Quickest Change Detection of a Markov Process Across a Sensor Array

TL;DR

This paper develops a Bayesian quickest change detection method for sensor arrays where the change propagates as a Markov process, providing an optimal stopping rule and demonstrating asymptotic optimality with improved performance over naive methods.

Contribution

It introduces a Bayesian framework for detecting propagating changes modeled as a Markov process across sensors, with a threshold-based optimal stopping rule and asymptotic optimality analysis.

Findings

Optimal stopping rule reduces to thresholding the a posteriori probability in rare change scenarios.

The proposed test outperforms naive single-sensor and instant propagation assumptions.

Asymptotic optimality established under certain K-L divergence conditions.

Abstract

Recent attention in quickest change detection in the multi-sensor 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, with a fusion center that has perfect information about the observations and a priori knowledge of the statistics of the change process, is considered. The problem of minimizing the average detection delay subject to false alarm constraints is formulated as a partially observable Markov decision process (POMDP). Insights into the structure of the optimal stopping rule are presented. In the limiting case of rare disruptions, we show that the structure of the optimal test reduces to thresholding the a posteriori probability of the hypothesis that no change has happened. We establish the asymptotic optimality (in the vanishing false alarm probability regime) of this threshold test under a certain condition on the Kullback-Leibler (K-L) divergence between the post- and the pre-change densities. In the special case of near-instantaneous change propagation across the sensors, this condition reduces to the mild condition that the K-L divergence be positive. Numerical studies show that this low complexity threshold test results in a substantial improvement in performance over naive tests such as a single-sensor test or a test that wrongly assumes that the change propagates instantaneously.