A New Parametrization of Correlation Matrices
Ilya Archakov, Peter Reinhard Hansen
TL;DR
This paper presents a novel parametrization of correlation matrices that simplifies modeling by ensuring positive definiteness through an unconstrained vector, extending Fisher's Z-transformation to higher dimensions.
Contribution
It introduces a new parametrization method for correlation matrices that guarantees positive definiteness and generalizes Fisher's Z-transformation to higher dimensions.
Findings
Provides an algorithm for reconstructing correlation matrices from vectors
Ensures positive definiteness inherently in the parametrization
Analyzes the numerical complexity of the reconstruction algorithm
Abstract
We introduce a novel parametrization of the correlation matrix. The reparametrization facilitates modeling of correlation and covariance matrices by an unrestricted vector, where positive definiteness is an innate property. This parametrization can be viewed as a generalization of Fisther's Z-transformation to higher dimensions and has a wide range of potential applications. An algorithm for reconstructing the unique n x n correlation matrix from any d-dimensional vector (with d = n(n-1)/2) is provided, and we derive its numerical complexity.
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