A spectral approach for online covariance change point detection

TL;DR

This paper introduces a spectral CUSUM-type method for online detection of covariance change points in high-dimensional data, leveraging random matrix theory to promptly identify structural changes with high sensitivity.

Contribution

It develops a novel spectral approach using linear spectral statistics and martingale convergence to detect change points online, addressing a gap in high-dimensional sequential data analysis.

Findings

Method detects change points shortly after they occur.

Outperforms existing approaches in sensitivity.

Effective in high-dimensional settings.

Abstract

Change point detection in covariance structures is a fundamental and crucial problem for sequential data. Under the high-dimensional setting, most of the existing research has focused on identifying change points in historical data. However, there is a significant lack of studies on the practically relevant online change point problem, which means promptly detecting change points as they occur. In this paper, applying the limiting theory of linear spectral statistics for random matrices, we propose a class of spectrum based CUSUM-type statistic. We first construct a martingale from the difference of linear spectral statistics of sequential sample Fisher matrices, which converges to a Brownian motion. Our CUSUM-type statistic is then defined as the maximum of a variant of this process. Finally, we develop our detection procedure based on the invariance principle. Simulation results show that our detection method is highly sensitive to the occurrence of change point and is able to identify it shortly after they arise, outperforming the existing approaches.