Multiple Change-point Detection Research Articles

Rough-Fuzzy CPD: a gradual change point detection algorithm

Changepoint detection is the problem of finding abrupt or gradual changes in time series data when the distribution of the time series changes significantly. There are many sophisticated statistical algorithms for solving changepoint detection problem, although there is not much work devoted towards gradual changepoints as compared to abrupt ones. Here we present a new approach to solve the changepoint detection problem using the fuzzy rough set theory which is able to detect such gradual changepoints. An expression for the rough-fuzzy estimate of changepoints is derived along with its mathematical properties concerning fast computation. In a statistical hypothesis testing framework, the asymptotic distribution of the proposed statistic on both single and multiple changepoints is derived under the null hypothesis enabling multiple changepoint detection. Extensive simulation studies have been performed to investigate how simple crude statistical measures of disparity can be subjected to improve their efficiency in the estimation of gradual changepoints. Also, the said rough-fuzzy estimate is robust to signal-to-noise ratio, a high degree of fuzziness in true changepoints, and also to hyperparameter values. Simulation studies reveal that the proposed method beats other methods of gradual changepoint detection (including MJPD, HSMUCE, fuzzy methods like FCP, FCMLCP etc) and also popular crisp methods like Binary Segmentation, PELT, and BOCD in detecting gradual changepoints. The applicability of the estimate is demonstrated using multiple real-life datasets including Covid-19. We have developed the python package roufcp for broader dissemination of the methods.

Clustering Gene Expressions Using the Table Invitation Prior.

A prior for Bayesian nonparametric clustering called the Table Invitation Prior (TIP) is used to cluster gene expression data. TIP uses information concerning the pairwise distances between subjects (e.g., gene expression samples) and automatically estimates the number of clusters. TIP's hyperparameters are estimated using a univariate multiple change point detection algorithm with respect to the subject distances, and thus TIP does not require an analyst's intervention for estimating hyperparameters. A Gibbs sampling algorithm is provided, and TIP is used in conjunction with a Normal-Inverse-Wishart likelihood to cluster 801 gene expression samples, each of which belongs to one of five different types of cancer.

Bayesian multiple changepoint detection with missing data and its application to the magnitude‐frequency distributions

AbstractThe detection of abrupt changes in an evolving pattern of time series in the presence of missing data still poses a challenge to real applications. We formulate the multiple changepoint problem into a latent Markov model on a countably infinite state space. For efficiency‐enhancing, we propose a partially collapsed Gibbs sampler for the inference of the joint posterior of the number of changepoints and their locations. Variants of Viterbi algorithms are suggested for obtaining the MAP estimates of random changepoints in the presence of missing data, which provides better performances in these varying‐dimensional problems. The method is generally applicable for multiple changepoint detection under a variety of missing data mechanism. The method is applied to a case study of the magnitude‐frequency distribution of the 2010 Darfield M7.1 earthquake sequence in New Zealand. We find out some unusual features of the seismic b‐value in the Darfield earthquake sequence. It is noted that two changepoints are detected and in contrast to the background seismic b‐value, relatively low b‐values in the early aftershock propagation period are identified. We suggest that this might be a forewarning of potentially devastatingly strong aftershocks. The advance in the method of b‐value changepoint detection will enhance our understanding of earthquake occurrence and potentially lead to improved risk forecasting.

Efficient multiple change point detection for high-dimensional generalized linear models.

Change point detection for high-dimensional data is an important yet challenging problem for many applications. In this paper, we consider multiple change point detection in the context of high-dimensional generalized linear models, allowing the covariate dimension p to grow exponentially with the sample size n. The model considered is general and flexible in the sense that it covers various specific models as special cases. It can automatically account for the underlying data generation mechanism without specifying any prior knowledge about the number of change points. Based on dynamic programming and binary segmentation techniques, two algorithms are proposed to detect multiple change points, allowing the number of change points to grow with n. To further improve the computational efficiency, a more efficient algorithm designed for the case of a single change point is proposed. We present theoretical properties of our proposed algorithms, including estimation consistency for the number and locations of change points as well as consistency and asymptotic distributions for the underlying regression coefficients. Finally, extensive simulation studies and application to the Alzheimer's Disease Neuroimaging Initiative data further demonstrate the competitive performance of our proposed methods.

