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.

该研究探讨了一维高斯过程数据的均值变化检测问题，并提出了基于广义似然比检验的检测方法，该方法在固定域和增长域的两个设置中都几乎渐近地达到最优水平，主要特点是充分利用了高斯过程协方差结构所捕捉到的数据依赖性，即使协方差未知，我们也提出了插件 GLRT 方法，并推导出了该方法仍然渐近近似最优的条件。