mixAK (version 5.8)

TMVN: Truncated multivariate normal distribution

Description

Random generation for the truncated multivariate normal distribution. The mean and covariance matrix of the original multivariate normal distribution are mean and Sigma. Truncation limits are given by a, b, type of truncation is given by trunc.

This function uses a Gibbs algorithm to produce a Markov chain whose stationary distribution is the targeted truncated multivariate normal distribution, see Geweke (1991) for more details. Be aware that the sampled values are not i.i.d.!

Usage

rTMVN(n, mean=c(0, 0), Sigma=diag(2), a, b, trunc, xinit)

Value

A matrix with the sampled values (Markov chain) in rows.

Arguments

mean

a numeric vector of the mean of the original multivariate normal distribution.

Sigma

covariance matrix of the original multivariate normal distribution.

a

a numeric vector of the same length as mean of truncation limits 1.

b

a numeric vector of the same length as mean of truncation limits 2.

trunc

a numeric vector of the same length as mean describing the type of truncation in each margin.

trunc=0

normal distribution is truncated on the interval \((a,\,\infty).\). Value of \(b\) is ignored.

trunc=1

degenerated normal distribution, all values are with probability 1 equal to \(a\), \(b\) is ignored.

trunc=2

normal distribution is truncated on the interval \((-\infty,\,a).\) Value of \(b\) is ignored.

trunc=3

normal distribution is truncated on the interval \((a,\,b).\)

trunc=4

there is no truncation, values of \(a\) and \(b\) are ignored.

If trunc is not given, it is assumed that it is equal to 4. Note that a, b and trunc must have the same length, with exception that b does not have to be supplied if all trunc values 0, 1, 2 or 4.

xinit

a numeric vector of the same length as mean with the initial value for the Gibbs sampler. If it is not supplied, the function determines itself the initial value.

n

number of observations to be sampled.

Author

Arnošt Komárek

References

Geweke, J. (1991). Efficient simulation from the multivariate normal and Student-t distributions subject to linear constraints and the evaluation of constraint probabilities. Computer Sciences and Statistics, 23, 571--578.

See Also

rTNorm.

Examples

if (FALSE) {
set.seed(1977)

exam2 <- function(n, mu, sigma, rho, a, b, trunc)
{
  Sigma <- matrix(c(sigma[1]^2, rho*sigma[1]*sigma[2], rho*sigma[1]*sigma[2], sigma[2]^2), nrow=2)
  x <- rTMVN(n, mean=mu, Sigma=Sigma, a=a, b=b, trunc=trunc)
  x1.gr <- seq(mu[1]-3.5*sigma[1], mu[1]+3.5*sigma[1], length=100)
  x2.gr <- seq(mu[2]-3.5*sigma[2], mu[2]+3.5*sigma[2], length=100)    
  z <- cbind(rep(x1.gr, 100), rep(x2.gr, each=100))
  dens.z <- matrix(dMVN(z, mean=mu, Sigma=Sigma), ncol=100)

  MEAN <- round(apply(x, 2, mean), 3)
  SIGMA <- var(x)
  SD <- sqrt(diag(SIGMA))
  RHO <- round(SIGMA[1,2]/(SD[1]*SD[2]), 3)
  SD <- round(SD, 3)
  
  layout(matrix(c(0,1,1,0, 2,2,3,3), nrow=2, byrow=TRUE))
  contour(x1.gr, x2.gr, dens.z, col="darkblue", xlab="x[1]", ylab="x[2]")
  points(x[,1], x[,2], col="red")
  title(sub=paste("Sample mean = ", MEAN[1], ", ", MEAN[2], ",  Sample SD = ", SD[1], ", ", SD[2],
                  ",  Sample rho = ", RHO, sep=""))
  plot(1:n, x[,1], type="l", xlab="Iteration", ylab="x[1]", col="darkgreen")
  plot(1:n, x[,2], type="l", xlab="Iteration", ylab="x[2]", col="darkgreen")

  return(x)  
}  

x1 <- exam2(1000, mu=c(-1, 1), sigma=c(1, sqrt(2)), rho=0, a=c(-6, -9), b=c(4, 11), trunc=c(3, 3))
x2 <- exam2(1000, mu=c(-1, 1), sigma=c(1, sqrt(2)), rho=0.7, a=c(-6, -9), b=c(4, 11), trunc=c(3, 3))
x3 <- exam2(1000, mu=c(-1, 1), sigma=c(1, sqrt(2)), rho=0.7, a=c(-100, -100), b=c(100, 100),
            trunc=c(3, 3))
x4 <- exam2(1000, mu=c(-1, 1), sigma=c(1, sqrt(2)), rho=-0.7, a=c(-6, -9), b=c(4, 11),
            trunc=c(3, 3))
x5 <- exam2(1000, mu=c(-1, 1), sigma=c(1, sqrt(2)), rho=-0.9, a=c(-6, -9), b=c(4, 11),
            trunc=c(3, 3))

x6 <- exam2(1000, mu=c(-1, 1), sigma=c(1, sqrt(2)), rho=0.7, a=c(0, 0), trunc=c(0, 2))
x7 <- exam2(1000, mu=c(-1, 1), sigma=c(1, sqrt(2)), rho=0.7, a=c(-1, 1), trunc=c(0, 2))
x8 <- exam2(1000, mu=c(-1, 1), sigma=c(1, sqrt(2)), rho=0.7, a=c(-1, 1), trunc=c(1, 2))
x9 <- exam2(1000, mu=c(-1, 1), sigma=c(1, sqrt(2)), rho=0.7, a=c(-1.5, 0.5), b=c(-0.5, 1.5),
            trunc=c(3, 3))
x10 <- exam2(1000, mu=c(-1, 1), sigma=c(1, sqrt(2)), rho=0.7, a=c(-1.5, 0.5), b=c(-0.5, 1.5),
             trunc=c(4, 3))
}