A spectral approach for online covariance change point detection

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.

💡 Research Summary

This paper addresses the problem of detecting changes in the covariance structure of high‑dimensional data streams in an online (sequential) setting. While a large body of literature exists for offline change‑point detection, relatively few methods can monitor covariance matrices in real time when the dimension p is comparable to the sample size. The authors propose a novel procedure that leverages linear spectral statistics (LSS) of the Fisher matrix and a CUSUM‑type statistic built from the one‑step differences of these LSS.

Model and notation.
Observations are p‑dimensional vectors (y_i = \Sigma_i^{1/2} x_i), where (\Sigma_i) is a deterministic positive‑definite covariance matrix that may change at an unknown time (k^\star). The innovations (x_i) are i.i.d. with zero mean, unit variance, and finite fourth moment. Under the null hypothesis (H_0) all (\Sigma_i = I_p). At a given time k the data are split into a reference window of size (k_1) and a monitoring window of size (k-k_1). Sample covariance matrices for the two windows are \