Robustness of the ${N}$-CUSUM stopping rule in a Wiener disorder problem

TL;DR

This paper analyzes the robustness and asymptotic optimality of the N-CUSUM stopping rule for quickest detection of the earliest change-point among multiple correlated observation channels with different change strengths.

Contribution

It establishes the asymptotic optimality of the N-CUSUM rule under partial information and correlated noises in a multi-channel Wiener disorder detection problem.

Findings

N-CUSUM is asymptotically optimal as false alarm rate increases.

The approach applies under partial post-change drift information and general noise correlation.

Results have implications for decentralized and centralized detection systems in engineering.

Abstract

We study a Wiener disorder problem of detecting the minimum of $N$ change-points in $N$ observation channels coupled by correlated noises. It is assumed that the observations in each dimension can have different strengths and that the change-points may differ from channel to channel. The objective is the quickest detection of the minimum of the $N$ change-points. We adopt a min-max approach and consider an extended Lorden's criterion, which is minimized subject to a constraint on the mean time to the first false alarm. It is seen that, under partial information of the post-change drifts and a general nonsingular stochastic correlation structure in the noises, the minimum of $N$ cumulative sums (CUSUM) stopping rules is asymptotically optimal as the mean time to the first false alarm increases without bound. We further discuss applications of this result with emphasis on its implications to the efficiency of the decentralized versus the centralized systems of observations which arise in engineering.