mixAK (version 5.8)

MVN: Multivariate normal distribution

Description

Density and random generation for the multivariate normal distribution with mean equal to mean, precision matrix equal to Q (or covariance matrix equal to Sigma).

Function rcMVN samples from the multivariate normal distribution with a canonical mean \(b\), i.e., the mean is \(\mu = Q^{-1}\,b.\)

Usage

dMVN(x, mean=0, Q=1, Sigma, log=FALSE)
rMVN(n, mean=0, Q=1, Sigma)
rcMVN(n, b=0, Q=1, Sigma)

Value

Some objects.

Arguments

mean

vector of mean.

b

vector of a canonical mean.

Q

precision matrix of the multivariate normal distribution. Ignored if Sigma is given.

Sigma

covariance matrix of the multivariate normal distribution. If Sigma is supplied, precision is computed from \(\Sigma\) as \(Q = \Sigma^{-1}\).

n

number of observations to be sampled.

x

vector or matrix of the points where the density should be evaluated.

log

logical; if TRUE, log-density is computed

Value for dMVN

A vector with evaluated values of the (log-)density

Value for rMVN

A list with the components:

x

vector or matrix with sampled values

log.dens

vector with the values of the log-density evaluated in the sampled values

Value for rcMVN

A list with the components:

x

vector or matrix with sampled values

mean

vector or the mean of the normal distribution

log.dens

vector with the values of the log-density evaluated in the sampled values

Author

Arnošt Komárek

References

Rue, H. and Held, L. (2005). Gaussian Markov Random Fields: Theory and Applications. Boca Raton: Chapman and Hall/CRC.

See Also

dnorm, Mvnorm.

Examples

set.seed(1977)

### Univariate normal distribution
### ==============================
c(dMVN(0), dnorm(0))
c(dMVN(0, log=TRUE), dnorm(0, log=TRUE))

rbind(dMVN(c(-1, 0, 1)), dnorm(c(-1, 0, 1)))
rbind(dMVN(c(-1, 0, 1), log=TRUE), dnorm(c(-1, 0, 1), log=TRUE))

c(dMVN(1, mean=1.2, Q=0.5), dnorm(1, mean=1.2, sd=sqrt(2)))
c(dMVN(1, mean=1.2, Q=0.5, log=TRUE), dnorm(1, mean=1.2, sd=sqrt(2), log=TRUE))

rbind(dMVN(0:2, mean=1.2, Q=0.5), dnorm(0:2, mean=1.2, sd=sqrt(2)))
rbind(dMVN(0:2, mean=1.2, Q=0.5, log=TRUE), dnorm(0:2, mean=1.2, sd=sqrt(2), log=TRUE))

### Multivariate normal distribution
### ================================
mu <- c(0, 6, 8)
L <- matrix(1:9, nrow=3)
L[upper.tri(L, diag=FALSE)] <- 0
Sigma <- L %*% t(L)
Q <- chol2inv(chol(Sigma))
b <- solve(Sigma, mu)

dMVN(mu, mean=mu, Q=Q)
dMVN(mu, mean=mu, Sigma=Sigma)
dMVN(mu, mean=mu, Q=Q, log=TRUE)
dMVN(mu, mean=mu, Sigma=Sigma, log=TRUE)

xx <- matrix(c(0,6,8, 1,5,7, -0.5,5.5,8.5, 0.5,6.5,7.5), ncol=3, byrow=TRUE)
dMVN(xx, mean=mu, Q=Q)
dMVN(xx, mean=mu, Sigma=Sigma)
dMVN(xx, mean=mu, Q=Q, log=TRUE)
dMVN(xx, mean=mu, Sigma=Sigma, log=TRUE)

zz <- rMVN(1000, mean=mu, Sigma=Sigma)
rbind(apply(zz$x, 2, mean), mu)
var(zz$x)
Sigma
cbind(dMVN(zz$x, mean=mu, Sigma=Sigma, log=TRUE), zz$log.dens)[1:10,]

zz <- rcMVN(1000, b=b, Sigma=Sigma)
rbind(apply(zz$x, 2, mean), mu)
var(zz$x)
Sigma
cbind(dMVN(zz$x, mean=mu, Sigma=Sigma, log=TRUE), zz$log.dens)[1:10,]

zz <- rMVN(1000, mean=rep(0, 3), Sigma=Sigma)
rbind(apply(zz$x, 2, mean), rep(0, 3))
var(zz$x)
Sigma
cbind(dMVN(zz$x, mean=rep(0, 3), Sigma=Sigma, log=TRUE), zz$log.dens)[1:10,]


### The same using the package mvtnorm
### ==================================
# require(mvtnorm)
# c(dMVN(mu, mean=mu, Sigma=Sigma), dmvnorm(mu, mean=mu, sigma=Sigma))
# c(dMVN(mu, mean=mu, Sigma=Sigma, log=TRUE), dmvnorm(mu, mean=mu, sigma=Sigma, log=TRUE))
#
# rbind(dMVN(xx, mean=mu, Sigma=Sigma), dmvnorm(xx, mean=mu, sigma=Sigma))
# rbind(dMVN(xx, mean=mu, Sigma=Sigma, log=TRUE), dmvnorm(xx, mean=mu, sigma=Sigma, log=TRUE))