[Paper Review] Multivariate analysis of short time series in terms of ensembles of correlation matrices

Manan Vyas, Thomas Guhr+1 more|arXiv (Cornell University)|Jan 23, 2018

Complex Systems and Time Series Analysis43 references7 citations

TL;DR

This paper introduces a novel ensemble technique to analyze multivariate short time series by randomly sampling non-singular subsets of time series from a large but singular data matrix, enabling robust statistical analysis of eigenvalues in correlation matrices. The method overcomes poor statistics in singular correlation matrices by generating an ensemble of non-singular correlation matrices, successfully capturing bulk eigenvalue distributions and outliers, with strong agreement between numerical simulations and analytic results for correlated Wishart ensembles.

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.

Motivation & Objective

• To address the poor statistical reliability of correlation matrices derived from short time series where N >> T, leading to singular matrices with only T−1 non-zero eigenvalues.
• To develop a method that enables robust statistical analysis of eigenvalue distributions—especially outliers and bulk spectra—without assuming stationarity.
• To test the method on model systems with block-diagonal and constant correlations, comparing numerical results with analytic predictions from correlated Wishart ensembles.
• To investigate the impact of repeated time series usage in ensemble construction on eigenvalue outlier distributions.
• To demonstrate the method's utility for detecting early warning signals in non-stationary or quasi-stationary systems, such as near-critical transitions.

Proposed method

• Construct a large N × T data matrix with N time series of length T, where N >> T, leading to a singular correlation matrix with only T−1 non-zero eigenvalues.
• Form an ensemble of m × T data matrices (m << N) by randomly selecting m non-repeating time series from the original N, ensuring no repeated time series within a matrix and no identical matrices in the ensemble.
• Compute correlation matrices for each selected subset, resulting in a set of non-singular correlation matrices that form the Non-Singular Random Selection Ensemble (NSRSE).
• Analyze the ensemble’s eigenvalue density and outlier distributions (largest and second-largest eigenvalues), comparing with the Empirical Random Selection Ensemble (ERSE) where time series are sampled with replacement.
• Use saddle point approximations and microcanonical/canonical normalization to model the distribution of eigenvalues, particularly in the tails and for outliers.
• Validate results against analytic solutions for correlated Wishart ensembles under constant and block-diagonal correlation structures.

Experimental results

Research questions

• RQ1 How can eigenvalue statistics be reliably extracted from a singular correlation matrix formed by N >> T short time series?
• RQ2 What is the effect of sampling without replacement (NSRSE) versus with replacement (ERSE) on the distribution of eigenvalue outliers?
• RQ3 To what extent do the ensemble’s eigenvalue distributions match analytic predictions for correlated Wishart ensembles with block-diagonal or constant correlations?
• RQ4 How does the repetition of time series in ensemble construction distort the statistical properties of eigenvalue outliers?
• RQ5 Can this ensemble method detect early signs of instability or critical transitions in non-stationary systems, even when individual correlation matrices are singular?

Key findings

• The NSRSE method successfully generates a large, non-singular ensemble of correlation matrices from a singular data matrix, enabling reliable statistical analysis of eigenvalue distributions.
• The ensemble’s bulk eigenvalue density closely matches analytic predictions from correlated Wishart ensembles for both constant and block-diagonal correlation structures.
• Outlier eigenvalues (largest and second-largest) in the NSRSE are well-approximated by Gaussian distributions, with moments differing significantly from those in the ERSE due to repeated time series usage.
• The repetition of time series in ERSE leads to stronger correlations among ensemble members, distorting the outlier distribution and increasing variance compared to NSRSE.
• The method effectively captures spectral features such as power-law behavior in eigenvalue spectra, even when using only a fraction of the full time series, as demonstrated in Ising model simulations.
• The ensemble approach enables the analysis of higher-order correlation functions (e.g., two- and three-point functions), which are otherwise inaccessible from a single singular correlation matrix.