Sequential Change-point Detection Research Articles

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 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.

Structural Damage Detection and Localization with Unknown Postdamage Feature Distribution Using Sequential Change-Point Detection Method

The high structural deficient rate poses serious risks to the operation of many bridges and buildings. To prevent critical damage and structural collapse, a quick structural health diagnosis tool is needed during normal operation or immediately after extreme events. In structural health monitoring (SHM), many existing methods will have limited usefulness in the quick damage identification process because (1) the damage event needs to be identified quickly, and (2) postdamage information is usually unavailable. To address these drawbacks, we propose a new damage detection and localization approach based on stochastic time series analysis. Specifically, damage sensitive features, which are extracted from vibration signals, follow different distributions before and after a damage event. Hence, we use optimal change-point detection theory to find the time of damage occurrence. Because existing change-point detectors require the postdamage feature distribution, which is unavailable in SHM, we propose a maximum likelihood method for learning the distribution parameters from the time-series data. The proposed damage detection using estimated parameters achieves optimal performance. Also, we utilize the detection results to find damage location without any further computation. Validation results show highly accurate damage identification in American Society of Civil Engineers benchmark structures and two shake table experiments.

Special Issue on Recent Trends in Stochastic Models, Statistics and Their Applications

This special issue was organized on the occasion of the 13th Workshop on Stochastic Models, Statistics and Their Applications (SMSA 2017) held at Humboldt University of Berlin on February 20-24, 2017. Key areas of the workshop focused on models and methods for problems arising in industry, finance, natural sciences, engineering, and technology. In these areas, solutions are needed, which are able to deal with complex and possibly high-dimensional data, make use of innovative methodologies, and have proven applicability to real-world data. In this special issue, we have 13 papers. The papers consider stochastic models for diverse data sets and streams such as bridge vehicle loads, watermark identification data, text documents, images of digitized handwritten digits, financial asset returns, option prices, and equity data. The methods and algorithms required to handle come from machine learning, stochastic processes, sequential (on-line) change-point detection, regression analysis, image analysis, and robust divergence measures. The work of Barbulescu analyzes the properties of Generalized Regression Neural Network (GRNN) model applied to the prediction of the selected Romanian stock market prices. In the work of Luo et al, a nonlinear PDE model to estimate the American call options and put options under financial crisis scenario is implemented with additional penalty introduced to Black-Scholes formula. The work of Liu et al proposes a compound Poisson bivariate process model of vehicle load with Levy copula function and validates it in a reliability analysis of bridges in China. Migalska proposes in her paper a density-free symmetry verification test for arbitrary symmetries in images. In their work, Darkhovsky et al propose a novel method based on the notion of ϵ-complexity of continuous vector functions for an off-line detection of multiple change-points in multidimensional time series. The work of Rafajlowicz et al proposes a method of spike detection and suppression (SDS method), which extends the jump detector proposed by the Authors in their earlier work and applies it for a laser power control in a 3D additive manufacturing process. Sliwinski et al propose two algorithms for nonlinearity recovery in Hammerstein systems, one based on orthogonal series expansions and the other using aggregation and validate them in numerical simulations. The work of Sousa et al provides an insight into the behavior of GARCH process describing financial markets affected by rumours and other random factors through an analysis of process mean and variance by estimating run length. Kisslinger et al describe a new toolkit based on scaled Bregman divergences for detecting distributional changes in random data, including streams and clouds. The work of Sarnowski et al derives an optimal decision function for a sudden change detection in political business cycle model based on the Markovian process. The work of Walkowiak et al proposes a two-step procedure for open-set classification of text documents and validates it on simulated data and on the collection of Wikipedia documents. The work by Xie et al combines two-tier stochastic frontier model with the traditional Richardson Model and applies it to the Chinese stock market analysis. Finally, the work of Yuan et al derives the Hamilton-Jacobi-Bellman equation for an optimal consumption problem of maximizing the expected discounted value of utility in an economic growth model with random technological shocks. We thank numerous anonymous reviewers for helping in evaluating all submitted manuscripts.

Sequential Change-Point Detection via Online Convex Optimization.

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 tackling complex situations where recursive maximum likelihood estimators cannot be found. When the underlying distributions belong to a exponential family and the estimators satisfy the logarithm regret property, we show that this approach is nearly second-order asymptotically optimal. This means that the upper bound for the false alarm rate of the algorithm (measured by the average-run-length) meets the lower bound asymptotically up to a log-log factor when the threshold tends to infinity. Our proof is achieved by making a connection between sequential change-point and online convex optimization and leveraging the logarithmic regret bound property of online mirror descent algorithm. Numerical and real data examples validate our theory.

Real-time particle filtering and smoothing algorithms for detecting abrupt changes in neural ensemble spike activity.

