Quickest Change Detection in the Presence of a Nuisance Change

Abstract

In the quickest change detection problem in which both nuisance and critical changes may occur, the objective is to detect the critical change as quickly as possible without raising an alarm when either there is no change or a nuisance change has occurred. A window-limited sequential change detection procedure based on the generalized likelihood ratio test statistic is proposed. A recursive update scheme for the proposed test statistic is developed and is shown to be asymptotically optimal under mild technical conditions. In the scenario where the post-change distribution belongs to a parametrized family, a generalized stopping time and a lower bound on its average run length are derived. The proposed stopping rule is compared with the finite moving average (FMA) stopping time and the naive 2-stage procedure that detects the nuisance or critical change using separate CuSum stopping procedures for the nuisance and critical changes. Simulations demonstrate that the proposed rule outperforms the FMA stopping time and the 2-stage procedure, and experiments on a real dataset on bearing failure verify the performance of the proposed stopping time. Index TermsQuickest change detection, nuisance change, Generalized Likelihood Ratio Test (GLRT), average run length, average detection delay under a nuisance change, and propose a window-limited stopping time that ignores the nuisance change but detects the critical change as quickly as possible. A. Related WorkExisting works in QCD that consider the problem where observations are not generated i.i.d. before and after the change-point can be categorized into three main categories. In the first category, the papers [31]-[33] consider the problem where the pre-change distribution and the post-change distribution are modeled as hidden Markov models (HMMs). In [31], the authors proved the asymptotic optimality of the CuSum procedure for the HMM signal model in the sense of Lorden. In [32], the authors developed the Shiryayev-Roberts-Pollak (SRP) rule for the HMM signal model and proved its optimality in the sense of Pollak. The authors of [33] consider the problem where the vector parameter of a two-state HMM changes at some unknown time. The second category of papers [34], [35] considers a QCD problem which relaxes the i.i.d. assumption. In [34], the authors established the optimality of CuSum and the Shiryayev-Roberts stopping rule in the class of random processes with likelihood ratios that satisfy certain independence and stationary conditions. The class of random processes includes Markov chains, AR processes, and processes evolving on a circle. In [35], the authors considered the Bayesian QCD problem where conditions on the asymptotic behavior of the likelihood process are assumed. Unlike all the aforementioned papers, the signal model in our QCD problem with nuisance change cannot be modeled by an HMM, and the likelihood ratios generated by our signal model are non-stationary. In the third category, the papers [36]-[41] consider QCD of transient changes, where the change i...

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