An updated version of the ETAS model based on multiple change points detection

An updated version of the ETAS model based on multiple change points detection

A Quasi-Bayesian change point detection with exchangeable weights

A Quasi-Bayesian change point test statistic is derived, under the fixed and random exchangeable priors, which is asymptotically close to an important subclass of Poisson–Dirichlet weights. While detecting change at each point, the random prior can be updated for future change points in a sequential sampling manner. Asymptotic behaviors of quasi-Bayesian test statistics, under the null and alternative hypothesis are represented in terms of stochastic integrals. Also, the M-estimate approaches for change in mean for both cases of finite and infinite variance observations are discussed. Aside to the application in change point detections, the asymptotic analysis reveals some interesting probabilistic properties. Multiple change point detection and simulation of finite sample distribution of test statistics are studied Finally, a conclusion section is also given.

Multiple Change-Point Detection in a Functional Sample via the 𝒢-Sum Process

We first define the G-CUSUM process and investigate its theoretical aspects including asymptotic behavior. By choosing different sets G, we propose some tests for multiple change-point detections in a functional sample. We apply the proposed testing procedures to the real-world neurophysiological data and demonstrate how it can identify the existence of the multiple change-points and localize them.

Scalable Bayesian Multiple Changepoint Detection via Auxiliary Uniformisation

SummaryIn this paper, we perform a sparse filtering recursion for efficient changepoint detection for discrete‐time observations. We attach auxiliary event times to the chronologically ordered observations and formulate multiple changepoint problems of discrete‐time observations into continuous‐time observations. Ideally, both the computational and memory costs of the proposed auxiliary uniformisation forward‐filtering backward‐sampling algorithm can be quadratically scaled down to the number of changepoints instead of the number of observations, which would otherwise be prohibitive for a long sequence of observations. To avoid model bias, a time‐varying changepoint recurrence rate across different segments is assumed to characterise diverse scales of run lengths of the changepoints. We demonstrate the methods through simulation studies and real data analysis.

On optimal segmentation and parameter tuning for multiple change-point detection and inference

Change-point analysis is the task of finding abrupt (and significant) changes in the underlying model of a signal or time series. Change-point detection methods typically involve specifying the maximum number of segments to search for and the minimum segment length. However, there is no objective way to pre-specify these two parameters, and it mostly depends upon the particular application. Within this framework, a recursive optimization algorithm is developed that is capable of exploring and fine tuning these two input parameters, and optimally segmenting a time series. This multiple change-point detection technique therefore addresses a wide class of real-life contexts and problems where the identification of optimal level shifts in a time series is the main goal. Extensive simulation results are presented and a real-life example is given to illustrate the implementation of the developed scheme in practice and to unfold its capabilities. Concluding remarks and suggestions for future research are also provided.

Multiple Change Point Detection in Reduced Rank High Dimensional Vector Autoregressive Models

ABSTRACT We study the problem of detecting and locating change points in high-dimensional Vector Autoregressive (VAR) models, whose transition matrices exhibit low rank plus sparse structure. We first address the problem of detecting a single change point using an exhaustive search algorithm and establish a finite sample error bound for its accuracy. Next, we extend the results to the case of multiple change points that can grow as a function of the sample size. Their detection is based on a two-step algorithm, wherein the first step, an exhaustive search for a candidate change point is employed for overlapping windows, and subsequently a backward elimination procedure is used to screen out redundant candidates. The two-step strategy yields consistent estimates of the number and the locations of the change points. To reduce computation cost, we also investigate conditions under which a surrogate VAR model with a weakly sparse transition matrix can accurately estimate the change points and their locations for data generated by the original model. This work also addresses and resolves a number of novel technical challenges posed by the nature of the VAR models under consideration. The effectiveness of the proposed algorithms and methodology is illustrated on both synthetic and two real datasets. Supplementary materials for this article are available online.

A Weighted Error Distance Metrics (WEDM) for Performance Evaluation on Multiple Change-Point (MCP) Detection in Synthetic Time Series.

Change-point detection (CPD) is to find abrupt changes in time-series data. Various computational algorithms have been developed for CPD applications. To compare the different CPD models, many performance metrics have been introduced to evaluate the algorithms. Each of the previous evaluation methods measures the different aspects of the methods. Based on the existing weighted error distance (WED) method on single change-point (CP) detection, a novel WED metrics (WEDM) was proposed to evaluate the overall performance of a CPD model across not only repetitive tests on single CP detection, but also successive tests on multiple change-point (MCP) detection on synthetic time series under the random slide window (RSW) and fixed slide window (FSW) frameworks. In the proposed WEDM method, a concept of normalized error distance was introduced that allows comparisons of the distance between the estimated change-point (eCP) position and the target change point (tCP) in the synthetic time series. In the successive MCPs detection, the proposed WEDM method first divides the original time-series sample into a series of data segments in terms of the assigned tCPs set and then calculates a normalized error distance (NED) value for each segment. Next, our WEDM presents the frequency and WED distribution of the resultant eCPs from all data segments in the normalized positive-error distance (NPED) and the normalized negative-error distance (NNED) intervals in the same coordinates. Last, the mean WED (MWED) and MWTD (1-MWED) were obtained and then dealt with as important performance evaluation indexes. Based on the synthetic datasets in the Matlab platform, repetitive tests on single CP detection were executed by using different CPD models, including ternary search tree (TST), binary search tree (BST), Kolmogorov–Smirnov (KS) tests, t-tests (T), and singular spectrum analysis (SSA) algorithms. Meanwhile, successive tests on MCPs detection were implemented under the fixed slide window (FSW) and random slide window (RSW) frameworks. These CPD models mentioned above were evaluated in terms of our WED metrics, together with supplementary indexes for evaluating the convergence of different CPD models, including rates of hit, miss, error, and computing time, respectively. The experimental results showed the value of this WEDM method.

