how to generate pseudo-random positive definite matrix with constraints on the off-diagonal elements?

Question

The user wants to impose a unique, non-trivial, upper/lower bound on the correlation between every pair of variable in a var/covar matrix.

For example: I want a vari

Answer 1

You can create a set of N random vectors of size M and unit variance. And add to them a random vector (size N and unit variance) multiplied by a certain number k. Then you take the correlation between all those vectors, that will be a positive definite matrix. If M is very big then there will be no variance in the correlation distribution and the correlation will be: k^2/(1+k^2). The smaller M gets the wider the distribution of the off diagonal elements. Alternatively, you can let M be very large and multiply the "common vector" by a different k each. You might get tighter control if you play with these parameters properly. Here goes some Matlab code to do that:

clear all;
vecLarg=10;
theDim=1000;
corrDist=0*randn(theDim,1);
Baux=randn(vecLarg,theDim)+  (corrDist*randn(1,vecLarg))'+(k*ones(theDim,1)*randn(1,vecLarg))'  ;
A=corrcoef(Baux);
hist(A(:),100);