[R] Generate multivariate normal data with a random correlation matrix

[Szumiloski, John]

The knee jerk thought I had was to express the correlation matrix as a generic Choleski decomposition, then randomly populate the triangular decomposed matrix.  When you remultiply, you can simply rescale to 1s on the diagonals.  Then rmnorm as usual.

In R, see ?chol

If you want to get fancy, you could look at the random distribution you would use for the triangular matrix and play with that, including different distributions for different elements, elements' distributions being conditional on values of previously randomized elements, etc.  

John

-----Original Message-----
From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org] On Behalf Of Rick DeShon
Sent: Wednesday, 09 February, 2011 11:06 AM
To: r-help at stat.math.ethz.ch
Subject: [R] Generate multivariate normal data with a random correlation matrix

Hi All.

I'd like to generate a sample of n observations from a k dimensional multivariate normal distribution with a random correlation matrix.

My solution:
The lower (or upper) triangle of the correlation matrix has n.tri=(d/2)(d+1)-d entries.
Take a uniform sample of n.tri possible correlations (runi(n.tr,-.99,.99) Populate a triangle of the matrix with the sampled correlations Mirror the triangle to populate the other triangle forming a symmetric matrix, cormat Sample n observations from a multivariate normal distribution with mean vector=0 and varcov=cormat


Problem:
This approach violates the triangle inequality property of correlation matrices.  So, the matrix I've constructed is certainly a valid matrix but it is not a valid correlation matrix and it blows up when you submit it to a random number generator such as rmnorm.  With a small matrix you sometimes get lucky and generate a valid correlation matrix but as you increase d the probability of obtaining a valid correlation matrix drops off quickly.

So, any ideas on how to construct a correlation matrix with random entries that cover the range (or most of the range) or the correlation [-1,1]?

Here's the code I've used that won't work.
************************************************
library(mnormt)
n <- 1000
d <- 50

n.tri <- ((d*(d+1))/2)-d
r       <- runif(n.tri, min=-.5, max=.5)

cormat <- diag(c)
count1=1
for (i in 1:c){
       for (j in 1:c){
               if (i<j) {
                               cormat[i,j]=r[count1]
                               cormat[j,i]=cormat[i,j]
                               count1=count1+1
                            }
       }
}
eigen(cormat)     # if negative eigenvalue, then the matrix violates the triangle inequality

x <-  rmnorm(n, rep(0, c), cormat)  # Sample the data



Thanks in advance,

Rick DeShon

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