On Thu, Jun 02, 2011 at 04:42:59PM -0700, Ned Dochtermann wrote:
> List members,
>
> Via searches I've seen similar discussion of this topic but have not seen
> resolution of the particular issue I am experiencing. If my search on this
> topic failed, I apologize for the redundancy. I am attempting to generate
> random covariance matrices but would like the corresponding correlations to
> be uniformly distributed between -1 and 1.
>
> The approach I have been using is:
>
> > st.dev<-.5 #this and k are aspects I would like to vary
> > k<-6 #number of... things
> > y1<-matrix(rnorm(k^2,mean=0,sd=st.dev),k)
> > y1<-t(y1)%*%y1
>
> This produces positive definite symmetrical matrices, which is what I want.
> However, the following:
>
> > k<-6; kk<-(k*(k-1))/2; st.dev<-.5
> > x<-matrix(0,5000,kk)
> > for(i in 1:5000){
> > y1<-matrix(rnorm(k^2,mean=0,sd=st.dev),k)
> > y1<-t(y1)%*%y1
> > y1<-y1/k #this keeps the variances similar across different k's
> > y1<-cov2cor(y1)
> > x[i,]<- y1[lower.tri(y1)]
> > }
> > hist(c(x))
>
> demonstrates that the distribution of corresponding correlations is not
> uniformly distributed between -1 and 1 (of course there is no reason to
> suspect that it would be). I have tried using both rcorrmatrix and
> genPositiveDefMat in the "clusterGeneration" package but even when picking
> the "unifcorrmat" option and setting "alphad" to 1, the distribution of
> correlations is not uniform.
>
> Any recommendations on how to generate the desired covariance matrices would
> be appreciated.
Hello.
Let me suggest the following procedure.
1. Generate a symmetric matrix A with the desired distribution of the
non-diagonal elements and with zeros on the diagonal.
2. Compute the smallest eigenvalue lambda_1 of A.
3. Replace A by A + t I, where I is the identity matrix and t is a
number such that t + lambda_1 > 0.
The resulting matrix will have the same non-diagonal elements as A,
but will be positive definite.
Petr Savicky.
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