Real time change-point detection in a nonlinear quantile model

Most 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.     

Change-Point Detection in Binomial Thinning Processes,with Applications in Epidemiology

Abstract

Most of the classical change-point detection schemes are designed for the sequences of independent and identically distributed (i.i.d.) random variables. In this article, motivated by the outbreak of 2009 H1N1 pandemic influenza, we develop change-point detection procedures for the susceptible–infected–recovered (SIR) epidemic model, where a change-point in the infection rate parameter signifies either the beginning or the end of an epidemic trend.

The considered model falls into a general class of binomial thinning processes, which is a Markov chain. The cumulative sum (CUSUM) change-point detection procedure is developed for this class, and its performance is evaluated. Apparently, the CUSUM stopping rule is no longer optimal for this non-i.i.d. case. It can be improved by introducing a non constant adaptive threshold. The resulting modified scheme attains a shorter mean delay and at the same time a longer expected time of a false alarm, given that a false alarm eventually occurs.

Proposed detection procedures are applied to the 2001–2012 influenza data published by the Centers for Disease Control and Prevention.     

Sequential Detection of Transient Changes

Abstract

This article addresses the transient change detection problem. It is assumed that a change occurs at an unknown (but nonrandom) change-point and the duration of post-change period is finite and known. A latent detection—that is, a detection that occurs after signal disappearance—is considered as a missed detection. A new optimality criterion adapted to the detection of transient changes involves the minimization of the worst-case probability of missed detection under constraint on the false alarm rate for a given period. A suboptimal sequential transient change detection algorithm is proposed. It is based on a window-limited cumulative sum (CUSUM) test. An upper bound for the worst-case probability of missed detection and a lower and an upper bound for the false alarm rate are proposed. Based on these bounds, the window-limited CUSUM test is optimized with respect to the proposed criterion. The developed algorithm and theoretical findings are applied to drinking water distribution network monitoring.     

Testing for Change in Long‐Memory Stochastic Volatility Time Series

In this article, change‐point problems for long‐memory stochastic volatility (LMSV) models are considered. A general testing problem which includes various alternative hypotheses is discussed. Under the hypothesis of stationarity the limiting behavior of CUSUM‐ and Wilcoxon‐type test statistics is derived. In this context, a limit theorem for the two‐parameter empirical process of LMSV time series is proved. In particular, it is shown that the asymptotic distribution of CUSUM test statistics may not be affected by long memory, unlike Wilcoxon test statistics which are typically influenced by long‐range dependence. To avoid the estimation of nuisance parameters in applications, the usage of self‐normalized test statistics is proposed. The theoretical results are accompanied by an analysis of Standard & Poor's 500 daily closing indices with respect to structural changes and by simulation studies which characterize the finite sample behavior of the considered testing procedures when testing for changes in mean and in variance.     

Structural breaks in time series

This paper gives an account of some of the recent work on structural breaks in time series models. In particular, we show how procedures based on the popular cumulative sum, CUSUM, statistics can be modified to work also for data exhibiting serial dependence. Both structural breaks in the unconditional and conditional mean as well as in the variance and covariance/correlation structure are covered. CUSUM procedures are nonparametric by design. If the data allows for parametric modeling, we demonstrate how likelihood approaches may be utilized to recover structural breaks. The estimation of multiple structural breaks is discussed. Furthermore, we cover how one can disentangle structural breaks (in the mean and/or the variance) on one hand and long memory or unit roots on the other. Several new lines of research are briefly mentioned.     

Asymptotically Pointwise Optimal Change Detection in Multiple Channels

Abstract

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.     

ON THE PARTIAL SUMS OF RESIDUALS IN AUTOREGRESSIVE AND MOVING AVERAGE MODELS

Abstract. The limiting process of partial sums of residuals in stationary and invertible autoregressive moving-average models is studied. It is shown that the partial sums converge to a standard Brownian motion under the assumptions that estimators of unknown parameters are root- n consistent and that innovations are independent and identically distributed random variables with zero mean and finite variance or, more generally, are martingale differences with moment restrictions specified in Theorem 1. Applications for goodness-of-fit and change-point problems are considered. The use of residuals for constructing nonparametric density estimation is discussed.     

Estimation of common change point and isolation of changed panels after sequential detection

Abstract

Quick detection of common changes is critical in sequential monitoring of multistream data where a common change is a change that only occurs in a portion of panels. After a common change is detected by using a combined cumulative sum Shiryaev-Roberts (CUSUM-SR) procedure, we first study the joint distribution for values of the CUSUM process and the estimated delay detection time for the unchanged panels. A Benjamini-Hochberg (BH) method using the asymptotic exponential property for the CUSUM process is developed to isolate the changed panels with control on the false discovery rate (FDR). The common change point is then estimated based on the isolated changed panels. Simulation results show that the proposed method can also control the false non-discovery rate (FNR) by properly selecting the FDR.     