Sequential change-point detection from time series data is a common problem in many neuroscience applications, such as seizure detection, anomaly detection, and pain detection. In our previous work (Chen Z, Zhang Q, Tong AP, Manders TR, Wang J. J Neural Eng 14: 036023, 2017), we developed a latent state-space model, known as the Poisson linear dynamical system, for detecting abrupt changes in neuronal ensemble spike activity. In online brain-machine interface (BMI) applications, a recursive filtering algorithm is used to track the changes in the latent variable. However, previous methods have been restricted to Gaussian dynamical noise and have used Gaussian approximation for the Poisson likelihood. To improve the detection speed, we introduce non-Gaussian dynamical noise for modeling a stochastic jump process in the latent state space. To efficiently estimate the state posterior that accommodates non-Gaussian noise and non-Gaussian likelihood, we propose particle filtering and smoothing algorithms for the change-point detection problem. To speed up the computation, we implement the proposed particle filtering algorithms using advanced graphics processing unit computing technology. We validate our algorithms, using both computer simulations and experimental data for acute pain detection. Finally, we discuss several important practical issues in the context of real-time closed-loop BMI applications. NEW & NOTEWORTHY Sequential change-point detection is an important problem in closed-loop neuroscience experiments. This study proposes novel sequential Monte Carlo methods to quickly detect the onset and offset of a stochastic jump process that drives the population spike activity. This new approach is robust with respect to spike sorting noise and varying levels of signal-to-noise ratio. The GPU implementation of the computational algorithm allows for parallel processing in real time.

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 (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 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 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.

On the Bayesian Sequential Change-Point Detection

On the Bayesian Sequential Change-Point Detection

Nonparametric Tests and Nested Sequential Sampling Plans for Change-Point Detection

We propose a new approach for solving nonparametric problems of sequential change-point detection. Nested sequential sampling plans allow using the Kolmogorov–Smirnov test and other nonparametric tests for sequential change-point detection. Exact formulas for the mathematical expectation and the variance of the false detection moment are based on the exact distributions of nonparametric statistics for small sample sizes.

Real time change-point detection in a nonlinear quantile model

ABSTRACTMost studies in real-time change-point detection either focus on the linear model or use the cumulative sum (CUSUM) method under classical assumptions on model errors. This article considers the sequential change-point detection in a nonlinear quantile model. A test statistic based on the CUSUM of the quantile process subgradient is proposed and studied. Under the null hypothesis that the model does not change, the asymptotic distribution of the test statistic is determined. Under the alternative hypothesis that at some unknown observation there is a change in the model, the proposed test statistic converges in probability to ∞. These results allow building the critical regions on open-end and on closed-end procedures. Simulation results, using a Monte Carlo technique, investigate the performance of the test statistic, especially for heavy-tailed error distributions. We also compare it with the classical CUSUM test statistic. An example on real data is also given.

Sequential change-point detection in time series models based on pairwise likelihood

Sequential change-point detection in time series models based on pairwise likelihood

Precision of sequential change point detection

A random sequence having two segments being the homogeneous Markov processes is registered. Each segment has his own transition probability law and the length of the segment is unknown and random. The transition probabilities of each process are known and a priori distribution of the disorder moment is given. The decision maker aim is to detect the moment of the transition probabilities change. The detection of the disorder rarely is precise. The decision maker accepts some deviation in estimation of the disorder moment. In the considered model the aim is to indicate the change point with fixed, bounded error with maximal probability. The case with various precision for over and under estimation of this point is analysed. The case when the disorder does not appears with positive probability is also included. The results insignificantly extends range of application, explain the structure of optimal detector in various circumstances and shows new details of the solution construction. The motivation for this investigation is the modelling of the attacks in the node of networks. The objectives is to detect one of the attack immediately or in very short time before or after it appearance with highest probability. The problem is reformulated to optimal stopping of the observed sequences. The detailed analysis of the problem is presented to show the form of optimal decision function.

Discussion on “Sequential detection/isolation of abrupt changes” by Igor V. Nikiforov

ABSTRACTSimultaneous sequential changepoint detection and isolation is a challenging theoretical and practical direction of research. Professor Igor Nikiforov's article provides an excellent review of the state-of-the-art literature in this important field. This discussion addresses several issues and extensions important for modern applications of sequential methodology.

Sequential detection of learning in cognitive diagnosis.

In order to look more closely at the many particular skills examinees utilize to answer items, cognitive diagnosis models have received much attention, and perhaps are preferable to item response models that ordinarily involve just one or a few broadly defined skills, when the objective is to hasten learning. If these fine-grained skills can be identified, a sharpened focus on learning and remediation can be achieved. The focus here is on how to detect when learning has taken place for a particular attribute and efficiently guide a student through a sequence of items to ultimately attain mastery of all attributes while administering as few items as possible. This can be seen as a problem in sequential change-point detection for which there is a long history and a well-developed literature. Though some ad hoc rules for determining learning may be used, such as stopping after M consecutive items have been successfully answered, more efficient methods that are optimal under various conditions are available. The CUSUM, Shiryaev-Roberts and Shiryaev procedures can dramatically reduce the time required to detect learning while maintaining rigorous Type I error control, and they are studied in this context through simulation. Future directions for modelling and detection of learning are discussed.

