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

Area of Science:

• Statistics
• Computational Mathematics
• Financial Modeling

Background:

• Generating correlation matrices with specific properties is crucial for various statistical and financial applications.
• Existing methods may lack efficiency or control over the degree of deviation from a target matrix.
• Understanding model-approximation error is key in robust analysis and stress testing.

Purpose of the Study:

• To present a simple and fast algorithm for generating uniformly sampled correlation matrices (R) with a fixed mean absolute discrepancy (MAD) relative to a target population matrix (R_pop).
• To demonstrate the utility of this algorithm in applications such as model robustness studies, investment portfolio stress testing, and dynamic model-fit analyses.
• To provide a theoretical underpinning using higher-dimensional geometry to characterize the properties of matrices generated by the algorithm.

Main Methods:

• Development of a novel algorithm for sampling correlation matrices with a controlled MAD.
• Application of higher-dimensional geometry to define the mathematical space where matrices with a fixed MAD reside.
• Implementation of the algorithm using R code, with accompanying materials for reproducibility.

Main Results:

• The algorithm efficiently generates correlation matrices with a specified MAD.
• Matrices with a fixed MAD are shown to lie in the intersection of an elliptope and the surface of a cross-polytope.
• For n=3, these geometric sets can be visualized as an elliptical tetrahedron and an octahedron surface.

Conclusions:

• The developed algorithm provides a practical tool for generating correlation matrices with precise control over approximation error.
• The theoretical framework offers insights into the geometric structure of correlation matrices with bounded discrepancies.
• The algorithm and its implementation are valuable for researchers and practitioners in statistics, finance, and computational modeling.