Optimal change point detection in Gaussian processes

TL;DR

This paper develops a nearly optimal change point detection method for Gaussian processes that leverages covariance structure, outperforming traditional methods like CUSUM especially in fixed domain settings.

Contribution

It introduces a generalized likelihood ratio test for Gaussian processes that is nearly asymptotically optimal across different domain settings and covariance structures.

Findings

Proposed GLRT achieves near minimax optimality in both increasing and fixed domain settings.

Plug-in GLRT remains near optimal even with unknown covariance, under certain conditions.

Standard CUSUM is only optimal in increasing domain scenarios.

Abstract

We study the problem of detecting a change in the mean of one-dimensional Gaussian process data. This problem is investigated in the setting of increasing domain (customarily employed in time series analysis) and in the setting of fixed domain (typically arising in spatial data analysis). We propose a detection method based on the generalized likelihood ratio test (GLRT), and show that our method achieves nearly asymptotically optimal rate in the minimax sense, in both settings. The salient feature of the proposed method is that it exploits in an efficient way the data dependence captured by the Gaussian process covariance structure. When the covariance is not known, we propose the plug-in GLRT method and derive conditions under which the method remains asymptotically near optimal. By contrast, the standard CUSUM method, which does not account for the covariance structure, is shown to be asymptotically optimal only in the increasing domain. Our algorithms and accompanying theory are applicable to a wide variety of covariance structures, including the Matern class, the powered exponential class, and others. The plug-in GLRT method is shown to perform well for maximum likelihood estimators with a dense covariance matrix.