Optimal change point detection in Gaussian processes
This work addresses change point detection for time series and spatial data analysis, offering a novel method that efficiently exploits covariance structures, though it is incremental as it builds on existing GLRT approaches.
The authors tackled the problem of detecting a change in the mean of one-dimensional Gaussian process data, proposing a generalized likelihood ratio test (GLRT) method that achieves nearly asymptotically optimal rates in both increasing and fixed domain settings, with concrete results showing it outperforms standard methods like CUSUM in certain scenarios.
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.