Multisource Bayesian sequential change detection

arXiv:0708.0224v3 [math.ST] 1 Apr 2008 The Annals of Applied Probability 2008, Vol. 18, No. 2, 552–590 DOI: 10.1214/07-AAP463 c ⃝Institute of Mathematical Statistics, 2008 MULTISOURCE BAYESIAN SEQUENTIAL CHANGE DETECTION By Savas Dayanik,1 H. Vincent Poor2 and Semih O. Sezer2 Princeton University, Princeton University and University of Michigan Suppose that local characteristics of several independent com- pound Poisson and Wiener processes change suddenly and simulta- neously at some unobservable disorder time. The problem is to detect the disorder time as quickly as possible after it happens and minimize the rate of false alarms at the same time. These problems arise, for example, from managing product quality in manufacturing systems and preventing the spread of infectious diseases. The promptness and accuracy of detection rules improve greatly if multiple independent information sources are available. Earlier work on sequential change detection in continuous time does not provide optimal rules for situa- tions in which several marked count data and continuously changing signals are simultaneously observable. In this paper, optimal Bayesian sequential detection rules are developed for such problems when the marked count data is in the form of independent compound Pois- son processes, and the continuously changing signals form a multi- dimensional Wiener process. An auxiliary optimal stopping problem for a jump-diffusion process is solved by transforming it first into a sequence of optimal stopping problems for a pure diffusion by means of a jump operator. This method is new and can be very useful in other applications as well, because it allows the use of the powerful optimal stopping theory for diffusions. 1. Introduction. Suppose that at some unobservable disorder time Θ, the local characteristics of several independent compound Poisson and Wiener processes undergo a sudden and simultaneous change. More precisely, the Received November 2006; revised July 2007. 1Supported in part by the Air Force Office of Scientific Research, Grant AFOSR- FA9550-06-1-0496, and by the U.S. Department of Homeland Security through the Center for Dynamic Data Analysis for Homeland Security administered through ONR Grant N00014-07-1-0150 to Rutgers University. 2Supported by the U.S. Army Pantheon Program. AMS 2000 subject classifications. Primary 62L10; secondary 62L15, 62C10, 60G40. Key words and phrases. Sequential change detection, jump-diffusion processes, optimal stopping. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Applied Probability, 2008, Vol. 18, No. 2, 552–590. This reprint differs from the original in pagination and typographic detail. 1 2 S. DAYANIK, H. V. POOR AND S. O. SEZER pairs (λ(i) 0 ,ν(i) 0 ), 1 ≤i ≤m, consisting of the arrival rate and mark distri- bution of m compound Poisson processes (T (i) n ,Z(i) n )n≥1, 1 ≤i ≤m, become (λ(i) 1 ,ν(i) 1 ), 1 ≤i ≤m, and d Wiener processes W (j) t , 1 ≤j ≤d gain drifts µ(j), 1 ≤j ≤d at time Θ. We assume that Θ is a random variable with the zero-modified exponential distribution P{Θ = 0} = π and P{Θ > t} = (1 −π)e−λt, t ≥0, (1.1) and (λ(i) 0 ,ν(i) 0 )1≤i≤m, (λ(i) 1 ,ν(i) 1 )1≤i≤m, (µ(j))1≤j≤d, π, and λ are known. The objective is to detect the disorder time Θ as soon as possible after disorder happens by using the observations of (T (i) n ,Z(i) n )n≥1, 1 ≤i ≤m, and X(j) t = X(j) 0 + µ(j)(t −Θ)+ + W (j) t , t ≥0,1 ≤j ≤d. More precisely, if F = {Ft}t≥0 denotes the observation filtration, then we would like to find, if it exists, an F-stopping time τ whose Bayes risk Rτ(π) ≜P{τ < Θ} + cE(τ −Θ)+, 0 ≤π < 1 (1.2) is the smallest for any given constant cost parameter c > 0 and calculate its Bayes risk. If such a stopping time exists, then it provides the best trade-off between false alarm frequency P{τ < Θ} and expected detection delay cost cE(τ −Θ)+. Important applications of this problem are the quickest detection of man- ufacturing defects during product quality assurance, online fault detection and identification for condition-based equipment maintenance, prompt de- tection of shifts in the riskiness of various financial instruments, early detec- tion of the onset of an epidemic to protect public health, quickest detection of a threat to homeland security, and online detection of unauthorized access to privileged resources in the fight against fraud. In many of those applica- tions, a range of data, changing over time either continuously or by jumps or both, are collected from multiple sources/sensors in order to detect a sudden unobserved change as quickly as possible after it happens, and the problems can be modeled as the quickest detection of a change in the local character- istics of several Wiener and compound Poisson processes. For example, in condition-based maintenance, an equipment is monitored continuously by a web of sensors for both continuously-changing data (such as oil level, tem- perature,

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