Sequential Change Detection by Optimal Weighted ℓ₂ Divergence

Detecting emergence of a low-rank signal from high-dimensional data is an important problem arising from many applications such as camera surveillance and swarm monitoring using sensors. We consider a procedure based on the largest eigenvalue of the sample covariance matrix over a sliding window to detect the change. To achieve dimensionality reduction, we present a sketching-based approach for rank change detection using the low-dimensional linear sketches of the original high-dimensional observations. The premise is that when the sketching matrix is a random Gaussian matrix, and the dimension of the sketching vector is sufficiently large, the rank of sample covariance matrix for these sketches equals the rank of the original sample covariance matrix with high probability. Hence, we may be able to detect the low-rank change using sample covariance matrices of the sketches without having to recover the original covariance matrix. We character the performance of the largest eigenvalue statistic in terms of the false-alarm-rate and the expected detection delay, and present an efficient online implementation via subspace tracking.

Detecting change points sequentially in a streaming setting, especially when both the mean and the variance of the signal can change, is often a challenging task. A key difficulty in this context often involves setting an appropriate detection threshold, which for many standard change statistics may need to be tuned depending on the prechange and postchange distributions. This presents a challenge in a sequential change detection setting when a signal switches between multiple distributions. Unfortunately, change point detection schemes that use the log-likelihood ratio, such as cumulative sum (CUSUM) and the generalized log-likelihood ratio (GLR), are quick to react to changes but are not symmetric when both the mean and the variance of the signal change. This makes it difficult to set a single threshold to detect multiple change points sequentially in a streaming setting. We propose a modified version of CUSUM that we call data-adaptive symmetric CUSUM (DAS-CUSUM). The DAS-CUSUM procedure is symmetric for changes between distributions, making it suitable to set a single threshold to detect multiple change points sequentially in a streaming setting. We provide results that relate the expected detection delay and average run length for our proposed procedure when both prechange and postchange distributions are normally distributed. Experiments on simulated and real-world data show the utility of DAS-CUSUM.

Nous proposons une methode « operationnelle » pour le probleme de tests d’hypotheses composites convexes lorsque les distributions sous-jacentes possedent des queues lourdes. Elle s’appuie sur la separation euclidienne des ensembles convexes et peut etre vue comme une extension de la methode developpee dans (Electron. J. Stat. 9 (2015) 1645–1712 ; Electron. J. Stat. 10 (2016) 2204–2242) pour l’etude des tests d’hypotheses reposant sur des techniques d’optimisation convexe. En particulier, nous montrons comment construire des tests quasi-optimaux pour des familles de distributions qui sont majorees, dans un sens precis, par une distribution symetrique quasi-spherique et etudions la relation entre les tests bases sur la separation euclidienne et les tests utilisant des potentiels. Nous appliquons la methodologie proposee au probleme de la detection sequentielle et decrivons sa mise en oeuvre pour la detection sequentielle de ruptures dans l’entree d’un systeme dynamique.

Sequential change detection is a fundamental problem in statistics and signal processing, with the CUSUM procedure widely used to achieve minimax detection delay under a prescribed false alarm rate when pre- and post-change distributions are fully known. However, in many practical settings, raw observations cannot be shared with a trusted central curator, and privacy must be enforced at the data source, which prevents the computation of exact CUSUM statistics. We therefore introduce a local differentially private (DP) variant called LDP-CUSUM, which first applies a local DP mechanism to transform the raw data into privatized observations and then applies a CUSUM procedure to detect the change. We derive closed-form bounds on the average run length to false alarm and on the worst-case average detection delay, explicitly characterizing the tradeoff among privacy level, false alarm rate, and detection efficiency. Numerical simulations and a real-data case study were conducted to demonstrate the detection efficiency of our proposed LDP-CUSUM across various scenarios.

Control charts are used to detect assignable causes in different manufacturing and nonmanufacturing processes. This study presents a new maximum exponentially weighted moving average (Max-EWMA) chart for joint unit interval time and magnitude monitoring. To this end, beta distribution is considered for time whereas simplex distribution is used for magnitude. Average run length (ARL), standard deviation of run length (SDRL), and different quantiles are used to evaluate the performance of the Max-EWMA chart. A real data example is also included in the study to show the application of the proposed chart. The results provide evidences that the Max-EWMA chart is efficient in detecting small- to medium-sized shifts.

Sequential change detection in high-dimensional dynamic networks has attracted growing attention in modern applications. A major challenge is that as network scale increases, limited sensing resources make it difficult to fully observe links at each time point, resulting in partial observability and complicating detection. To tackle this, we propose a Latent Bi-Transit Network (LBTN) that learns unobserved edge formation, latent node states, and the evolution mechanisms of dynamic networks. Based on LBTN, we design a variational Bayesian method to infer sparse node-level changes and employ a likelihood ratio test as the detection statistic. By further formulating the statistic as the reward function of a combinatorial multi-armed bandit (CMAB) problem, we develop a Thompson sampling strategy to adaptively select edges for observation, balancing exploration and exploitation. Comprehensive simulations and real-world experiments show that our method consistently outperforms existing baselines and its variants across diverse scenarios.