Bayesian nonparametric change point detection for multivariate time series with missing observations

A time series is a commonly observed type of data, and it is analyzed in several ways in real applications. Within the possible analysis, change point detection is one of the crucial inferential targets for studying the behavior of a time series. We consider a multiple change point detection model for a multivariate time series. Among the possible approaches to perform multiple change point detection, we propose an extension to the multivariate case of one of the main state-of-the-art approaches, working in a Bayesian nonparametric framework. We combine a combinatorial prior distribution, which relies on a model-based clustering approach to detect the change points, with a multivariate kernel for time-dependent realizations in a general fashion. We further extend the model to the case of missing observations and derive opportune quantities to perform data imputation. Thereafter, we investigate the properties of the multivariate model with an extensive simulation study, and we apply the model to perform change point detection in two real data applications.

Bayesian Quickest Detection of Propagating Spatial Events

Rapid detection of spatial events that propagate across a sensor network is of wide interest in many modern applications. In particular, in communications, radar, IoT, environmental monitoring, and biosurveillance, we may observe propagating fields or particles. In this paper, we propose Bayesian sequential single and multiple change-point detection procedures for the rapid detection of such phenomena. Using a dynamic programming framework we derive the structure of the optimal single-event quickest detection procedure, which minimizes the average detection delay (ADD) subject to a false alarm probability upper bound. The multi-sensor system configuration is arbitrary and sensors may be mobile. In the rare event regime, the optimal procedure converges to a more practical threshold test on the posterior probability of the change point. A convenient recursive computation of this posterior probability is derived by using the propagation characteristics of the spatial event. The ADD of the posterior probability threshold test is analyzed in the asymptotic regime, and specific analysis is conducted in the setting of detecting random Gaussian signals affected by path loss. Then, we show how the proposed procedure is easy to extend for detecting multiple propagating spatial events in parallel in a multiple hypothesis testing setting. A method that provides strict false discovery rate (FDR) control is proposed. In the simulation section, it is demonstrated that exploiting the spatial properties of the event decreases the ADD compared to procedures that do not utilize this information, even under model mismatch.

Multi-Threshold Structural Equation Model

In this article, we consider the instrumental variable estimation for causal regression parameters with multiple unknown structural changes across subpopulations. We propose a multiple change point detection method to determine the number of thresholds and estimate the threshold locations in the two-stage least square procedure. After identifying the estimated threshold locations, we use the Wald method to estimate the parameters of interest, that is, the regression coefficients of the endogenous variable. Based on some technical assumptions, we carefully establish the consistency of estimated parameters and the asymptotic normality of causal coefficients. Simulation studies are included to examine the performance of the proposed method. Finally, our method is illustrated via an application of the Philippine farm households data for which some new findings are discovered.

A novel RSW&TST framework of MCPs detection for abnormal pattern recognition on large-scale time series and pathological signals in epilepsy.

To quickly and efficiently recognize abnormal patterns from large-scale time series and pathological signals in epilepsy, this paper presents here a preliminary RSW&TST framework for Multiple Change-Points (MCPs) detection based on the Random Slide Window (RSW) and Trigeminal Search Tree (TST) methods. To avoid the remaining local optima, the proposed framework applies a random strategy for selecting the size of each slide window from a predefined collection, in terms of data feature and experimental knowledge. For each data segment to be diagnosed in a current slide window, an optimal path towards a potential change point is detected by TST methods from the top root to leaf nodes with O(log3(N)). Then, the resulting MCPs vector is assembled by means of TST-based single CP detection on data segments within each of the slide windows. In our experiments, the RSW&TST framework was tested by using large-scale synthetic time series, and then its performance was evaluated by comparing it with existing binary search tree (BST), Kolmogorov-Smirnov (KS)-statistics, and T-test under the fixed slide window (FSW) approach, as well as the integrated method of wild binary segmentation and CUSUM test (WBS&CUSUM). The simulation results indicate that our RSW&TST is both more efficient and effective, with a higher hit rate, shorter computing time, and lower missed, error and redundancy rates. When the proposed RSW&TST framework is executed for MCPs detection on pathological ECG (electrocardiogram)/EEG (electroencephalogram) recordings of people in epileptic states, the abnormal patterns are roughly recognized in terms of the number and position of the resultant MCPs. Furthermore, the severity of epilepsy is roughly analyzed based on the strength and period of signal fluctuations among multiple change points in the stage of a sudden epileptic attack. The purpose of our RSW&TST framework is to provide an encouraging platform for abnormal pattern recognition through MCPs detection on large-scale time series quickly and efficiently.

