Generating random correlation matrices based on partial correlations

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 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