Two-sided change detection under unknown initial state

Abstract

The problem of detecting a change in distribution of a sequence of independent and identically distributed (IID) random variables is addressed. Unlike previous approaches to sequential change detection, which assume a known initial probability density function (PDF) for the sequence, in this paper we address the case where the initial distribution of the sequence is unknown. An optimal stopping approach based on Bayesian hypothesis testing with exponential delay cost is proposed. The tradeoffs among average detection delay, probability of false alarm and probability of detecting a change in the incorrect direction are investigated. It is shown that the proposed test's probability of change detection in the incorrect direction can be made arbitrarily small without significantly increasing average detection delay for change times larger than a minimum value determined by the hypothesis testing problem itself. The proposed test also has a recursive algorithm to track the minimum risk hypotheses with fixed complexity per sample. Simulation results confirm the derived properties and reveal that the average delay, after an initial transient period, approaches that of the CUSUM test, which is delay-optimal if the initial state were known.