Quickest Change Detection of a Markov Process Across a Sensor Array

Abstract

Recent attention in quickest change detection in the multisensor setting has been on the case where the densities of the observations change at the same instant at all the sensors due to the disruption. In this work, a more general scenario is considered where the change propagates across the sensors, and its propagation can be modeled as a Markov process. A centralized, Bayesian version of this problem is considered, with a fusion center that has perfect information about the observations and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> knowledge of the statistics of the change process. The problem of minimizing the average detection delay subject to false alarm constraints is formulated in a dynamic programming framework. Insights into the structure of the optimal stopping rule are presented. In the limiting case of rare disruptions, it is shown that the structure of the optimal test reduces to thresholding the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a posteriori</i> probability of the hypothesis that no change has happened. Under a certain condition on the Kullback-Leibler (K-L) divergence between the post- and the pre-change densities, it is established that the threshold test is asymptotically optimal (in the vanishing false alarm probability regime). It is shown via numerical studies that this <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">low-complexity</i> threshold test results in a substantial improvement in performance over <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">naive</i> tests such as a single-sensor test or a test that incorrectly assumes that the change propagates instantaneously.