultivariateProbabilityDistribution

Gets the cumulative probability F(x) = Pr(X ≤ x).

Specified by:

cdf in interface MultivariateProbabilityDistribution

Parameters:

x - x

Returns:

F(x) = Pr(X ≤ x)

density

public double density(Vector x)

Description copied from interface: MultivariateProbabilityDistribution

The density function, which, if exists, is the derivative of F. It describes the density of probability at each point in the sample space. \[ f(x) = dF(X) / dx \] This may not always exist.

For the discrete cases, this is the probability mass function. It gives the probability that a discrete random variable is exactly equal to some value.

Specified by:

density in interface MultivariateProbabilityDistribution

Parameters:

x - x

Returns:

f(x)

mean

public Vector mean()

Description copied from interface: MultivariateProbabilityDistribution

Gets the mean of this distribution.

Specified by:

mean in interface MultivariateProbabilityDistribution

Returns:

the mean

mode

public Vector mode()

Description copied from interface: MultivariateProbabilityDistribution

Gets the mode of this distribution.

Specified by:

mode in interface MultivariateProbabilityDistribution

Returns:

the mean

covariance

public Matrix covariance()

Description copied from interface: MultivariateProbabilityDistribution

Gets the covariance matrix of this distribution.

Specified by:

covariance in interface MultivariateProbabilityDistribution

Returns:

the covariance

entropy

public double entropy()

Description copied from interface: MultivariateProbabilityDistribution

Gets the entropy of this distribution.

Specified by:

entropy in interface MultivariateProbabilityDistribution

Returns:

the entropy

See Also:

Wikipedia: Entropy (information theory)

moment

public double moment(Vector t)

Description copied from interface: MultivariateProbabilityDistribution

The moment generating function is the expected value of \(e^{tX}\). That is, \[ E(e^{tX}) \] This may not always exist.

Specified by:

moment in interface MultivariateProbabilityDistribution

Parameters:

t - t

Returns:

E(exp(tX))

See Also:

Wikipedia: Moment-generating function