Precision of sequential change point detection

IEEE Transactions on Information Theory, 1994

Abstruct-This paper proposes a new procedure for quickest detection of an abrupt change in a random sequence, where a change is known to occur with probability one. Applications include on-line speech segmentation, edge detection in image processing, and communications channel monitoring. In contrast to Shiryayev's problem formulation, prior knowledge of a change-time distribution is not required. The optimality criterion considered is similar to Shiryayev's, except that the expected delay is to be minimized subject to both overall false alarm probability and false alarm average run length constraints. Under this criterion, theoretical study shows that the new procedure approximates an optimal but unrealizable Bayesian procedure, particularly for small signal changes or for low probability of false alarm. Simulations confirm that under such conditions, the new procedure compares favorably with Page's CUSUM, an optimized moving-window fixed-sample-size (FSS) procedure, and a special case of the Girshick-Rubin-Shiryayev (GRS) procedure.

Statistical Methodology, 2006

The discussion points on the inadequacy of the ARL as an index of efficiency of a detection procedure for change points. It is shown that the FAR = 1/ARL might be small, while the probability of false alarm (PFA) is at the same time considerable. This is illustrated with simulation runs, using the Shiryayev-Roberts detection procedure. The need to develop procedures based on continuous monitoring is also mentioned.

arXiv (Cornell University), 2008

We register a random sequence constructed based on Markov processes by switching between them. At unobservable random moment a change in distribution of observed sequence takes place. Using probability maximizing approach the optimal stopping rule for detecting the disorder is identified. Some explicit solution for example is also obtained. The result is generalization of Bojdecki's model where before and after the change independent processes are observed. Keywords.

2015

This paper deals with the problem of detecting transient changes in stochastic-dynamical systems. A statistical observation model which depends on unknown system states (often regarded as the nuisance parameter) is developed. The negative impact of nuisance parameter is then eliminated from the observation model by utilizing the invariant statistics. The Variable Threshold Window Limited CUmulative SUM (VTWL CUSUM) test, previously developed for independent observations, is adapted to the novel observation model. Taking into account the transient change detection criterion, minimizing the worst-case probability of missed detection subject to an acceptable level of the worst-case probability of false alarm within a given time period, the thresholds of the VTWL CUSUM test are optimized. It is shown that the optimized VTWL CUSUM algorithm is equivalent to the Finite Moving Average (FMA) detection rule. A numerical method for estimating the probability of false alarm and missed detectio...

Lecture Notes in Computer Science, 2011

The paper deals with a mathematical model of a surveillance system based on a net of sensors. The signals acquired by each node of the net are Markovian process, have two different transition probabilities, which depends on the presence or absence of a intruder nearby. The detection of the transition probability change at one node should be confirmed by a detection of similar change at some other sensors. Based on a simple game the model of a fusion center is then constructed. The aggregate function defined on the net is the background of the definition of a non-cooperative stopping game which is a model of the multivariate disorder detection.

2015 IEEE International Symposium on Information Theory (ISIT), 2015

The problem of detecting a transient change in distribution of a discrete time series is investigated when there is a constraint on the number of observed samples. Under a minimax setting where the change time is unknown, the objective is to design a statistical test that minimizes a measure of worst case delay under a constraint on the average time to false alarm as well as a constraint on the sampling rate. Leveraging the results in the non-transient setting, it is shown that under full sampling there exists an asymptotic threshold on the minimum duration of a change that can be detected reliably with such false alarm constrained tests. Next, given a transient change with duration above this asymptotic threshold, the smallest sampling rate for which the change can be detected as efficiently as under full sampling is characterized asymptotically.

2012

Abstract We consider the problem of efficient on-line anomaly detection in computer network traffic. The problem is approached statistically, as that of sequential (quickest) changepoint detection. A multi-cyclic setting of quickest change detection is a natural fit for this problem. We propose a novel scorebased multi-cyclic detection algorithm. The algorithm is based on the so-called ShiryaevRoberts procedure.

53rd IEEE Conference on Decision and Control, 2014

We consider the problem of detection of abrupt changes when there is uncertainty about the post-change distribution. In particular we examine this problem in the prototypical model of continuous time in which the drift of a Wiener process changes at an unknown time from zero to a random value. It is assumed that the change time is an unknown constant while the drift assumed after the change has a Bernoulli distribution with all values of the same sign independent of the process observed. We set up the problem as a stochastic optimization in which the objective is to minimize a measure of detection delay subject to a frequency of false alarm constraint. As a measure of detection delay we consider that of a worst detection delay weighed by the probabilities of the different possible drift values assumed after the change point to which we are able to compute a lower bound amongst the class of all stopping times. Our objective is to then construct low complexity, easy to implement decision rules, that achieve this lower bound exactly, while maintaining the same frequency of false alarms as the family of stopping times. In this effort, we consider a special class of decision rules that are delayed versions of CUSUM algorithm. In this enlarged collection, we are able to construct a family of computationally efficient decision rules that achieve the lower bound with equality, and then choose a best one whose performance is as close to the performance of a stopping time as possible.

IEEE Transactions on Information Theory

In the 1960s, Shiryaev developed a Bayesian theory of change-point detection in the i.i.d. case, which was generalized in the beginning of the 2000s by Tartakovsky and Veeravalli for general stochastic models assuming a certain stability of the log-likelihood ratio process. Hidden Markov models represent a wide class of stochastic processes that are very useful in a variety of applications. In this paper, we investigate the performance of the Bayesian Shiryaev change-point detection rule for hidden Markov models. We propose a set of regularity conditions under which the Shiryaev procedure is first-order asymptotically optimal in a Bayesian context, minimizing moments of the detection delay up to certain order asymptotically as the probability of false alarm goes to zero. The developed theory for hidden Markov models is based on Markov chain representation for the likelihood ratio and r-quick convergence for Markov random walks. In addition, applying Markov nonlinear renewal theory, we present a high-order asymptotic approximation for the expected delay to detection of the Shiryaev detection rule. Asymptotic properties of another popular change detection rule, the Shiryaev-Roberts rule, is studied as well. Some interesting examples are given for illustration.

2007

This paper examines the joint problem of detection and identification of a sudden and unobservable change in the probability distribution function (pdf) of a sequence of independent and identically distributed (i.i.d.) random variables to one of finitely many alternative pdf's. The objective is quick detection of the change and accurate inference of the ensuing pdf. Following a Bayesian approach, a new sequential decision strategy for this problem is revealed and is proven optimal. Geometrical properties of this strategy are demonstrated via numerical examples.