Optimal change point detection in Gaussian processes

We study the problem of detecting a change in the mean of one-dimensional Gaussian process data in the fixed domain regime. We propose a detection procedure based on the generalized likelihood ratio test (GLRT), and show that our method achieves asymptotically near-optimal rate in a minimax sense. The notable 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 suboptimal. Our algorithms and asymptotic analysis are applicable to a number 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.