Detecting changes in real-time data: a user's guide to optimal detection.

The real-time detection of changes in a noisily observed signal is an important problem in applied science and engineering. The study of parametric optimal detection theory began in the 1930s, motivated by applications in production and defence. Today this theory, which aims to minimize a given measure of detection delay under accuracy constraints, finds applications in domains including radar, sonar, seismic activity, global positioning, psychological testing, quality control, communications and power systems engineering. This paper reviews developments in optimal detection theory and sequential analysis, including sequential hypothesis testing and change-point detection, in both Bayesian and classical (non-Bayesian) settings. For clarity of exposition, we work in discrete time and provide a brief discussion of the continuous time setting, including recent developments using stochastic calculus. Different measures of detection delay are presented, together with the corresponding optimal solutions. We emphasize the important role of the signal-to-noise ratio and discuss both the underlying assumptions and some typical applications for each formulation.This article is part of the themed issue 'Energy management: flexibility, risk and optimization'.