Optimal Change Point Detection Research Articles

Optimal change-point detection and localization

Optimal change-point detection and localization

Constructing outlier-free histograms with variable bin-width based on distance minimization

We propose a new method of constructing a variable bin width histogram that can accommodate the unbalanced distribution of the samples yet retaining, as a whole, the good aspect of both equal width (EW) and equal-area (EA) histograms that are being used popularly for data visualization and analysis. We formulate this as an optimal change point detection problem in which the bin boundaries are determined by minimizing the sum of the absolute error or the squared error in each bin. The former is based on Distance Minimization (DM) and new, and the latter is based on Variance Minimization (VM) and is considered the state-of-the-art. The constructed histograms can effectively be used to detect and visualize hidden outliers/anomalies by applying the interquartile range method in each bin. The final histograms are obtained by adjusting bin boundaries and heights accordingly after removing the detected outliers/anomalies. We further propose a method to annotate the constructed bins if the data for annotation is given for each sample as a set of nominal variables, using z-score with respect to their distribution within each bin. We applied our method to both real vinyl greenhouse datasets and two different sets of three synthetic datasets, and confirmed that both DM and VM methods work as intended, both can represent the sample distribution with a smaller number of bins than those by EW and EA methods, The use of interquartile range method can detect anomalies as well as outliers, and the terms selected for annotation are interpretable and reasonable. EW and EA methods have contrasting properties. DM and VM methods lie in between, but the former is closer to EA method and the latter to EW method. DM method runs substantially faster than VM method and performs slightly better than VM method in outlier detection and annotation tasks.

Seeded binary segmentation: a general methodology for fast and optimal changepoint detection

Summary We propose seeded binary segmentation for large-scale changepoint detection problems. We construct a deterministic set of background intervals, called seeded intervals, in which single changepoint candidates are searched for. The final selection of changepoints based on these candidates can be done in various ways, adapted to the problem at hand. The method is thus easy to adapt to many changepoint problems, ranging from univariate to high dimensional. Compared to recently popular random background intervals, seeded intervals lead to reproducibility and much faster computations. For the univariate Gaussian change in mean set-up, the methodology is shown to be asymptotically minimax optimal when paired with appropriate selection criteria. We demonstrate near-linear runtimes and competitive finite sample estimation performance. Furthermore, we illustrate the versatility of our method in high-dimensional settings.

Labeled optimal partitioning

In data sequences measured over space or time, an important problem is accurate detection of abrupt changes. In partially labeled data, it is important to correctly predict presence/absence of changes in positive/negative labeled regions, in both the train and test sets. One existing dynamic programming algorithm is designed for prediction in unlabeled test regions (and ignores the labels in the train set); another is for accurate fitting of train labels (but does not predict changepoints in unlabeled test regions). We resolve these issues by proposing a new optimal changepoint detection model that is guaranteed to fit the labels in the train data, and can also provide predictions of unlabeled changepoints in test data. We propose a new dynamic programming algorithm, Labeled Optimal Partitioning, and we provide a formal proof that it solves the resulting non-convex optimization problem. We provide theoretical and empirical analysis of the time complexity of our algorithm, in terms of the number of labels and the size of the data sequence to segment. Finally, we provide empirical evidence that our algorithm is more accurate than the existing baselines, in terms of train and test label error.

Optimal Change-Point Detection With Training Sequences in the Large and Moderate Deviations Regimes

This paper investigates a novel offline change-point detection problem from an information-theoretic perspective. In contrast to most related works, we assume that the knowledge of the underlying pre- and post-change distributions are not known and can only be learned from the training sequences which are available. We further require the probability of the estimation error to decay either exponentially or sub-exponentially fast (corresponding respectively to the large and moderate deviations regimes in information theory parlance). Based on the training sequences as well as the test sequence consisting of a single change-point, we design a change-point estimator and further show that this estimator is optimal by establishing matching (strong) converses. This leads to a full characterization of the optimal confidence width (i.e., half the width of the confidence interval within which the true change-point is located at with high probability) as a function of the undetected error, under both the large and moderate deviations regimes.

