Eigenvalue Density of the Doubly Correlated Wishart Model: Exact Results

TL;DR

This paper derives exact eigenvalue density formulas for the doubly correlated Wishart model, accounting for correlations over time and space, using supersymmetry techniques, and provides results for both complex and real cases.

Contribution

It introduces new exact formulas for eigenvalue densities of doubly correlated Wishart matrices, including a closed form for complex cases and an integral expression for real cases.

Findings

New closed form for complex eigenvalue density

Integral expression for real eigenvalue density

Asymptotic density in large matrix limit

Abstract

Data sets collected at different times and different observing points can possess correlations at different times $and$ at different positions. The doubly correlated Wishart model takes both into account. We calculate the eigenvalue density of the Wishart correlation matrices using supersymmetry. In the complex case we obtain a new closed form expression which we compare to previous results in the literature. In the more relevant and much more complicated real case we derive an expression for the density in terms of a fourfold integral. Finally, we calculate the density in the limit of large correlation matrices.