An exact approach to Bayesian sequential change point detection

Change point models seek to fit a piecewise regression model with unknown breakpoints to a data set whose parameters are suspected to change through time. However, the exponential number of possible solutions to a multiple change point problem requires an efficient algorithm if long time series are to be analyzed. A sequential Bayesian change point algorithm is introduced that provides uncertainty bounds on both the number and location of change points. The algorithm is able to quickly update itself in linear time as each new data point is recorded and uses the exact posterior distribution to infer whether or not a change point has been observed. Simulation studies illustrate how the algorithm performs under various parameter settings, including detection speeds and error rates, and allow for comparison with several existing multiple change point algorithms. The algorithm is then used to analyze two real data sets, including global surface temperature anomalies over the last 130 years.

Congestion detection of pedestrians using the velocity entropy: A case study of Love Parade 2010 disaster

Gatherings of large human crowds often result in crowd disasters such as the Love Parade Disaster in Duisburg, Germany on July 24, 2010. To avoid these tragedies, video surveillance and early warning are becoming more and more significant. In this paper, the velocity entropy is first defined as the criterion for congestion detection, which represents the motion magnitude distribution and the motion direction distribution simultaneously. Then the detection method is verified by the simulation data based on AnyLogic software. To test the generalization performance of this method, video recordings of a real-world case, the Love Parade disaster, are also used in the experiments. The velocity histograms of the foreground object in the videos are extracted by the Gaussian Mixture Model (GMM) and optical flow computation. With a sequential change-point detection algorithm, the velocity entropy can be applied to detect congestions of the Love Parade festival. It turned out that without recognizing and tracking individual pedestrian, our method can detect abnormal crowd behaviors in real-time.

Sequential change point detection in linear quantile regression models

We develop a method for sequential detection of structural changes in linear quantile regression models. We establish the asymptotic properties of the proposed test statistic, and demonstrate the advantages of the proposed method over existing tests through simulation.

On the use of estimating functions in monitoring time series for change points

A large class of estimators including maximum likelihood, least squares and M-estimators are based on estimating functions. In sequential change point detection related monitoring functions can be used to monitor new incoming observations based on an initial estimator, which is computationally efficient because possible numeric optimization is restricted to the initial estimation. In this work, we give general regularity conditions under which we derive the asymptotic null behavior of the corresponding tests in addition to their behavior under alternatives, where conditions become particularly simple for sufficiently smooth estimating and monitoring functions. These regularity conditions unify and extend a large amount of existing procedures in the literature, while they also allow us to derive monitoring schemes in time series that have not yet been considered in the literature including non-linear autoregressive time series and certain count time series such as binary or Poisson autoregressive models. We do not assume that the estimating and monitoring functions are equal or even of the same dimension, therefore, we allow for example to combine a non-robust but more precise initial estimator with a robust monitoring scheme. Some simulations and data examples illustrate the usefulness of the described procedures.

Hybrid algorithms for multiple change-point detection in biological sequences.

Array comparative genomic hybridization (aCGH) is one of the techniques that can be used to detect copy number variations in DNA sequences in high resolution. It has been identified that abrupt changes in the human genome play a vital role in the progression and development of many complex diseases. In this study we propose two distinct hybrid algorithms that combine efficient sequential change-point detection procedures (the Shiryaev-Roberts procedure and the cumulative sum control chart (CUSUM) procedure) with the Cross-Entropy method, which is an evolutionary stochastic optimization technique to estimate both the number of change-points and their corresponding locations in aCGH data. The proposed hybrid algorithms are applied to both artificially generated data and real aCGH experimental data to illustrate their usefulness. Our results show that the proposed methodologies are effective in detecting multiple change-points in biological sequences of continuous measurements.

Asymptotically Pointwise Optimal Change Detection in Multiple Channels

A Bayesian multichannel change-point detection problem is studied in the following general setting. A multidimensional stochastic process is observed; some or all of its components may experience changes in distribution, simultaneously or not. The loss function penalizes for false alarms and detection delays, and the penalty increases with each missed change-point. For wide classes of stochastic processes, with or without nuisance parameters and practically any joint prior distribution of change-points, asymptotically pointwise optimal (APO) rules are obtained, translating the classical concept of Bickel and Yahav to the sequential change-point detection. These APO rules are attractive because of their simple analytic form and straightforward computation. An application to a multidimensional autoregressive time series is shown.

An Automatic Extraction Approach of Worm Signatures Based on Behavioral Footprint Analysis

This paper presents an automatic extraction approach of worm signatures based on behavioral footprint analysis. Firstly, the suspicious worm traffic is detected based on worm behavior and is an instance of the Sequential Change Point Detection. To make the detection mechanism insensitive to site and access pattern, a non-parametric Cumulative Sum (CUSUM) method is applied, thus making the detection mechanism much more generally applicable and its deployment much easier. Secondly, worm behavioral footprint is defined and classified. The chronicle formalism is applied to correlate the footprint in suspicious worm traffic. Finally, worm signatures are extracted and ascertained by the evaluation function. The experiment shows that the approach can extract worm signatures effectively and accurately. There is a conclusion that worm behavioral footprint can't identify the worm accurately, but it helps to locate worm signatures, so signatures can be extracted effectively.