Generating random correlation matrices based on partial correlations

Abstract

A d-dimensional positive definite correlation matrix R = ( ρ ij ) can be parametrized in terms of the correlations ρ i , i + 1 for i = 1 , … , d - 1 , and the partial correlations ρ ij | i + 1 , … j - 1 for j - i ⩾ 2 . These d 2 parameters can independently take values in the interval ( - 1 , 1 ) . Hence we can generate a random positive definite correlation matrix by choosing independent distributions F ij , 1 ⩽ i < j ⩽ d , for these d 2 parameters. We obtain conditions on the F ij so that the joint density of ( ρ ij ) is proportional to a power of det ( R ) and hence independent of the order of indices defining the sequence of partial correlations. As a special case, we have a simple construction for generating R that is uniform over the space of positive definite correlation matrices. As a byproduct, we determine the volume of the set of correlation matrices in d 2 -dimensional space. To prove our results, we obtain a simple remarkable identity which expresses det ( R ) as a function of ρ i , i + 1 for i = 1 , … , d - 1 , and ρ ij | i + 1 , … j - 1 for j - i ⩾ 2 .