Generating random correlation matrices based on partial correlation vines and the onion method

Partial correlation vines and the onion method are presented for generating random correlation matrices. As a special case, a uniform distribution over the set of d× d positive definite correlation matrices obtains. Byproducts are: (a) For a uniform distribution over the space of d × d correlation matrices, the marginal distribution of each correlation is Beta(d/2, d/2) on (−1, 1). (b) An identity is obtained for the determinant of a correlation matrix R via partial correlations in a vine. (c) A formula is obtained for the volume of the set of d× d positive definite correlation matrices in d2 ) -dimensional space. Outline 1. Statement of key results on generating random correlation matrices 2. Parametrization with partial correlations 3. Regular vines, D-vines, C-vines 4. Random correlation matrices based on partial correlations 5. Random correlation matrices based on onion method Key results: R is a d× d correlation matrix 1. Several simple ways to generate a random R that is uniform over set of positive definite d× d correlation matrices; more generally with density ∝ [det(R)]α−1. 2. Identity: (1− ρ12)(1− ρ23)(1− ρ13;2) for d = 3