Emerging spectra of singular correlation matrices under small power-map deformations

TL;DR

This paper investigates how small power-map deformations affect the spectra of highly singular correlation matrices, revealing their sensitivity to underlying correlations in complex systems like financial markets.

Contribution

It introduces an analysis of the emerging spectra of singular correlation matrices under small non-linear power-map deformations, highlighting their sensitivity to correlations.

Findings

Power-map deformations break eigenvalue degeneracy.

Emerging spectra are sensitive to correlations.

Analysis applies to Wishart ensembles.

Abstract

Correlation matrices are a standard tool in the analysis of the time evolution of complex systems in general and financial markets in particular. Yet most analysis assume stationarity of the underlying time series. This tends to be an assumption of varying and often dubious validity. The validity of the assumption improves as shorter time series are used. If many time series are used this implies an analysis of highly singular correlation matrices. We attack this problem by using the so called {\it power map} which was introduced to reduce noise. Its non-linearity breaks the degeneracy of the zero eigenvalues and we analyze the sensitivity of the so emerging spectra to correlations. This sensitivity will be demonstrated for uncorrelated and correlated Wishart ensembles.