If you want a random correlation matrix, why not just generate random data and accept the correlation matrix that you get? The standard normal distribution in k dimensions is (hyper)spherically symmetric. If you generate k standard normal N(0,1) variates, you have a point in k-space with direction uniformly distributed on the (k-1)sphere and Gaussian magnitude. If you generate k such, you have a random linear transformation with all sorts of desirable symmetries. So, if you generate a kxk matrix of standard normal variates, and another nxk standard normal variates, and multiply the two matrices to get n points in k space, that seems to be a pretty good definition of random correlation to me. I'm sure you can decompose the kxk matrix to get the theoretical distribution, maybe by multiplying it by its transpose and doing an SVD; I'd have to think about that part.
... unless you have a particular distribution of correlation matrices in mind to begin with, which doesn't seem to be the case.
-----Original Message-----
From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org] On Behalf Of Szumiloski, John
Sent: Wednesday, February 09, 2011 11:30 AM
To: r-help at r-project.org
Cc: Rick DeShon
Subject: Re: [R] Generate multivariate normal data with a random correlation matrix
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
______________________________________________
R-help at r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
Notice: This e-mail message, together with any attachme...{{dropped:11}}
______________________________________________
R-help at r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
message may contain confidential information. If you are not the designated recipient, please notify the sender immediately, and delete the original and any copies. Any use of the message by you is prohibited.