In sequential change detection, existing performance measures differ significantly in the way they treat the time of change. By modeling this quantity as a random time, we introduce a general framework capable of capturing and better understanding most well-known criteria and also propose new ones. For a specific new criterion that constitutes an extension to Lorden's performance measure, we offer the optimum structure for detecting a change in the constant drift of a Brownian motion and a formula for the corresponding optimum performance.
The problem of sequential change diagnosis is considered, where observations are obtained on-line, an abrupt change occurs in their distribution, and the goal is to quickly detect the change and accurately identify the post-change distribution, while controlling the false alarm rate. A finite set of alternatives is postulated for the post-change regime, but no prior information is assumed for the unknown change-point. A drawback of many algorithms that have been proposed for this problem is the implicit use of pre-change data for determining the post-change distribution. This can lead to very …
Online detection of changes in stochastic systems, referred to as sequential change detection or quickest change detection, is an important research topic in statistics, signal processing, and information theory, and has a wide range of applications. This survey starts with the basics of sequential change detection, and then moves on to generalizations and extensions of sequential change detection theory and methods. We also discuss some new dimensions that emerge at the intersection of sequential change detection with other areas, along with a selection of modern applications and remarks on o…
Quick detection of common changes is critical in sequential monitoring of multi-stream data where a common change is referred as a change that only occurs in a portion of panels. After a common change is detected by using a combined CUSUM-SR procedure, we first study the joint distribution for values of the CUSUM process and the estimated delay detection time for the unchanged panels. The BH method by using the asymptotic exponential property for the CUSUM process is developed to isolate the changed panels with the control on FDR. The common change point is then estimated based on the isolated…
This paper considers the change-point problem for finite sequences of networks. To avoid the difficulty of computing the normalization coefficient, such as in Exponential random graphical models (ERGMs) and Markov networks, we construct a finite measure space with measure ratio statistics. A new performance measure of detection delay is proposed to detect the changes in distribution of the network. And an optimal sequential test is proposed under the performance measure. The good performance of the optimal sequential test is illustrated numerically on ERGMs and Erdos-Rényi network sequences.
Sequential change point tests aim at giving an alarm as soon as possible after a structural break occurs while controlling the asymptotic false alarm error. For such tests it is of particular importance to understand how quickly a break is detected. While this is often assessed by simulations only, in this paper, we derive the asymptotic distribution of the delay time for sequential change point procedures based on U-statistics. This includes the difference-of-means (DOM) sequential test, that has been discussed previously, but also a new robust Wilcoxon sequential change point test. Similar t…
We consider the sequential change-point detection problem of detecting changes that are characterized by a subspace structure. Such changes are frequent in high-dimensional streaming data altering the form of the corresponding covariance matrix. In this work we present a Subspace-CUSUM procedure and demonstrate its first-order asymptotic optimality properties for the case where the subspace structure is unknown and needs to be simultaneously estimated. To achieve this goal we develop a suitable analytical methodology that includes a proper parameter optimization for the proposed detection sche…
We present a new non-parametric statistic, called the weighed $\ell_2$ divergence, based on empirical distributions for sequential change detection. We start by constructing the weighed $\ell_2$ divergence as a fundamental building block for two-sample tests and change detection. The proposed statistic is proved to attain the optimal sample complexity in the offline setting. We then study the sequential change detection using the weighed $\ell_2$ divergence and characterize the fundamental performance metrics, including the average run length (ARL) and the expected detection delay (EDD). We al…
We propose a general framework of sequential testing procedures based on $U$-statistics which contains as an example a sequential CUSUM test based on differences in mean but also includes a robust sequential Wilcoxon change point procedure. Within this framework, we consider several monitoring schemes that take different observations into account to make a decision at a given time point. Unlike the originally proposed scheme that takes all observations of the monitoring period into account, we also consider a modified moving-sum-version as well as a version of a Page-monitoring scheme. The lat…
This paper considers the problem of joint change detection and identification assuming multiple composite postchange hypotheses. We propose a multihypothesis changepoint detection-identification procedure that controls the probabilities of false alarm and wrong identification. We show that the proposed procedure is asymptotically minimax and pointwise optimal, minimizing moments of the detection delay as probabilities of false alarm and wrong identification approach zero. The asymptotic optimality properties hold for general stochastic models with dependent observations. We illustrate general …
After obtaining an accurate approximation for $ARL_0$, we first consider the optimal design of weight parameter for a multivariate EWMA chart that minimizes the stationary average delay detection time (SADDT). Comparisons with moving average (MA), CUSUM, generalized likelihood ratio test (GLRT), and Shiryayev-Roberts (S-R) charts after obtaining their $ARL_0$ and SADDT's are conducted numerically. To detect the change with sparse signals, hard-threshold and soft-threshold EWMA charts are proposed. Comparisons with other charts including adaptive techniques show that the EWMA procedure should b…
We consider the online monitoring of multivariate streaming data for changes that are characterized by an unknown subspace structure manifested in the covariance matrix. In particular, we consider the covariance structure changes from an identity matrix to an unknown spiked covariance model. We assume the post-change distribution is unknown, and propose two detection procedures: the Largest-Eigenvalue Shewhart chart and the Subspace-CUSUM detection procedure. We present theoretical approximations to the average run length (ARL) and the expected detection delay (EDD) for the Largest-Eigenvalue …
