@article { ISI:000241970400006,
title = {Generating random correlation matrices based on partial correlations},
journal = {Journal of Multivariate Analysis},
volume = {97},
number = {10},
year = {2006},
month = {NOV},
pages = {2177-2189},
publisher = {Elsevier Inc},
type = {article},
abstract = {A d-dimensional positive definite correlation matrix R = (rho(ij)) can be parametrized in terms of the correlations rho(i,i+1) for i = 1,..., d - 1, and the partial correlations rho(ij\textbackslashi+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 (rho(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 rho(i,i+1) for i = 1,..., d - 1, and p(ij\textbackslashi+1,... j-1) for j - i >= 2. (C) 2005 Elsevier Inc. All rights reserved.},
keywords = {Beta distribution, determinant of correlation matrix},
issn = {0047-259X},
doi = {10.1016/j.jmva.2005.05.010},
author = {Joe, Harry}
}