Asymptotically Optimal Search for a Change Point Anomaly under a Composite Hypothesis Model

TL;DR

This paper develops an asymptotically optimal sequential search algorithm for detecting a change point in a set of processes with unknown parameters, balancing detection accuracy and sample complexity.

Contribution

It introduces a deterministic search strategy that is proven to be asymptotically optimal under composite hypothesis models with unknown distributions.

Findings

Algorithm is asymptotically optimal when distributions are unknown.

Improved detection time when null hypothesis parameters are known.

Simulation results validate theoretical asymptotic optimality.

Abstract

We address the problem of searching for a change point in an anomalous process among a finite set of M processes. Specifically, we address a composite hypothesis model in which each process generates measurements following a common distribution with an unknown parameter (vector). This parameter belongs to either a normal or abnormal space depending on the current state of the process. Before the change point, all processes, including the anomalous one, are in a normal state; after the change point, the anomalous process transitions to an abnormal state. Our goal is to design a sequential search strategy that minimizes the Bayes risk by balancing sample complexity and detection accuracy. We propose a deterministic search algorithm with the following notable properties. First, we analytically demonstrate that when the distributions of both normal and abnormal processes are unknown, the algorithm is asymptotically optimal in minimizing the Bayes risk as the error probability approaches zero. In the second setting, where the parameter under the null hypothesis is known, the algorithm achieves asymptotic optimality with improved detection time based on the true normal state. Simulation results are presented to validate the theoretical findings.