Page 111 - Computational Statistics Handbook with MATLAB

98 Computational Statistics Handbook with MATLAB

Example 4.10
The function csmvrnd generates multivariate normal random variables
using the Cholesky factorization. Note that we are transposing the transfor-
mation given in Equation 4.18, yielding the following

X = ZR + µ T ,

where X is an n × d matrix of d-dimensional random variables and Z is an
n × d matrix of standard normal random variables.

% function X = csmvrnd(mu,covm,n);
% This function will return n multivariate random
% normal variables with d-dimensional mean mu and
% covariance matrix covm. Note that the covariance
% matrix must be positive definite (all eigenvalues
% are greater than zero), and the mean
% vector is a column

function X = csmvrnd(mu,covm,n)
d = length(mu);
% Get Cholesky factorization of covariance.
R = chol(covm);
% Generate the standard normal random variables.
Z = randn(n,d);
X = Z*R + ones(n,1)*mu';
We illustrate its use by generating some multivariate normal random vari-
,
T
ables with µ = – ( 23) and covariance

10.7
Σ = .
0.7 1

% Generate the multivariate random normal variables.
mu = [-2;3];
covm = [1 0.7 ; 0.7 1];
X = csmvrnd(mu,covm,500);

To check the results, we plot the random variables in a scatterplot in
Figure 4.7. We can also calculate the sample mean and sample covariance
matrix to compare with what we used as input arguments to csmvrnd. By
typing mean(X) at the command line, we get

-2.0629 2.9394

Similarly, entering corrcoef(X)at the command line yields

© 2002 by Chapman & Hall/CRC