This article develops a method to construct the optimal sequential test for monitoring the changes in the distribution of finite observation sequences with a general dependence structure. This method allows us to prove that different optimal sequential tests can be constructed for different performance measures of detection delay times. We also provide a formula to calculate the value of the generalized out-of-control average run length for every optimal sequential test. Moreover, we show that there is an equivalent optimal control limit which does not depend on the test statistic directly whe…
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 n…
We consider sequential change-point detection in parallel data streams, where each stream has its own change point. Once a change is detected in a data stream, this stream is deactivated permanently. The goal is to maximize the normal operation of the pre-change streams, while controlling the proportion of post-change streams among the active streams at all time points. Taking a Bayesian formulation, we develop a compound decision framework for this problem. A procedure is proposed that is uniformly optimal among all sequential procedures which control the expected proportion of postchange str…
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 tru…
We study the problem of detecting an abrupt change to the signal covariance matrix. In particular, the covariance changes from a "white" identity matrix to an unknown spiked or low-rank matrix. Two sequential change-point detection procedures are presented, based on the largest and the smallest eigenvalues of the sample covariance matrix. To control false-alarm-rate, we present an accurate theoretical approximation to the average-run-length (ARL) and expected detection delay (EDD) of the detection, leveraging the extreme eigenvalue distributions from random matrix theory and by capturing a non…
We construct a Bayesian sequential test of two simple hypotheses about the value of the unobservable drift coefficient of a Brownian motion, with a possibility to change the initial decision at subsequent moments of time for some penalty. Such a testing procedure allows to correct the initial decision if it turns out to be wrong. The test is based on observation of the posterior mean process and makes the initial decision and, possibly, changes it later, when this process crosses certain thresholds. The solution of the problem is obtained by reducing it to joint optimal stopping and optimal sw…
Sequential change-point detection when the distribution parameters are unknown is a fundamental problem in statistics and machine learning. When the post-change parameters are unknown, we consider a set of detection procedures based on sequential likelihood ratios with non-anticipating estimators constructed using online convex optimization algorithms such as online mirror descent, which provides a more versatile approach to tackle complex situations where recursive maximum likelihood estimators cannot be found. When the underlying distributions belong to a exponential family and the estimator…
This paper considers the detection of change points in parallel data streams, a problem widely encountered when analyzing large-scale real-time streaming data. Each stream may have its own change point, at which its data has a distributional change. With sequentially observed data, a decision maker needs to declare whether changes have already occurred to the streams at each time point.Once a stream is declared to have changed, it is deactivated permanently so that its future data will no longer be collected. This is a compound decision problem in the sense that the decision maker may want to …
We propose a new sequential monitoring scheme for changes in the parameters of a multivariate time series. In contrast to procedures proposed in the literature which compare an estimator from the training sample with an estimator calculated from the remaining data, we suggest to divide the sample at each time point after the training sample. Estimators from the sample before and after all separation points are then continuously compared calculating a maximum of norms of their differences. For open-end scenarios our approach yields an asymptotic level $α$ procedure, which is consistent under th…
The aim of online monitoring is to issue an alarm as soon as there is significant evidence in the collected observations to suggest that the underlying data generating mechanism has changed. This work is concerned with open-end, nonparametric procedures that can be interpreted as statistical tests. The proposed monitoring schemes consist of computing the so-called retrospective CUSUM statistic (or minor variations thereof) after the arrival of each new observation. After proposing suitable threshold functions for the chosen detectors, the asymptotic validity of the procedures is investigated i…
The Bradley-Terry-Luce (BTL) model is a classic and very popular statistical approach for eliciting a global ranking among a collection of items using pairwise comparison data. In applications in which the comparison outcomes are observed as a time series, it is often the case that data are non-stationary, in the sense that the true underlying ranking changes over time. In this paper we are concerned with localizing the change points in a high-dimensional BTL model with piece-wise constant parameters. We propose novel and practicable algorithms based on dynamic programming that can consistentl…
In the multisensor sequential change detection problem, a disruption occurs in an environment monitored by multiple sensors. This disruption induces a change in the observations of an unknown subset of sensors. In the Byzantine version of this problem, which is the focus of this work, it is further assumed that the postulated change-point model may be misspecified for an unknown subset of sensors. The problem then is to detect the change quickly and reliably, for any possible subset of affected sensors, even if the misspecified sensors are controlled by an adversary. Given a user-specified upp…
We present a simple reduction from sequential estimation to sequential changepoint detection (SCD). In short, suppose we are interested in detecting changepoints in some parameter or functional $θ$ of the underlying distribution. We demonstrate that if we can construct a confidence sequence (CS) for $θ$, then we can also successfully perform SCD for $θ$. This is accomplished by checking if two CSs -- one forwards and the other backwards -- ever fail to intersect. Since the literature on CSs has been rapidly evolving recently, the reduction provided in this paper immediately solves several old …
