How to Get MAD: Generating Uniformly Sampled Correlation Matrices with a Fixed Mean Absolute Discrepancy

This article describes a simple and fast algorithm for generating uniformly sampled correlation matrices (R) with a fixed mean absolute discrepancy (MAD) relative to a target (population) $R_{\text{pop}}\text{.}$ The algorithm can be profitably used in many settings including model robustness studies and stress testing of investment portfolios, or in dynamic model-fit analyses to generate R matrices with a known degree of model-approximation error (as operationalized by the MAD). Using results from higher-dimensional geometry, I show that $R_{n\times n}$ matrices with a fixed MAD lie in the intersection of two sets that represent: (a) an elliptope and (b) the surface of a cross-polytope. When n = 3, these sets can be visualized as an elliptical tetrahedron and the surface of an octahedron. An online supplement includes $R$ code for implementing the algorithm and for reproducing all of the results in the article.