Sequential detection of three-dimensional signals under dependent noise

We study detection methods for multivariable signals under dependent noise. The main focus is on three-dimensional signals; that is, on signals in the space-time domain. Examples for such signals are multifaceted. They include geographic and climatic data as well as image data that are observed over a fixed time horizon. We assume that the signal is observed as a finite block of noisy samples whereby we are interested in detecting changes from a given reference signal. Our detector statistic is based on a sequential partial sum process, related to classical signal decomposition and reconstruction approaches applied to the sampled signal. We show that this detector process converges weakly under the no change null hypothesis that the signal coincides with the reference signal, provided that the spatial-temporal partial sum process associated with the random field of the noise terms disturbing the sampled signal converges to a Brownian motion. More generally, we also establish the limiting distribution under a wide class of local alternatives that allows for smooth as well as discontinuous changes. Our results also cover extensions to the case in which the reference signal is unknown. We conclude with an extensive simulation study of the detection algorithm.