A greedy graph search algorithm based on changepoint analysis for automatic QRS complex detection

The electrocardiogram (ECG) signal is the most widely used non-invasive tool for the investigation of cardiovascular diseases. Automatic delineation of ECG fiducial points, in particular the R-peak, serves as the basis for ECG processing and analysis. This study proposes a new method of ECG signal analysis by introducing a new class of graphical models based on optimal changepoint detection models, named the graph-constrained changepoint detection (GCCD) model. The GCCD model treats fiducial points delineation in the non-stationary ECG signal as a changepoint detection problem. The proposed model exploits the sparsity of changepoints to detect abrupt changes within the ECG signal; thereby, the R-peak detection task can be relaxed from any preprocessing step. In this novel approach, prior biological knowledge about the expected sequence of changes is incorporated into the model using the constraint graph, which can be defined manually or automatically. First, we define the constraint graph manually; then, we present a graph learning algorithm that can search for an optimal graph in a greedy scheme. Finally, we compare the manually defined graphs and learned graphs in terms of graph structure and detection accuracy. We evaluate the performance of the algorithm using the MIT-BIH Arrhythmia Database. The proposed model achieves an overall sensitivity of 99.64%, positive predictivity of 99.71%, and detection error rate of 0.19 for the manually defined constraint graph and overall sensitivity of 99.76%, positive predictivity of 99.68%, and detection error rate of 0.55 for the automatic learning constraint graph.

Quick line outage identification in urban distribution grids via smart meters

The growing integration of distributed energy resources (DERs) in distribution grids raises various reliability issues due to DER's uncertain and complex behaviors. With large-scale DER penetration in distribution grids, traditional outage detection methods, which rely on customers report and smart meters' “last gasp” signals, will have poor performance, because renewable generators and storage and the mesh structure in urban distribution grids can continue supplying power after line outages. To address these challenges, we propose a data-driven outage monitoring approach based on the stochastic time series analysis with a theoretical guarantee. Specifically, we prove via power flow analysis that dependency of time-series voltage measurements exhibits significant statistical changes after line outages. This makes the theory on optimal change-point detection suitable to identify line outages. However, existing change point detection methods require post-outage voltage distribution, which are unknown in distribution systems. Therefore, we design a maximum likelihood estimator to directly learn distribution pa-rameters from voltage data. We prove the estimated parameters-based detection also achieves optimal performance, making it extremely useful for fast distribution grid outage identifications. Furthermore, since smart meters have been widely installed in distribution grids and advanced infrastructure (e.g., PMU) has not widely been available, our approach only requires voltage magnitude for quick outage identification. Simulation results show highly accurate outage identification in eight distribution grids with 17 configurations with and without DERs using smart meter data.

Asymptotically Optimal Change Point Detection for Composite Hypothesis in State Space Models

This paper investigates change point detection in state space models, in which the pre-change distribution f <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">θ0</sup> is given, while the post-distribution f <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">θ</sup> after change is unknown. The problem is to raise an alarm as soon as possible after the distribution changes from f <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">θ0</sup> to f <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">θ</sup> , under a restriction on the false alarms. We investigate theoretical properties of a weighted Shiryayev-Roberts-Pollak (SRP) change point detection rule in state space models. By making use of a Markov chain representation for the likelihood function, exponential embedding of the induced Markovian transition operator, nonlinear Markov renewal theory, and sequential hypothesis testing theory for Markov random walks, we show that the weighted SRP procedure is second-order asymptotically optimal. To this end, we derive an asymptotic approximation for the expected stopping time of such a stopping scheme when the change time w = 1. To illustrate our method we apply the results to two types of state space models: general state Markov chains and linear state space models. Simulation study is given to illustrate the method.

