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 (<b><i>R</i></b>) with a fixed mean absolute discrepancy (MAD) relative to a target (population) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mtext>pop</mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mtext>.</mml:mtext></mml:math> 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 <b><i>R</i></b> matrices with a known degree of model-approximation error (as operationalized by the MAD). Using results from higher-dimensional geometry, I show that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>×</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> 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 <i>n</i> = 3, these sets can be visualized as an elliptical tetrahedron and the surface of an octahedron. An online supplement includes <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>R</mml:mi></mml:mrow></mml:math> code for implementing the algorithm and for reproducing all of the results in the article.