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A spectral approach for online covariance change point detection

Zhigang Bao, Kha Man Cheong, Yuji Li, Jiaxin Qiu

TL;DR

This paper tackles online detection of changes in high-dimensional covariance structures by leveraging a spectrum-based approach anchored in random matrix theory. It constructs Fisher matrices from adjacent data blocks and studies the resulting linear spectral statistics (LSS), deriving explicit mean and variance for the one-step LSS difference under no-change, and proving a Brownian-motion limit for the cumulative LSS process. The authors then design an online CUSUM-type detector with a burn-in period and weight functions to control false alarms while achieving prompt detection when a change occurs. Through extensive simulations and an application to S&P 500 data, the method demonstrates high detection power, short detection delays, and robustness across distributions, with guidance recommending a log test function and the rho_{1,0} weight as a default configuration.

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.

A spectral approach for online covariance change point detection

TL;DR

This paper tackles online detection of changes in high-dimensional covariance structures by leveraging a spectrum-based approach anchored in random matrix theory. It constructs Fisher matrices from adjacent data blocks and studies the resulting linear spectral statistics (LSS), deriving explicit mean and variance for the one-step LSS difference under no-change, and proving a Brownian-motion limit for the cumulative LSS process. The authors then design an online CUSUM-type detector with a burn-in period and weight functions to control false alarms while achieving prompt detection when a change occurs. Through extensive simulations and an application to S&P 500 data, the method demonstrates high detection power, short detection delays, and robustness across distributions, with guidance recommending a log test function and the rho_{1,0} weight as a default configuration.

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.
Paper Structure (4 sections, 3 theorems, 24 equations, 1 figure, 1 table)

This paper contains 4 sections, 3 theorems, 24 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Main Results
  3. One-step LSS difference
  4. Online change point detection procedure

Key Result

Proposition 2.1

Suppose that Assumptions asm:data and asm:sample_dimension hold. Under $H_0$, for any analytic function $f$, the mean and variance functions of $L_k(f)$ are given by

Figures (1)

  • Figure 1: Sample paths (blue) of the standardized LSS difference process $\{\frac{1}{\sqrt{n}} \sum_{k=n+1}^{n+\lfloor nt\rfloor} \widetilde{L}_k(f), t\geq 0\}$ with test function $f(x)=x$, together with boundary curves. The simulation setting is $p=100$, $n=300$ ($k_1=k_2^{\circ}=150$). The red dashed lines represent the boundary $\pm \rho_2^{-1}(t)$, while the green, orange, and purple lines depict the boundaries $\pm c_{\alpha} \rho_{1, \gamma}^{-1}(t)$ for $\gamma = 0.0, 0.25, 0.45$ at significance level $\alpha=0.05$. Reject $H_0$ whenever the sample path crosses the selected boundaries.

Theorems & Definitions (4)

  • Proposition 2.1
  • Theorem 2.2: Weak Convergence of $W_{n,t}$ under $H_0$
  • Theorem 2.3: Limiting null distribution
  • Remark 2.4