Optimal change point detection and localization in sparse dynamic networks

We study the problem of change point localization in dynamic networks models. We assume that we observe a sequence of independent adjacency matrices of the same size, each corresponding to a realization of an unknown inhomogeneous Bernoulli model. The underlying distribution of the adjacency matrices are piecewise constant, and may change over a subset of the time points, called change points. We are concerned with recovering the unknown number and positions of the change points. In our model setting, we allow for all the model parameters to change with the total number of time points, including the network size, the minimal spacing between consecutive change points, the magnitude of the smallest change and the degree of sparsity of the networks. We first identify a region of impossibility in the space of the model parameters such that no change point estimator is provably consistent if the data are generated according to parameters falling in that region. We propose a computationally-simple algorithm for network change point localization, called network binary segmentation, that relies on weighted averages of the adjacency matrices. We show that network binary segmentation is consistent over a range of the model parameters that nearly cover the complement of the impossibility region, thus demonstrating the existence of a phase transition for the problem at hand. Next, we devise a more sophisticated algorithm based on singular value thresholding, called local refinement, that delivers more accurate estimates of the change point locations. Under appropriate conditions, local refinement guarantees a minimax optimal rate for network change point localization while remaining computationally feasible.

A Graph-constrained Changepoint Detection Approach for ECG Segmentation.

Electrocardiogram (ECG) signal is the most commonly used non-invasive tool in the assessment of cardiovascular diseases. Segmentation of the ECG signal to locate its constitutive waves, in particular the R-peaks, is a key step in ECG processing and analysis. Over the years, several segmentation and QRS complex detection algorithms have been proposed with different features; however, their performance highly depends on applying preprocessing steps which makes them unreliable in realtime data analysis of ambulatory care settings and remote monitoring systems, where the collected data is highly noisy. Moreover, some issues still remain with the current algorithms in regard to the diverse morphological categories for the ECG signal and their high computation cost. In this paper, we introduce a novel graph-based optimal changepoint detection (GCCD) method for reliable detection of Rpeak positions without employing any preprocessing step. The proposed model guarantees to compute the globally optimal changepoint detection solution. It is also generic in nature and can be applied to other time-series biomedical signals. Based on the MIT-BIH arrhythmia (MIT-BIH-AR) database, the proposed method achieves overall sensitivity Sen = 99.76, positive predictivity PPR = 99.68, and detection error rate DER = 0.55 which are comparable to other state-of-the-art approaches.1 2.

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.

Optimal change point detection in Gaussian processes

We study the problem of detecting a change in the mean of one-dimensional Gaussian process data in the fixed domain regime. We propose a detection procedure based on the generalized likelihood ratio test (GLRT), and show that our method achieves asymptotically near-optimal rate in a minimax sense. The notable feature of the proposed method is that it exploits in an efficient way the data dependence captured by the Gaussian process covariance structure. When the covariance is not known, we propose the plug-in GLRT method and derive conditions under which the method remains asymptotically near-optimal. By contrast, the standard CUSUM method, which does not account for the covariance structure, is shown to be suboptimal. Our algorithms and asymptotic analysis are applicable to a number of covariance structures, including the Matern class, the powered exponential class, and others. The plug-in GLRT method is shown to perform well for maximum likelihood estimators with a dense covariance matrix.

Tracking Intentions from Streaming Video: A Double Hmm Approach

In this paper we discuss problems related to the identification and prediction of abstract behavioural characteristics of a intelligent agent based on physical behaviour recorded by surveillance cameras. Our general approach is to use Bayesian nonlinear filtering-type algorithms and optimal change-point detection criterion, but specifically we ex- ploit hierarchical dynamics of the agent’s behaviour by modelling with a double Markov chain (DMC), which iteratively conditions the dis- tribution of physical attributes on the values taken by the abstract ones.

