Thanks for the response, Rex. This is an interesting approach. The
Choleski decomposition approach that John suggested seems to be an
obvious and direct approach to this problem. Your approach is less
obvious to me but may be equal or superior to the Choleski
decomposition.
Are all possible correlation matrices of size k equally likely using
your approach? It would seem so based on your description. If so, it
is a way cool solution.
Rick
On Thu, Feb 10, 2011 at 12:18 PM, <rex.dwyer at syngenta.com> wrote:
> 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
>
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