loglike_mvnorm: Log-Likelihood Value of a Multivariate Normal Distribution

In LAM: Some Latent Variable Models

Log-Likelihood Value of a Multivariate Normal Distribution

Description

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

Usage

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).

Value

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)

LAM documentation built on Sept. 11, 2024, 9:16 p.m.