Generating random correlation matrices based on partial correlations

• Authors: Harry Joe
• Venue: Journal of Multivariate Analysis
• Year: 2006

A d-dimensional positive definite correlation matrix R = (ρij) can be parametrized in terms of the correlations ρi,i+1 for i = 1,..., d-1, and the partial correlations ρij|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 Fij, 1 ≤ i j ≤ d, for these (d 2) parameters. We obtain conditions on the Fij so that the joint density of (ρij) 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 ρi,i+1 for i = 1,..., d - 1, and ρij|i+1,..., j-1 for j-i ≥ 2.