arXiv subjects

Alexander G. Tartakovsky

Publications and source records attributed to Alexander G. Tartakovsky.

At least 19 recordsLinked to original sources

arXiv2025

Change, dependence, and discovery: Celebrating the work of T.L. Lai

Tze Leung Lai made seminal contributions to sequential analysis, particularly in sequential hypothesis testing, changepoint detection and nonlinear renewal theory. His work established fundamental optimality results for the sequential probability ratio test and its extensions, and provided a general framework for testing composite hypotheses. In changepoint detection, he introduced new optimality criteria and computationally efficient procedures that remain influential. He applied these and related tools to problems in biostatistics. In this article, we review these key results in the broader context of sequential analysis.

stat.OT↗

arXiv2024

Beyond boundaries: Gary Lorden's groundbreaking contributions to sequential analysis

Gary Lorden provided several fundamental and novel insights into sequential hypothesis testing and changepoint detection. In this article, we provide an overview of Lorden's contributions in the context of existing results in those areas, and some extensions made possible by Lorden's work. We also mention some of Lorden's significant consulting work, including as an expert witness and for NASA, the entertainment industry, and Major League Baseball.

math.ST↗

arXiv2024

Nearly Optimum Properties of Certain Multi-Decision Sequential Rules for General Non-i.i.d. Stochastic Models

Dedicated to the memory of Professor Tze Leung Lai, this paper introduces three multi-hypothesis sequential tests. These tests are derived from one-sided versions of the sequential probability ratio test and its modifications. They are proven to be first-order asymptotically optimal for testing simple and parametric composite hypotheses when error probabilities are small. These tests exhibit near optimality properties not only in classical i.i.d. observation models but also in general non-i.i.d. models, provided that the log-likelihood ratios between hypotheses converge r-completely to positive and finite numbers. These findings extend the seminal work of Lai (1981) on two hypotheses.

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arXiv2023

Detecting an Intermittent Change of Unknown Duration

Oftentimes in practice, the observed process changes statistical properties at an unknown point in time and the duration of a change is substantially finite, in which case one says that the change is intermittent or transient. We provide an overview of existing approaches for intermittent change detection and advocate in favor of a particular setting driven by the intermittent nature of the change. We propose a novel optimization criterion that is more appropriate for many applied areas such as the detection of threats in physical-computer systems, near-Earth space informatics, epidemiology, pharmacokinetics, etc. We argue that controlling the local conditional probability of a false alarm, rather than the familiar average run length to a false alarm, and maximizing the local conditional probability of detection is a more reasonable approach versus a traditional quickest change detection approach that requires minimizing the expected delay to detection. We adopt the maximum likelihood (ML) approach with respect to the change duration and show that several commonly used detection rules (CUSUM, window-limited CUSUM, and FMA) are equivalent to the ML-based stopping times. We discuss how to choose design parameters for these rules and provide a comprehensive simulation study to corroborate intuitive expectations.

stat.AP↗

arXiv2022

Quickest Change Detection with Non-Stationary Post-Change Observations

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 distribution. 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.

eess.SP↗

arXiv2021

Optimal Sequential Detection of Signals with Unknown Appearance and Disappearance Points in Time

The paper addresses a sequential changepoint detection problem, assuming that the duration of change may be finite and unknown. This problem is of importance for many applications, e.g., for signal and image processing where signals appear and disappear at unknown points in time or space. In contrast to the conventional optimality criterion in quickest change detection that requires minimization of the expected delay to detection for a given average run length to a false alarm, we focus on a reliable maximin change detection criterion of maximizing the minimal probability of detection in a given time (or space) window for a given local maximal probability of false alarm in the prescribed window. We show that the optimal detection procedure is a modified CUSUM procedure. We then compare operating characteristics of this optimal procedure with popular in engineering the Finite Moving Average (FMA) detection algorithm and the ordinary CUSUM procedure using Monte Carlo simulations, which show that typically the later algorithms have almost the same performance as the optimal one. At the same time, the FMA procedure has a substantial advantage -- independence to the intensity of the signal, which is usually unknown. Finally, the FMA algorithm is applied to detecting faint streaks of satellites in optical images.

math.ST↗

arXiv2021

An Asymptotic Theory of Joint Sequential Changepoint Detection and Identification for General Stochastic Models

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.

