The problem of quickest detection of a change in the distribution of a sequence of independent observations is considered. The pre-change observations are assumed to be stationary with a known distribution, while the post-change observations are allowed to be non-stationary with some possible parametric uncertainty in their distributions. In particular, it is assumed that the cumulative Kullback-Leibler divergence between the post-change and the pre-change distributions grows in a certain manner with time after the change-point. For the case where the post-change distributions are known, a universal asymptotic lower bound on the delay is derived, as the false alarm rate goes to zero. Furthermore, a window-limited Cumulative Sum (CuSum) procedure is developed, and shown to achieve the lower bound asymptotically. For the case where the post-change distributions have parametric uncertainty, a window-limited (WL) generalized likelihood-ratio (GLR) CuSum procedure is developed and is shown to achieve the universal lower bound asymptotically. Extensions to the case with dependent observations are discussed. The analysis is validated through numerical results on synthetic data. The use of the WL-GLR-CuSum procedure in monitoring pandemics is also demonstrated.

The paper addresses a joint sequential changepoint detection and identification/isolation problem for a general stochastic model, assuming that the observed data may be dependent and non-identically distributed, the prior distribution of the change point is arbitrary, and the post-change hypotheses are composite. The developed detection–identification theory generalizes the changepoint detection theory developed by Tartakovsky (2019) to the case of multiple composite post-change hypotheses when one has not only to detect a change as quickly as possible but also to identify (or isolate) the true post-change distribution. We propose a multi-hypothesis change detection–identification rule and show that it is nearly optimal, minimizing moments of the delay to detection as the probability of a false alarm and the probabilities of misidentification go to zero.

In the 1960s, Shiryaev developed a Bayesian theory of change-point detection in the i.i.d. case, which was generalized in the early 2000s by Tartakovsky and Veeravalli and recently by Tartakovsky (2017) for general stochastic models assuming a certain stability of the log-likelihood ratio process. Hidden Markov models represent a wide class of stochastic processes 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 and a first-order asymptotic approximation for the probability of false alarm of the Shiryaev detection rule. We also study asymptotic properties of another popular change detection rule, the Shiryaev-Roberts rule, and provide some interesting examples.

The paper addresses a sequential changepoint detection problem for a general stochastic model, assuming that the observed data may be non-i.i.d. (i.e., dependent and non-identically distributed) and prior distribution of the change point is arbitrary. Tartakovsky and Veeravalli (2005), Baron and Tartakovsky (2006), and, more recently, Tartakovsky (2017) developed a general asymptotic theory of changepoint detection for non-i.i.d. stochastic models, assuming certain stability of the log-likelihood ratio process, in the case of simple hypotheses when both pre-change and post-change models are completely specified. However, in most applications, the post-change distribution is not completely known. In the present paper, we generalize previous results to the case of parametric uncertainty, assuming that the parameter of the post-change distribution is unknown. We introduce two detection rules based on mixtures-the mixture Shiryaev rule and the mixture Shiryaev-Roberts rule-and study their asymptotic properties in the Bayesian context. In particular, we provide sufficient conditions under which these rules are first-order asymptotically optimal, minimizing moments of the delay to detection as the probability of false alarm approaches zero.

The focus of this paper is on the activity profiles of certain networks, such as terrorist networks, that show frequent spurts and downfalls. Of particular interest is the quick detection of changes in two specific activity patterns corresponding to resilience and level of coordination in the network. Understanding changes in resilience and coordination could provide insights into the underlying organizational dynamics and aid in more informed decision-making. Prior work in tackling this problem is based on parametric approaches and relies on models developed with time-series analysis techniques, self-exciting hurdle models, or hidden Markov models. While these approaches detect spurts and downfalls reasonably accurately, they are all based on model learning-a task that is computationally difficult in practice because of the “rare” nature of terrorist attacks from a model learning perspective. In this paper, we pursue an alternate statistical nonparametric approach for spurt detection in activity profiles. Our approach is based on binning the count data of activity to form observation vectors that can be compared with each other. Motivated by a majorization theory framework, these vectors are then transformed via certain functionals and used in spurt detection and classification. While the parametric approaches often result in either a large number of missed detections of real changes or false alarms, the proposed approach is shown to result in a small number of missed detections and false alarms. Furthermore, since spurt detection is a problem of importance across multiple applications, the nonparametric nature of the approach makes it attractive in these applications.

