Optimum Multi-Stream Sequential Change-Point Detection With Sampling Control

Abstract

In multi-stream sequential change-point detection it is assumed that there are <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> processes in a system and at some unknown time, an occurring event changes the distribution of the samples of a particular process. In this article, we consider this problem under a sampling control constraint when one is allowed, at each point in time, to sample a single process. The objective is to raise an alarm as quickly as possible subject to a proper false alarm constraint. We show that under sampling control, a simple myopic-sampling-based sequential change-point detection strategy is second-order asymptotically optimal when the number <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> of processes is fixed. This means that the proposed detector, even by sampling with a rate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1/M$ </tex-math></inline-formula> of the full rate, enjoys the same detection delay, up to some additive finite constant, as the optimal procedure. Simulation experiments corroborate our theoretical results.