Sequential Detection of Abrupt Changes in ARMA Models

SummaryA review of methods suggested in the literature for sequential detection of changes in public health surveillance data is presented. Many researchers have noted the need for prospective methods. In recent years there has been an increased interest in both the statistical and the epidemiological literature concerning this type of problem. However, most of the vast literature in public health monitoring deals with retrospective methods, especially spatial methods. Evaluations with respect to the statistical properties of interest for prospective surveillance are rare. The special aspects of prospective statistical surveillance and different ways of evaluating such methods are described. Attention is given to methods that include only the time domain as well as methods for detection where observations have a spatial structure. In the case of surveillance of a change in a Poisson process the likelihood ratio method and the Shiryaev–Roberts method are derived.

In content-based video analysis, substantial research efforts have been focused on developing techniques to detect the boundaries between two successive shots, each comprising consecutive frames filmed with a single camera act. However, there is still room for further improvement in the detection performance. With the use of sequential change detection and the help of nonparametric density estimation principles, we propose A new shot boundary detection method that can maintain not only satisfactory detection accuracy, but also consistent detection performance based on the results of various test videos.

The problem of sequential detection of spontaneous changes in equations for unobserved state variables and observations of linear and nonlinear multivariate state-space models is considered. We consider the case of new additive terms and the case of changing coefficients of these equations. The proposed nonparametric method uses the idea of the “moving window” of observations. In Theorem 4.1 the exponential rate of convergence to zero for type 1 error probability (a wrong decision about a change) is proved. In Theorem 4.2 we consider type 2 error probability and the normalized (by the volume N of the moving window) delay time in change-point detection. Asymptotic optimality of the proposed method is proved in Theorem 5.1. Experimental study includes Monte Carlo examples of sequentially detected changes in coefficients of linear and nonlinear state-space models.

Research on U-statistics within the general framework of sequential and change-point literature is surveyed. Some recent developments are discussed and extended. New sequential testing strategies based on Wiener process approximations are proposed, and empirical studies explore the finite sample performance of these tests. It allows users to choose one that is appropriate for their application.

A change detection method for sequential patterns

The moving range control chart is usually established to monitor the variability of the process characteristic for situations where the individuals control chart is applied. In the design of a control chart, the average run length (ARL) of the chart is usually used as an important indicator for evaluating control chart performance. In the present paper, the out-of-control ARL of the moving range chart is investigated using computer simulation. Three types of data are generated for study: normal and independent data, non-normal data, and autocorrelated data, such that the effects of non-normality and autocorrelation on the detection ability of the chart can be investigated. From the simulation results, we find that positive skewness coefficient present in the underlying distribution would produce a shorter out-of-control ARL than the normal data. Also, if we are able to know that the data are positively highly autocorrelated, then “span two” should be used for the operation of the moving range chart to keep the chart's detection ability.

The cumulative sum (CUSUM) control chart can sensitively detect small-to-moderate shifts in the process mean. The average run length (ARL) is a popular technique used to determine the performance of a control chart. Recently, several researchers investigated the performance of processes on a CUSUM control chart by evaluating the ARL using either Monte Carlo simulation or Markov chain. As these methods only yield approximate results, we developed solutions for the exact ARL by using explicit formulas based on an integral equation (IE) for studying the performance of a CUSUM control chart running a long-memory process with exponential white noise. The long-memory process observations are derived from a seasonal fractionally integrated MAX model while focusing on X. The existence and uniqueness of the solution for calculating the ARL via explicit formulas were proved by using Banach's fixed-point theorem. The accuracy percentage of the explicit formulas against the approximate ARL obtained via the numerical IE method was greater than 99%, which indicates excellent agreement between the two methods. An important conclusion of this study is that the proposed solution for the ARL using explicit formulas could sensitively detect changes in the process mean on a CUSUM control chart in this situation. Finally, an illustrative case study is provided to show the efficacy of the proposed explicit formulas with processes involving real data.

Measurement error is a usually met distortion factor in real-world applications that influences the outcome of a process. The performance of the control charts can be affected in the presence of measurement error which also leads to erroneous conclusions regarding the average run length. In this paper, we proposed regression estimator-based mixed exponentially weighted moving average-cumulative sum (MxEMCUS) control chart to monitor the process mean and investigate the effect of measurement error on shift detection ability of the proposed control chart. The performance of MxEMCUS control chart is evaluated in the cases of mean shift with and without measurement error. The linear covariate model is used to evaluate the control chart in the presence of measurement error. The case of multiple measurements and linear increasing variance is also investigated in the presence of measurement error. The average run length and standard deviation of run length are used as the performance measuring tool. Based on simulation results, it is concluded that in the presence of measurement error, proposed chart has better performance as compared to other existing charts. An example with real data set is given to demonstrate the implementation of MxEMCUS control chart.