dtmvnorm.marginal2: Bivariate marginal density functions from a Truncated...

In tmvtnorm: Truncated Multivariate Normal and Student t Distribution

Bivariate marginal density functions from a Truncated Multivariate Normal distribution

Description

This function computes the bivariate marginal density function f(x_q, x_r) from a k-dimensional Truncated Multivariate Normal density function (k>=2). The bivariate marginal density is obtained by integrating out (k-2) dimensions as proposed by Tallis (1961). This function is basically an extraction of the Leppard and Tallis (1989) Fortran code for moments calculation, but extended to the double truncated case.

Usage

dtmvnorm.marginal2(xq, xr, q, r, 
 mean = rep(0, nrow(sigma)), 
 sigma = diag(length(mean)), 
 lower = rep(-Inf, length = length(mean)), 
 upper = rep(Inf, length = length(mean)),
 log = FALSE, pmvnorm.algorithm=GenzBretz())

Arguments

xq	Value x_q
xr	Value x_r
q	Index position for x_q within mean vector to calculate the bivariate marginal density for.
r	Index position for x_r within mean vector to calculate the bivariate marginal density for.
mean	Mean vector, default is rep(0, length = 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).
pmvnorm.algorithm	Algorithm used for pmvnorm

Details

The bivariate marginal density function f(x_q, x_r) for x \sim TN(\mu, \Sigma, a, b) and q \ne r is defined as

F_{q,r}(x_q=c_q, x_r=c_r) = \int^{b_1}_{a_1}...\int^{b_{q-1}}_{a_{q-1}}\int^{b_{q+1}}_{a_{q+1}}...\int^{b_{r-1}}_{a_{r-1}}\int^{b_{r+1}}_{a_{r+1}}...\int^{b_{k}}_{a_{k}} \varphi{_{\alpha}}_{\Sigma}(x_s, c_q, c_r) dx_s

Author(s)

Stefan Wilhelm <[e-mail]>, Manjunath B G <[e-mail]>

References

Tallis, G. M. (1961). The moment generating function of the truncated multinormal distribution. Journal of the Royal Statistical Society, Series B, 23, 223–229

Leppard, P. and Tallis, G. M. (1989). Evaluation of the Mean and Covariance of the Truncated Multinormal Applied Statistics, 38, 543–553

Manjunath B G and Wilhelm, S. (2009). Moments Calculation For the Double Truncated Multivariate Normal Density. Working Paper. Available at SSRN: https://www.ssrn.com/abstract=1472153

Examples

  
  lower = c(-0.5, -1, -1)
  upper = c( 2.2,  2,  2)
  
  mean  = c(0,0,0)
  sigma = matrix(c(2.0, -0.6,  0.7, 
                  -0.6,  1.0, -0.2, 
                   0.7, -0.2,  1.0), 3, 3)
  
  # generate random samples from untruncated and truncated distribution
  Y = rmvnorm(10000, mean=mean, sigma=sigma)
  X = rtmvnorm(500,  mean=mean, sigma=sigma, lower=lower, upper=upper, 
      algorithm="gibbs")
    
  # compute bivariate marginal density of x1 and x2
  xq <- seq(lower[1], upper[1], by=0.1)
  xr <- seq(lower[2], upper[2], by=0.1)
  
  grid <- matrix(NA, length(xq), length(xr))
  for (i in 1:length(xq))
  {
    for (j in 1:length(xr))
    {
      grid[i,j] = dtmvnorm.marginal2(xq=xq[i], xr=xr[j], 
        q=1, r=2, sigma=sigma, lower=lower, upper=upper)
    }
  }
  
  plot(Y[,1], Y[,2], xlim=c(-4, 4), ylim=c(-4, 4), 
     main=expression("bivariate marginal density ("*x[1]*","*x[2]*")"), 
     xlab=expression(x[1]), ylab=expression(x[2]), col="gray80")
  points(X[,1], X[,2], col="black")
  
  lines(x=c(lower[1], upper[1], upper[1], lower[1], lower[1]), 
        y=c(lower[2],lower[2],upper[2],upper[2],lower[2]), 
        lty=2, col="red")
  contour(xq, xr, grid, add=TRUE, nlevels = 8, col="red", lwd=2)
  
  # scatterplot matrices for untruncated and truncated points
  require(lattice)
  splom(Y)
  splom(X)

tmvtnorm documentation built on May 29, 2024, 1:14 a.m.