# SuanShu, a Java numerical and statistical library

com.numericalmethod.suanshu.stats.distribution.multivariate

## Class MultivariateNormalDistribution

• java.lang.Object
• com.numericalmethod.suanshu.stats.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.

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

All Methods
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 etX.
• ### 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
Wikipedia: Entropy (information theory)
• #### moment

public double moment(Vector t)
Description copied from interface: MultivariateProbabilityDistribution
The moment generating function is the expected value of etX. That is,
E(etX)
This may not always exist.
Specified by:
moment in interface MultivariateProbabilityDistribution
Parameters:
t - t
Returns:
E(exp(tX))