We consider the problem of sequential signal detection in a multichannel system assuming that the number and location of signals are either unknown or only partially known a priori. We focus on the design and analysis of two sequential hypothesis tests: the generalized sequential likelihood ratio test and the mixture sequential likelihood ratio test. We develop an asymptotic theory for a general stochastic model, where the various data streams can be coupled and correlated, and the data in each stream can be dependent and non-identically distributed. Specifically, we show that the two proposed sequential detection procedures asymptotically minimize the expected sample size and even higher moments of the sample size in the class of hypothesis tests with given probabilities of errors under weak distributional assumptions. We also propose efficient importance sampling algorithms for estimating error probabilities of the sequential tests by Monte Carlo simulation. The general theory is illustrated with several practical examples, such as the detection of signals, in Gaussian hidden Markov models, white Gaussian noises with unknown intensity, and testing of the first-order autoregression's correlation coefficient. Finally, we illustrate our asymptotic results and compare the two proposed procedures with a simulation study.

We consider a sequential Bayesian changepoint detection problem for a general stochastic model, assuming that the observed data may be dependent and non-identically distributed and the prior distribution of the change point is arbitrary, not necessarily geometric. Tartakovsky and Veeravalli (2005) developed a general asymptotic theory of changepoint detection in the case of non-identically distributed and dependent observations and discrete time, and Baron and Tartakovsky (2006) in continuous time assuming the certain stability of the loglikelihood ratio process. This stability property was formulated in terms of the r-quick convergence of the normalized loglikelihood ratio process to a positive and finite number, which can be interpreted as the limiting Kullback-Leibler information between the “change” and “no change” hypotheses. In these papers, it was conjectured that the r-quick convergence can be relaxed in the r-complete convergence, which is typically much easier to verify in particular examples. In the present paper, we justify this conjecture by showing that the Shiryaev change detection procedure is nearly optimal, minimizing asymptotically to first order (as the probability of false alarm vanishes) moments of the delay to detection up to order r whenever r-complete convergence holds. We also study asymptotic properties of the Shiryaev-Roberts detection procedure in the Bayesian context.

The focus of this work is on developing probabilistic models for temporal activity of users in social networks (e.g., posting and tweeting) by incorporating the social network influence as perceived by the user. Although prior work in this area has developed sophisticated models for user activity, these models either ignore social network influence completely or incorporate it in an implicit manner. We overcome the nontransparency of the network in the model at the individual scale by proposing a coupled hidden Markov model (HMM), where each user's activity evolves according to a Markov chain with a hidden state that is influenced by the collective activity of the friends of the user. We develop generalized Baum-Welch and Viterbi algorithms for parameter learning and state estimation for the proposed framework. We then validate the proposed model using a significant corpus of user activity on Twitter. Our numerical studies show that with sufficient observations to ensure accurate model learning, the proposed framework explains the observed data better than either a renewal process-based model or a conventional (uncoupled) HMM. We also demonstrate the utility of the proposed approach in predicting the time to the next tweet. Finally, clustering in the model parameter space is shown to result in distinct natural clusters of users characterized by the interaction dynamic between a user and his network.

The problem of decentralized quickest change detection is studied with an additional constraint on the cost of observations used at each sensor. Minimax problem formulations are proposed for the problem. A distributed algorithm called the DE-All algorithm is proposed in which on-off observation control is employed locally at each sensor. It is shown that the proposed algorithm is asymptotically optimal up to first order for the proposed formulations.

We design a quickest change detection procedure that is almost minimax in the sense of minimizing the maximal (over change points) expected delay to detection for a given low false alarm rate. This procedure represents a variation of the Shiryaev-Roberts procedure that starts off at a fixed specially designed point.

We compare several decentralized change-point detection procedures for multisensor distributed systems when the information available for decision-making is distributed across a set of sensors. Asymptotically optimal procedures for two scenarios are presented. In the first scenario, the sensors send quantized versions of their observations to a fusion center where change detection is performed based on all the sensor messages. If in particular, the quantizers are binary, then the proposed binary CUSUM detection test is optimal in the class of tests with binary quantized data. In the second scenario, the sensors perform local change detection using the CUSUM procedures and send their final decisions to the fusion center for combining. The decision in favor of the change occurrence is made whenever CUSUM statistics at all sensors exceed thresholds. The latter decentralized procedure has the same first order asymptotic (as the false alarm rate is low) minimax operating characteristics as the globally optimal centralized detection procedure that has access to all the sensor observations. However, the presented Monte Carlo experiments for the Poisson example show that despite the fact that the procedure with local decisions is globally asymptotically optimal for a low false alarm rate, it performs worse than the procedure with binary quantization unless the false alarm rate is extremely low. In addition, two voting-type local decision based detection procedures are proposed and evaluated. Applications to network security (rapid detection of computer intrusions) are discussed