Nearly Optimal Change-Point Detection with an Application to Cybersecurity

We address the sequential change-point detection problem for the Gaussian model where baseline distribution is Gaussian with variance σ2 and mean μ such that σ2 = aμ, where a > 0 is a known constant; the change is in μ from one known value to another. First, we carry out a comparative performance analysis of four detection procedures: the Cumulative Sum (CUSUM) procedure, the Shiryaev–Roberts (SR) procedure, and two its modifications—the Shiryaev–Roberts–Pollak and Shiryaev–Roberts–r procedures. The performance is benchmarked via Pollak's maximal average delay to detection and Shiryaev's stationary average delay to detection, each subject to a fixed average run length to false alarm. The analysis shows that in practically interesting cases the accuracy of asymptotic approximations is “reasonable” to “excellent”. We also consider an application of change-point detection to cybersecurity for rapid anomaly detection in computer networks. Using real network data we show that statistically traffic's intensity can be well described by the proposed Gaussian model with σ2 = aμ instead of the traditional Poisson model, which requires σ2 = μ. By successively devising the SR and CUSUM procedures to “catch” a low-contrast network anomaly (caused by an Internet Control Message Protocol reflector attack), we then show that the SR rule is quicker. We conclude that the SR procedure is a better cyber “watch dog” than the popular CUSUM procedure.

Asymptotic Optimality of Change-Point Detection Schemes in General Continuous-Time Models

In the early 1960s, Shiryaev obtained the structure of Bayesian stopping rules for detecting abrupt changes in independent and identically distributed sequences as well as in a constant drift of the Brownian motion. Since then, the methodology of optimal change-point detection has concentrated on the search for stopping rules that achieve the best balance of the mean detection delay and the rate of false alarms or minimize the mean delay under a fixed false alarm probability. In this respect, analysis of the performance of the Shiryaev procedure has been an open problem. Recently, Tartakovsky and Veeravalli (2005) investigated asymptotic performance of the Shiryaev Bayesian change detection procedure, the Page procedure, and the Shiryaev–Roberts procedure when the false alarm probability goes to zero for general discrete-time models. In this article, we investigate the asymptotic performance of Shiryaev and Shiryaev–Roberts procedures for general continuous-time stochastic models for a small false alarm probability and small cost of detection delay. We show that the Shiryaev procedure has asymptotic optimality properties under mild conditions, while the Shiryaev–Roberts procedure may or may not be asymptotically optimal depending on the type of the prior distribution. The presented asymptotic Bayesian detection theory substantially generalizes previous work in the field of change-point detection for continuous-time processes.

Asymptotic operating characteristics of an optimal change point detection in hidden Markov models

Let ξ0,ξ1,…,ξω−1 be observations from the hidden Markov model with probability distribution Pθ0, and let ξω,ξω+1,… be observations from the hidden Markov model with probability distribution Pθ1. The parameters θ0 and θ1 are given, while the change point ω is unknown. The problem is to raise an alarm as soon as possible after the distribution changes from Pθ0 to Pθ1, but to avoid false alarms. Specifically, we seek a stopping rule N which allows us to observe the ξ's sequentially, such that E∞N is large, and subject to this constraint, sup kEk(N−k|N≥k) is as small as possible. Here Ek denotes expectation under the change point k, and E∞ denotes expectation under the hypothesis of no change whatever. In this paper we investigate the performance of the Shiryayev–Roberts–Pollak (SRP) rule for change point detection in the dynamic system of hidden Markov models. By making use of Markov chain representation for the likelihood function, the structure of asymptotically minimax policy and of the Bayes rule, and sequential hypothesis testing theory for Markov random walks, we show that the SRP procedure is asymptotically minimax in the sense of Pollak [Ann. Statist. 13 (1985) 206–227]. Next, we present a second-order asymptotic approximation for the expected stopping time of such a stopping scheme when ω=1. Motivated by the sequential analysis in hidden Markov models, a nonlinear renewal theory for Markov random walks is also given.

BAYES STOPPING RULES IN A CHANGE-POINT MODEL WITH A RANDOM HAZARD RATE

In the Bayes sequential change-point problem, an assumption of a fully known prior distribution of a change-point is usually impracticable. At every moment, one often knows only the discrete hazard function, that is, the probability of a change occurring before the next observation is collected, given that it has not occurred so far. In the randomized model, the observed or predicted values of the hazard function are assumed to form a Markov chain. Under these assumptions, the optimal change-point detection stopping rules are derived for two popular loss functions introduced in Shiryaev (1978) and Ritov (1990). Derivations are based on the theory of optimal stopping of Markov sequences.