math.ST↗

arXiv2019

Asymptotically Optimal Pointwise and Minimax Change-point Detection for General Stochastic Models With a Composite Post-Change Hypothesis

A weighted Shiryaev-Roberts change detection procedure is shown to approximately minimize the expected delay to detection as well as higher moments of the detection delay among all change-point detection procedures with the given low maximal local probability of a false alarm within a window of a fixed length in pointwise and minimax settings for general non-i.i.d. data models and for the composite post-change hypothesis when the post-change parameter is unknown. We establish very general conditions for the models under which the weighted Shiryaev-Roberts procedure is asymptotically optimal. These conditions are formulated in terms of the rate of convergence in the strong law of large numbers for the log-likelihood ratios between the "change" and "no-change" hypotheses, and we also provide sufficient conditions for a large class of ergodic Markov processes. Examples, where these conditions hold, are given.

math.ST↗

arXiv2018

Asymptotic Optimality of Mixture Rules for Detecting Changes in General Stochastic Models

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 the 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 the 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 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.

math.ST↗

arXiv2016

Asymptotic Bayesian Theory of Quickest Change Detection for Hidden Markov Models

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.

math.ST↗

arXiv2016

Tracking Changes in Resilience and Level of Coordination in Terrorist Groups

Activity profiles of terrorist groups show frequent spurts and downfalls corresponding to changes in the underlying organizational dynamics. In particular, it is of interest in understanding changes in attributes such as intentions/ideology, tactics/strategies, capabilities/resources, etc., that influence and impact the activity. The goal of this work is the quick detection of such changes and in general, tracking of macroscopic as well as microscopic trends in group dynamics. Prior work in this area are based on parametric approaches and rely on time-series analysis techniques, self-exciting hurdle models (SEHM), or hidden Markov models (HMM). While these approaches detect spurts and downfalls reasonably accurately, they are all based on model learning --- a task that is difficult in practice because of the "rare" nature of terrorist attacks from a model learning perspective. In this paper, we pursue an alternate non-parametric approach for spurt detection in activity profiles. Our approach is based on binning the count data of terrorist 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 classification. While the parametric approaches often result in either a large number of missed detections of real changes or false alarms of unoccurred changes, the proposed approach is shown to result in a small number of missed detections and false alarms. Further, the non-parametric nature of the approach makes it attractive for ready applications in a practical context.

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arXiv2016

Asymptotically Optimal Pointwise and Minimax Quickest Change-point Detection for Dependent Data

We consider the quickest change-point detection problem in pointwise and minimax settings for general dependent data models. Two new classes of sequential detection procedures associated with the maximal "local" probability of a false alarm within a period of some fixed length are introduced. For these classes of detection procedures, we consider two popular risks: the expected positive part of the delay to detection and the conditional delay to detection. Under very general conditions for the observations, we show that the popular Shiryaev--Roberts procedure is asymptotically optimal, as the local probability of false alarm goes to zero, with respect to both these risks pointwise (uniformly for every possible point of change) and in the minimax sense (with respect to maximal over point of change expected detection delays). The conditions are formulated in terms of the rate of convergence in the strong law of large numbers for the log-likelihood ratios between the "change" and "no-change" hypotheses, specifically as a uniform complete convergence of the normalized log-likelihood ratio to a positive and finite number. We also develop tools and a set of sufficient conditions for verification of the uniform complete convergence for a large class of Markov processes. These tools are based on concentration inequalities for functions of Markov processes and the Meyn--Tweedie geometric ergodic theory. Finally, we check these sufficient conditions for a number of challenging examples (time series) frequently arising in applications, such as autoregression, autoregressive GARCH, etc.

math.ST↗

arXiv2016

On Asymptotic Optimality in Sequential Changepoint Detection: Non-iid Case

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 (2004) developed a general asymptotic theory of changepoint detection in the non-iid case and discrete time, and Baron and Tartakovsky (2006) in continuous time assuming certain stability of the log-likelihood ratio process. This stability property was formulated in terms of the r-quick convergence of the normalized log-likelihood 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 (as the probability of false alarm vanishes) the 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.

