Exact Multivariate Amplitude Distributions for Non-Stationary Gaussian or Algebraic Fluctuations of Covariances or Correlations

TL;DR

This paper develops a random matrix model to derive explicit multivariate amplitude distributions for non-stationary Gaussian or algebraic fluctuations in correlations, capturing heavy tails and non-stationarity.

Contribution

It introduces a novel ensemble-based approach to model non-stationary correlations and derives explicit multivariate amplitude distributions for various fluctuation types.

Findings

Derived closed-form multivariate distributions for non-stationary systems.

Provided a quantitative tool for measuring non-stationarity in correlations.

Extended previous models to include algebraic fluctuation cases.

Abstract

Complex systems are often non-stationary, typical indicators are continuously changing statistical properties of time series. In particular, the correlations between different time series fluctuate. Models that describe the multivariate amplitude distributions of such systems are of considerable interest. Extending previous work, we view a set of measured, non-stationary correlation matrices as an ensemble for which we set up a random matrix model. We use this ensemble to average the stationary multivariate amplitude distributions measured on short time scales and thus obtain for large time scales multivariate amplitude distributions which feature heavy tails. We explicitly work out four cases, combining Gaussian and algebraic distributions. The results are either of closed forms or single integrals. We thus provide, first, explicit multivariate distributions for such non-stationary systems and, second, a tool that quantitatively captures the degree of non-stationarity in the correlations.