Signal and Noise in Correlation Matrix

TL;DR

This paper derives an exact relation between the eigenvalue spectra of a covariance matrix and its estimator using random matrix theory, enabling practical computation of eigenvalue invariants in high-dimensional systems.

Contribution

It provides a novel exact relation connecting the eigenvalue spectra of the true covariance matrix and its estimator, applicable across multiple scientific fields.

Findings

Derived an exact spectral relation using random matrix theory

Enables computation of eigenvalue invariants from empirical data

Applicable to diverse high-dimensional systems

Abstract

Using random matrix technique we determine an exact relation between the eigenvalue spectrum of the covariance matrix and of its estimator. This relation can be used in practice to compute eigenvalue invariants of the covariance (correlation) matrix. Results can be applied in various problems where one experimentally estimates correlations in a system with many degrees of freedom, like in statistical physics, lattice measurements of field theory, genetics, quantitative finance and other applications of multivariate statistics.