Generate Random Samples from Multivariate Normal Distribution

In \(\mathbf{R}^p\), random samples are drawn $$X_1,X_2,\ldots,X_n~ \sim ~ \mathcal{N}(\mu, \Sigma)$$ where \(\mu \in \mathbf{R}^p\) is a mean vector and \(\Sigma \in \textrm{SPD}(p)\) is a positive definite covariance matrix.

Usage

rmvnorm(n = 1, mu, sigma)

Arguments

n

the number of samples to be generated.

mu

mean vector.

sigma

covariance matrix.

Value

either (1) a length-\(p\) vector (\(n=1\)) or (2) an \((n\times p)\) matrix where rows are random samples.

Validation status

This retained legacy interface is experimental. Its full numerical and statistical contract has not been independently verified across supported inputs. See riem-method-contracts and the installed contract table for method-specific assumptions, restrictions, and evidence scope.

Examples

#-------------------------------------------------------------------
#   Generate Random Data and Compare with Empirical Covariances
#
# In R^5 with zero mean and diagonal covariance, 
# generate 100 and 200 observations and compute MLE covariance.
#-------------------------------------------------------------------
## GENERATE DATA
mymu  = rep(0,5)
mysig = diag(5)

## MLE FOR COVARIANCE
smat1 = stats::cov(rmvnorm(n=100, mymu, mysig))
smat2 = stats::cov(rmvnorm(n=200, mymu, mysig))

## VISUALIZE
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3), pty="s")
image(mysig[,5:1], axes=FALSE, main="true covariance")
image(smat1[,5:1], axes=FALSE, main="empirical cov with n=100")
image(smat2[,5:1], axes=FALSE, main="empirical cov with n=200")

par(opar)