  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.
