U-Statistics in Sequential Tests and Change Detection

We present a new non-parametric statistic, called the weighed <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> divergence, based on empirical distributions for sequential change detection. We start by constructing the weighed <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> 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 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> divergence and characterize the fundamental performance metrics, including the average run length (ARL) and the expected detection delay (EDD). We also present practical algorithms to find the optimal projection to handle high-dimensional data and the optimal weights, which is critical to quick detection since, in such settings, there are not many post-change samples. Simulation results and real data examples are provided to validate the good performance of the proposed method.

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.

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.

A change detection method for sequential patterns

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.

Confronted with nonspecific symptoms, accurate screening tests would be useful to clinicians to distinguish between functional childhood disorders and inflammatory bowel disease (IBD), thus avoiding invasive diagnostic testing. Traditional ulcerative colitis-specific perinuclear antineutrophil cytoplasmic antibody (pANCA) and Crohn's disease-specific anti-Saccharomyces cerevisiae antibody (ASCA) serodiagnostic assays have recently been modified, with ELISA cut-off values recalculated to maximize sensitivity. The aim of this study was to determine whether the combination of these serodiagnostic tests could maximize diagnostic accuracy and minimize invasive investigations in pediatric patients presenting with nonspecific symptoms suggestive of IBD. With investigators blinded to clinical diagnoses, ASCA, ANCA, and pANCA profiles were obtained prospectively from 128 patients undergoing complete diagnostic evaluation for IBD. In phase I, diagnostic accuracy and predictive values of the modified and traditional assays were compared for the IBD (n = 54) and non-IBD groups (n = 74). In phase II, the overall accuracy of a novel sequential diagnostic testing strategy was determined. Additionally, the potential number of invasive investigations avoided with the hypothetical application of this strategy to the cohort was determined. For phase I, the modified serodiagnostic assay was more sensitive (81 vs 69%), whereas the traditional assay had a higher specificity (96 vs 72%) for IBD (p < 0.05) For phase II, false-positive diagnoses would have been reduced by 81%, yielding an overall sequential testing strategy accuracy of 84%. The incorporation of sequential noninvasive testing into a diagnostic strategy may avoid unnecessary and costly evaluations and facilitate clinical decision making when the diagnosis of IBD in children is initially uncertain.

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.

Sequential Detection of Abrupt Changes in ARMA Models

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.

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.

Statistical methods for sequential hypothesis testing and changepoint detection have applications across many fields, including quality control, biomedical engineering, communication networks, econometrics, image processing, security, etc. This book presents an overview of methodology in these related areas, providing a synthesis of research from the last few decades. The methods are illustrated through real data examples, and software is referenced where possible. The emphasis is on providing all the theoretical details in a unified framework, with pointers to new research directions.

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.

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.

Although similarity could improve visual working memory (VWM) performance, it remains unclear how the spatial configuration of visual information influences the similarity effect in VWM. We explored this question by manipulating the orderliness of spatial configuration (ordered vs. scrambled) in the simultaneous (Experiment 1) and sequential (Experiment 2) change detection tasks. The results showed that similarity improved VWM performance when memory items were presented simultaneously and sequentially. For the simultaneous memory array containing similar and dissimilar items, the performance of the ordered spatial configuration was better than that of the scrambled spatial configuration when probing dissimilar items, while no such difference was found when probing similar items. Further, the similarity effect value in the scrambled spatial configuration was higher than that in the ordered spatial configuration. For the sequential memory array containing similar and dissimilar items, spatial configuration did not affect the similarity effect in VWM. Taken together, these findings suggest that spatial configuration could modulate the similarity effect when memory items are presented simultaneously, in which the ordered spatial configuration protects representations of dissimilar items and reduces the similarity effect in VWM. Our study provides additional evidence for the role of spatial configuration in the similarity effect in VWM, and supports the hierarchical model.

SELECTING THRESHOLDS FOR SEQUENTIAL FAULT DETECTION TESTS