Detection of multiple change points in a Weibull accelerated failure time model using sequential testing

AbstractWith improvements to cancer diagnoses and treatments, incidences and mortality rates have changed. However, the most commonly used analysis methods do not account for such distributional changes. In survival analysis, change point problems can concern a shift in a distribution for a set of time‐ordered observations, potentially under censoring or truncation. We propose a sequential testing approach for detecting multiple change points in the Weibull accelerated failure time model, since this is sufficiently flexible to accommodate increasing, decreasing, or constant hazard rates and is also the only continuous distribution for which the accelerated failure time model can be reparameterized as a proportional hazards model. Our sequential testing procedure does not require the number of change points to be known; this information is instead inferred from the data. We conduct a simulation study to show that the method accurately detects change points and estimates the model. The numerical results along with real data applications demonstrate that our proposed method can detect change points in the hazard rate.

Scalable multiple changepoint detection for functional data sequences

AbstractWe propose the multiple changepoint isolation (MCI) method for detecting multiple changes in the mean and covariance of a functional process. We first introduce a pair of projections to represent the variability “between” and “within” the functional observations. We then present an augmented fused lasso procedure to split the projections into multiple regions robustly. These regions act to isolate each changepoint away from the others so that the powerful univariate CUSUM statistic can be applied region‐wise to identify the changepoints. Simulations show that our method accurately detects the number and locations of changepoints under many different scenarios. These include light and heavy tailed data, data with symmetric and skewed distributions, sparsely and densely sampled changepoints, and mean and covariance changes. We show that our method outperforms a recent multiple functional changepoint detector and several univariate changepoint detectors applied to our proposed projections. We also show that MCI is more robust than existing approaches and scales linearly with sample size. Finally, we demonstrate our method on a large time series of water vapor mixing ratio profiles from atmospheric emitted radiance interferometer measurements.

Strong convergence rates of multiple change-point estimator for ρ-mixing sequence

We consider the detection of multiple change-points for ρ-mixing sequence. A CUSUM-type method is proposed to estimate the multiple mean change-points and the strong consistency of the multiple mean change-point estimation is proved when the number of change-point is known. Furthermore, we establish the strong convergence rate of this change-point estimation under mild conditions and simulations of multiple change-point detection are also presented.

A novel group VIF regression for group variable selection with application to multiple change-point detection

ABSTRACT In this paper, we propose a novel group variance inflation factor (VIF) regression model for tackling large data sets where data follows a grouped structure. Unlike classical penalized methods, this approach can perform group variable selection in a sparse model, which is quite different from the classical penalized methods. We further adapt the proposed method associated with a two-stage procedure for detecting multiple change-point in linear models. We carry out extensive simulation studies to show that the proposed group variable selection and change-point detection methods are stable and efficient. Finally, we provide two real data examples, including a body fat data set and an air pollution data set, to illustrate the performance of our algorithms in group selection and change-point detection.

Efficient change-points detection for genomic sequences via cumulative segmented regression.

Knowing the number and the exact locations of multiple change points in genomic sequences serves several biological needs. The cumulative-segmented algorithm (cumSeg) has been recently proposed as a computationally efficient approach for multiple change-points detection, which is based on a simple transformation of data and provides results quite robust to model mis-specifications. However, the errors are also accumulated in the transformed model so that heteroscedasticity and serial correlation will show up, and thus the variations of the estimated change points will be quite different, while the locations of the change points should be of the same importance in the original genomic sequences. In this study, we develop two new change-points detection procedures in the framework of cumulative segmented regression. Simulations reveal that the proposed methods not only improve the efficiency of each change point estimator substantially but also provide the estimators with similar variations for all the change points. By applying these proposed algorithms to Coriel and SNP genotyping data, we illustrate their performance on detecting copy number variations. The proposed algorithms are implemented in R program and the codes are provided in the online supplementary material. Supplementary data are available at Bioinformatics online.