Conditional, Partial and Rank Correlation for the Elliptical Copula; Dependence Modelling in Uncertainty Analysis

The copula-vine method of specifying dependence in high dimensional distributions has been developed in Cooke [1], Bedford and Cooke [6], Kurowicka and Cooke ([2], [4]), and Kurowicka et all [3]. According to this method, a high dimensional distribution is constructed from two dimensional and conditional two dimensional distributions of uniform variaties. When the (conditional) two dimensional distributions are specified via (conditional) rank correlations, the distribution can be sampled on the fly. When instead we use partial correlations, the specifications are algebraically independent and uniquely determine the (rank) correlation matrix. We prove that for the elliptical copulae ([3]), the conditional and partial correlations are equal. This enables on-the-fly simulation of a full correlation structure; something which here to fore was not possible.