tmvtnorm (version 0.8-3)

tmvnorm: Truncated Multivariate Normal Density

Description

These functions provide the density function for the truncated multivariate normal distribution with mean equal to mean and covariance matrix sigma, lower and upper truncation points lower and upper

Usage

dtmvnorm(x, mean = rep(0, nrow(sigma)), 
  sigma = diag(length(mean)), 
  lower=rep(-Inf, length = length(mean)), 
  upper=rep( Inf, length = length(mean)),
  log=FALSE)

Arguments

x

Vector or matrix of quantiles. If x is a matrix, each row is taken to be a quantile.

mean

Mean vector, default is rep(0, length = ncol(x)).

sigma

Covariance matrix, default is diag(ncol(x)).

lower

Vector of lower truncation points,\ default is rep(-Inf, length = length(mean)).

upper

Vector of upper truncation points,\ default is rep( Inf, length = length(mean)).

log

Logical; if TRUE, densities d are given as log(d).

Details

The computation of truncated multivariate normal probabilities and densities is done using conditional probabilities from the standard/untruncated multivariate normal distribution. So we refer to the documentation of the mvtnorm package and the methodology is described in Genz (1992, 1993).

References

Genz, A. (1992). Numerical computation of multivariate normal probabilities. Journal of Computational and Graphical Statistics, 1, 141--150 Genz, A. (1993). Comparison of methods for the computation of multivariate normal probabilities. Computing Science and Statistics, 25, 400--405 Johnson, N./Kotz, S. (1970). Distributions in Statistics: Continuous Multivariate Distributions Wiley & Sons, pp. 70--73 Horrace, W. (2005). Some Results on the Multivariate Truncated Normal Distribution. Journal of Multivariate Analysis, 94, 209--221

See Also

ptmvnorm, pmvnorm, rmvnorm, dmvnorm

Examples

dtmvnorm(x=c(0,0))
dtmvnorm(x=c(0,0), mean=c(1,1), upper=c(0,0))

###########################################
#
# Example 1: 
# truncated multivariate normal density        
#
############################################

x1<-seq(-2, 3, by=0.1)
x2<-seq(-2, 3, by=0.1)

density<-function(x)
{
  sigma=matrix(c(1, -0.5, -0.5, 1), 2, 2)
  z=dtmvnorm(x, mean=c(0,0), sigma=sigma, lower=c(-1,-1))
  z
}

fgrid <- function(x, y, f)
{
    z <- matrix(nrow=length(x), ncol=length(y))
    for(m in 1:length(x)){
        for(n in 1:length(y)){
            z[m,n] <- f(c(x[m], y[n]))
        }
    }
    z
}

# compute density d for grid
d=fgrid(x1, x2, density)

# plot density as contourplot
contour(x1, x2, d, nlevels=5, main="Truncated Multivariate Normal Density", 
  xlab=expression(x[1]), ylab=expression(x[2]))
abline(v=-1, lty=3, lwd=2)
abline(h=-1, lty=3, lwd=2)

###########################################
#
# Example 2: 
# generation of random numbers
# from a truncated multivariate normal distribution        
#
############################################

sigma <- matrix(c(4,2,2,3), ncol=2)
x <- rtmvnorm(n=500, mean=c(1,2), sigma=sigma, upper=c(1,0))
plot(x, main="samples from truncated bivariate normal distribution",
  xlim=c(-6,6), ylim=c(-6,6), 
  xlab=expression(x[1]), ylab=expression(x[2]))
abline(v=1, lty=3, lwd=2, col="gray")
abline(h=0, lty=3, lwd=2, col="gray")