math.ST↗

arXiv2016

Multichannel Sequential Detection- Part I: Non-i.i.d. Data

We consider the problem of sequential signal detection in a multichannel system where the number and location of signals is a priori unknown. We assume that the data in each channel are sequentially observed and follow a general non-i.i.d. stochastic model. Under the assumption that the local log-likelihood ratio processes in the channels converge r-completely to positive and finite numbers, we establish the asymptotic optimality of a generalized sequential likelihood ratio test and a mixture-based sequential likelihood ratio test. Specifically, we show that both tests minimize the first r moments of the stopping time distribution asymptotically as the probabilities of false alarm and missed detection approach zero. Moreover, we show that both tests asymptotically minimize all moments of the stopping time distribution when the local log-likelihood ratio processes have independent increments and simply obey the Strong Law of Large Numbers. This extends a result previously known in the case of i.i.d. observations when only one channel is affected. We illustrate the general detection theory using several practical examples, including 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 the feasibility of both sequential tests when assuming an upper and a lower bound on the number of signals and compare their non-asymptotic performance using a simulation study.

math.ST↗

arXiv2014

Optimal Design and Analysis of the Exponentially Weighted Moving Average Chart for Exponential Data

We study optimal design of the Exponentially Weighted Moving Average (EWMA) chart by a proper choice of the smoothing factor and the initial value (headstart) of the decision statistic. The particular problem addressed is that of quickest detection of an abrupt change in the parameter of a discrete-time exponential model. Both pre- and post-change parameter values are assumed known, but the change-point is not known. For this change-point detection scenario, we examine the performance of the conventional one-sided EWMA chart with respect to two optimality criteria: Pollak's minimax criterion associated with the maximal conditional expected delay to detection and Shiryaev's multi-cyclic setup associated with the stationary expected delay to detection. Using the integral-equations approach, we derive the exact closed-form formulae for all of the required performance measures. Based on these formulae we find the optimal smoothing factor and headstart by solving the corresponding two bivariate constraint optimization problems. Finally, the performance of the optimized EWMA chart is compared against that of the Shiryaev--Roberts--$r$ procedure in the minimax setting, and against that of the original Shiryaev--Roberts procedure in the multi-cyclic setting. The main conclusion is that the EWMA chart, when fully optimized, turns out to be a very competitive procedure, with performance nearly indistinguishable from that of the known-to-be-best Shiryaev--Roberts--$r$ and Shiryaev--Roberts procedures.

stat.AP↗

arXiv2014

Hidden Markov models for the activity profile of terrorist groups

The main focus of this work is on developing models for the activity profile of a terrorist group, detecting sudden spurts and downfalls in this profile, and, in general, tracking it over a period of time. Toward this goal, a $d$-state hidden Markov model (HMM) that captures the latent states underlying the dynamics of the group and thus its activity profile is developed. The simplest setting of $d=2$ corresponds to the case where the dynamics are coarsely quantized as Active and Inactive, respectively. A state estimation strategy that exploits the underlying HMM structure is then developed for spurt detection and tracking. This strategy is shown to track even nonpersistent changes that last only for a short duration at the cost of learning the underlying model. Case studies with real terrorism data from open-source databases are provided to illustrate the performance of the proposed methodology.

stat.AP↗

arXiv2013

Modeling Temporal Activity Patterns in Dynamic Social Networks

The focus of this work is on developing probabilistic models for user activity in social networks by incorporating the social network influence as perceived by the user. For this, we propose a coupled Hidden Markov Model, 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 model 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 Hidden Markov Model. 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.

physics.soc-ph↗

arXiv2013

Almost optimal sequential tests of discrete composite hypotheses

We consider the problem of sequentially testing a simple null hypothesis versus a composite alternative hypothesis that consists of a finite set of densities. We study sequential tests that are based on thresholding of mixture-based likelihood ratio statistics and weighted generalized likelihood ratio statistics. It is shown that both sequential tests have several asymptotic optimality properties as error probabilities go to zero. First, for any weights, they minimize the expected sample size within a constant term under every scenario in the alternative hypothesis and at least to first order under the null hypothesis. Second, for appropriate weights that are specified up to a prior distribution, they minimize within an asymptotically negligible term a weighted expected sample size in the alternative hypothesis. Third, for a particular prior distribution, they are almost minimax with respect to the expected Kullback-Leibler divergence until stopping. Furthermore, based on high-order asymptotic expansions for the operating characteristics, we propose prior distributions that lead to a robust behavior. Finally, based on asymptotic analysis as well as on simulation experiments, we argue that both tests have the same performance when they are designed with the same weights.

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