Truncated Multivariate Normal Density

Share:

Description

This function provides the joint 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. For convenience, it furthermore serves as a wrapper function for the one-dimensional and bivariate marginal densities dtmvnorm.marginal() and dtmvnorm.marginal2() respectively when invoked with the margin argument.

Usage

1
2
3
4
5
6
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, 
  margin=NULL)

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, nrow(sigma)).

sigma

Covariance matrix, default is diag(length(mean)).

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).

margin

if NULL then the joint density is computed (the default), if MARGIN=1 then the one-dimensional marginal density in variate q (q = 1..length(mean)) is returned, if MARGIN=c(q,r) then the bivariate marginal density in variates q and r for q,r = 1..length(mean) and q != r is returned.

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).

Author(s)

Stefan Wilhelm <Stefan.Wilhelm@financial.com>

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, dtmvnorm.marginal and dtmvnorm.marginal2 for marginal density functions

Examples

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
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")