Monitoring for a Phase Transition in a Time Series of Wigner Matrices

TL;DR

This paper presents a real-time detection method for phase transitions in high-dimensional time-series data of Wigner matrices, combining random matrix theory and Gaussian approximations for effective monitoring.

Contribution

It introduces a self-normalized eigenvalue-based detector for phase transitions in high-dimensional matrix time series, with theoretical guarantees and practical applications.

Findings

Detector accurately identifies phase transitions in simulations

Method performs well across different matrix dimensions

Applications demonstrate usefulness in pollution and social data

Abstract

We develop methodology and theory for the detection of a phase transition in a time-series of high-dimensional random matrices. In the model we study, at each time point \( t = 1,2,\ldots \), we observe a deformed Wigner matrix \( \mathbf{M}_t \), where the unobservable deformation represents a latent signal. This signal is detectable only in the supercritical regime, and our objective is to detect the transition to this regime in real time, as new matrix--valued observations arrive. Our approach is based on a partial sum process of extremal eigenvalues of $\mathbf{M}_t$, and its theoretical analysis combines state-of-the-art tools from random-matrix-theory and Gaussian approximations. The resulting detector is self-normalized, which ensures appropriate scaling for convergence and a pivotal limit, without any additional parameter estimation. Simulations show excellent performance for varying dimensions. Applications to pollution monitoring and social interactions in primates illustrate the usefulness of our approach.