Multivariate analysis of short time series in terms of ensembles of correlation matrices

TL;DR

This paper introduces an ensemble method for analyzing short, non-stationary time series by generating multiple correlation matrices, enabling better statistical analysis of eigenvalues without assuming stationarity.

Contribution

The novel ensemble technique allows for effective analysis of short, singular correlation matrices by random subset selection, improving statistical reliability in multivariate time series analysis.

Findings

Ensemble approach yields stable eigenvalue statistics.

Comparison with Wishart models validates the method.

Technique extends to quasi-stationary epochs in non-stationary data.

Abstract

When dealing with non-stationary systems, for which many time series are available, it is common to divide time in epochs, i.e. smaller time intervals and deal with short time series in the hope to have some form of approximate stationarity on that time scale. We can then study time evolution by looking at properties as a function of the epochs. This leads to singular correlation matrices and thus poor statistics. In the present paper, we propose an ensemble technique to deal with a large set of short time series without any consideration of non-stationarity. Given a singular data matrix, we randomly select subsets of time series and thus create an ensemble of non-singular correlation matrices. As the selection possibilities are binomially large, we will obtain good statistics for eigenvalues of correlation matrices, which are typically not independent. Once we defined the ensemble, we analyze its behavior for constant and block-diagonal correlations and compare numerics with analytic results for the corresponding correlated Wishart ensembles. We discuss differences resulting from spurious correlations due to repetitive use of time-series. The usefulness of this technique should extend beyond the stationary case if, on the time scale of the epochs, we have quasi-stationarity at least for most epochs.