loglike_mvnorm function

Log-Likelihood Value of a Multivariate Normal Distribution

Computes log-likelihood value of a multivariate normal distribution given the empirical mean vector and the empirical covariance matrix as sufficient statistics.

loglike_mvnorm(M, S, mu, Sigma, n, log=TRUE, lambda=0, ginv=FALSE, eps=1e-30,
       use_rcpp=FALSE )

loglike_mvnorm_NA_pattern( suff_stat, mu, Sigma, log=TRUE, lambda=0, ginv=FALSE,
       eps=1e-30, use_rcpp=FALSE )

Arguments

• M: Empirical mean vector
• S: Empirical covariance matrix
• mu: Population mean vector
• Sigma: Population covariance matrix
• n: Sample size
• log: Optional logical indicating whether the logarithm of the likelihood should be calculated.
• lambda: Regularization parameter of the covariance matrix (see Details).
• ginv: Logical indicating whether generalized inverse matrix of $\bold{\Sigma}$ should be used
• eps: Threshold for determinant value of $\bold{\Sigma}$
• use_rcpp: Logical indicating whether Rcpp function should be used
• suff_stat: List with sufficient statistics as generated by suff_stat_NA_pattern.

Details

The population covariance matrix $\bold{\Sigma}$ is regularized if $\lambda$ (lambda) is chosen larger than zero. Let $\bold{\Delta _\Sigma}$ denote a diagonal matrix containing the diagonal entries of $\bold{\Sigma}$. Then, a regularized matrix $\bold{\Sigma}^\ast$ is defined as $\bold{\Sigma}^\ast=w \bold{\Sigma} + (1-w)\bold{\Delta _\Sigma }$

with $w=n/(n+\lambda)$.

Returns

Log-likelihood value

Examples

#############################################################################
# EXAMPLE 1: Multivariate normal distribution
#############################################################################

library(MASS)

#--- simulate data
Sigma <- c( 1, .55, .5, .55, 1, .5,.5, .5, 1 )
Sigma <- matrix( Sigma, nrow=3, ncol=3 )
mu <- c(0,1,1.2)
N <- 400
set.seed(9875)
dat <- MASS::mvrnorm( N, mu, Sigma )
colnames(dat) <- paste0("Y",1:3)
S <- stats::cov(dat)
M <- colMeans(dat)

#--- evaulate likelihood
res1 <- LAM::loglike_mvnorm( M=M, S=S, mu=mu, Sigma=Sigma, n=N, lambda=0 )
# compare log likelihood with somewhat regularized covariance matrix
res2 <- LAM::loglike_mvnorm( M=M, S=S, mu=mu, Sigma=Sigma, n=N, lambda=1 )
print(res1)
print(res2)

## Not run:

#############################################################################
# EXAMPLE 2: Multivariate normal distribution with missing data patterns
#############################################################################

library(STARTS)
data(data.starts01b, package="STARTS")
dat <- data.starts01b
dat1 <- dat[, paste0("E",1:3)]

#-- compute sufficient statistics
suff_stat <- LAM::suff_stat_NA_pattern(dat1)
#-- define some population mean and covariance
mu <- colMeans(dat1, na.rm=TRUE)
Sigma <- stats::cov(dat1, use="pairwise.complete.obs")

#-- compute log-likelihood
LAM::loglike_mvnorm_NA_pattern( suff_stat=suff_stat, mu=mu, Sigma=Sigma)
## End(Not run)