It is supposed that there is a multichannel sensor system which performs sequential detection of a target. Sequential detection is done by implementing a generalized Wald's sequential probability ratio test, which is based on the maximum-likelihood ratio statistic and allows one to fix the false-alarm rate and the rate of missed detections at specified levels. We present the asymptotic performance of this sequential detection procedure and show that it is asymptotically optimal in the sense of minimizing the expected sample size when the probabilities of erroneous decisions are small. We do not assume that the observations are independent and identically distributed (i.i.d.). The first-order asymptotic optimality result holds for general statistical models that are not confined to the restrictive i.i.d. assumption. However, for i.i.d. and quasi-i.i.d. cases, where the log-likelihood ratios can be represented in the form of sums of random walks and slowly changing sequences, we obtain much stronger results. Specifically, using the nonlinear renewal theory we are able to obtain both tight expressions for the error probabilities and higher order approximations for the average sample size up to a vanishing term. The performance of the multichannel sequential detection algorithm is illustrated by an example of detection of a deterministic signal in correlated (colored) Gaussian noise. In this example, we provide both the results of theoretical analysis and the results of a Monte Carlo experiment. These results allow us to conclude that the use of the sequential detection algorithm substantially reduces the required resources of the system compared to the best nonsequential algorithm.

The optimal detection procedure for detecting changes in independent and identically distributed sequences (i.i.d.) in a Bayesian setting was derived by Shiryaev in the nineteen sixties. However, the analysis of the performance of this procedure in terms of the average detection delay and false alarm probability has been an open problem. In this paper, we investigate the performance of Shiryaev's procedure in an asymptotic setting where the false alarm probability goes to zero. The asymptotic study is performed not only in. the i.d.d. case where the Shiryaev's procedure is optimal but also in a general, non-i.i.d. case. In the latter case, we show that Shiryaev's procedure is asymptotically optimum under mild conditions. We also show that the two popular non-Bayesian detection procedures, namely the Page and Shiryaev-Roberts-Pollak procedures, are not optimal (even asymptotically) under the Bayesian criterion. The results of this study are shown to be especially important in studying the asymptotics of decentralized quickest change detection procedures.

For pt. I see ibid. vol.45, p.2448-61, 1999. We proved in pt.I that two specific constructions of multihypothesis sequential tests, which we refer to as multihypothesis sequential probability ratio tests (MSPRTs), are asymptotically optimal as the decision risks (or error probabilities) go to zero. The MSPRTs asymptotically minimize not only the expected sample size but also any positive moment of the stopping time distribution, under very general statistical models for the observations. In this paper, based on nonlinear renewal theory we find accurate asymptotic approximations (up to a vanishing term) for the expected sample size that take into account the "overshoot" over the boundaries of decision statistics. The approximations are derived for the scenario where the hypotheses are simple, the observations are independent and identically distributed (i.i.d.) according to one of the underlying distributions, and the decision risks go to zero. Simulation results for practical examples show that these approximations are fairly accurate not only for large but also for moderate sample sizes. The asymptotic results given here complete the analysis initiated by Baum and Veeravalli (1994), where first-order asymptotics were obtained for the expected sample size under a specific restriction on the Kullback-Leibler distances between the hypotheses.

The problem of sequential testing of multiple hypotheses is considered, and two candidate sequential test procedures are studied. Both tests are multihypothesis versions of the binary sequential probability ratio test (SPRT), and are referred to as MSPRTs. The first test is motivated by Bayesian optimality arguments, while the second corresponds to a generalized likelihood ratio test. It is shown that both MSPRTs are asymptotically optimal relative not only to the expected sample size but also to any positive moment of the stopping time distribution, when the error probabilities or, more generally, risks associated with incorrect decisions are small. The results are first derived for the discrete-time case of independent and identically distributed (i.i.d.) observations and simple hypotheses. They are then extended to general, possibly continuous-time, statistical models that may include correlated and nonhomogeneous observation processes. It also demonstrated that the results can be extended to hypothesis testing problems with nuisance parameters, where the composite hypotheses, due to nuisance parameters, can be reduced to simple ones by using the principle of invariance. These results provide a complete generalization of the results given by Veeravalli and Baum (see ibid., vol.41, p.1994-97, 1995), where it was shown that the quasi-Bayesian MSPRT is asymptotically efficient with respect to the expected sample size for i.i.d. observations.