SEQUENTIAL METHODS FOR DETECTING CHANGES IN THE VARIANCE OF ECONOMIC TIME SERIES

In this paper EWMA charts and CUSUM charts are introduced for detecting changes in the variance of a GARCH process. The moments of the EWMA statistics are calculated. They permit a better understanding of the underlying control procedure. In an extensive simulation study all control schemes are compared with each other. “Optimal” smoothing parameters and “optimal” reference values are tabulated. It is shown how these charts can be applied to monitor stock market returns.     

A decision theoretic approach to change point estimation for binomial CUSUM control charts

Detecting when the process has changed is a classical problem in sequential analysis and is an important practical issue in statistical process control. This article is concerned about the binomial cumulative sum (CUSUM) control chart, which is extensively applied to industrial process control, health care, public health surveillance, and other fields. For the binomial CUSUM, a maximum likelihood estimator has been proposed to estimate the change point. In our article, following a decision theoretic approach, we develop a new estimator that aims to improve the existing methods. For interval estimation, we propose a parametric bootstrap procedure to construct the confidence set of the change point. We compare our proposed method with the maximum likelihood estimator and Page's last zero estimator in terms of mean squared error by simulations. We find that the proposed method gives more unbiased and robust results than the existing procedures under various parameter designs. We analyze jewelry manufacturing data for illustration.     

Kullback-Leibler Approach to CUSUM Quickest Detection Rule for Markovian Time Series

Abstract

Optimality properties of decision procedures are studied for the quickest detection of a change-point of parameters in autoregressive and other Markov type sequences. The limit of the normalized conditional log-likelihood ratios is shown to exist for Markov chains satisfying the ergodic theorem of information theory. Then closed-form expressions for this limit are derived by making use of the time average rate of Kullback-Leibler divergence. The good properties of the detection procedures based on a sequential analysis approach are proven to hold thanks to geometric ergodicity properties of the observation processes. In particular, the window-limited CUSUM rule is shown to be optimal for detecting the disruption point in autoregressive models. Sparre Andersen models are specifically studied.     

Likelihood inference for discriminating between long‐memory and change‐point models

We develop a likelihood ratio (LR) test procedure for discriminating between a short‐memory time series with a change‐point (CP) and a long‐memory (LM) time series. Under the null hypothesis, the time series consists of two segments of short‐memory time series with different means and possibly different covariance functions. The location of the shift in the mean is unknown. Under the alternative, the time series has no shift in mean but rather is LM. The LR statistic is defined as the normalized log‐ratio of the Whittle likelihood between the CP model and the LM model, which is asymptotically normally distributed under the null. The LR test provides a parametric alternative to the CUSUM test proposed by Berkes et al. (2006) . Moreover, the LR test is more general than the CUSUM test in the sense that it is applicable to changes in other marginal or dependence features other than a change‐in‐mean. We show its good performance in simulations and apply it to two data examples.     

Detection of Changes of Multiple Poisson Processes Monitored at Discrete Time Points Where the Arrival Rates are Unknown

Abstract

We observe a Poisson process in several categories where the arrival rates in each category change at some unknown integer. For some of these categories the arrival rates increase, whereas in other categores the arrival rates decrease. The point at which the process changes may be different for each category. We assume that both the arrival rates for each category as well as the change-point are unknown. We develop procedures for detecting when a change has occurred in at least one of these categories. We provide some numerical results to illustrate the effectiveness of the detection procedures.     

The Generalized Synthetic Chart

Abstract

The Generalized Synthetic chart is presented and mathematical expressions for its average run length and variance of the run length are developed. The methodology is applied to the EWMA and CUSUM charts and near-optimization procedures are discussed. The synthetic EWMA and CUSUM charts are compared with their standard counterparts, the original synthetic chart, and the Shewhart chart. Significant improvements in detecting power are reported.     

BIAS CALCULATIONS FOR ADAPTIVE URN DESIGNS

A clinical trial model is considered in which two treatments with immediate binary responses are to be compared. An adaptive urn design is used to assign patients to the treatments. The bias and variance of the maximum likelihood estimators of the probabilities of success are derived by differentiating the fundamental identity of sequential analysis. By embedding the design in a continuous-time process, probability generating functions are then calculated to obtain approximations for the bias and variance. Simulation is used to assess the accuracy of the approximations. It is shown that the bias cannot be ignored, and that the adaptive rules which are subcritical in nature have the most mathematically tractable bias and are the least variable. Methods for correcting for the bias are also addressed.     

Robust Wilcoxon‐Type Estimation of Change‐Point Location Under Short‐Range Dependence

We introduce a robust estimator of the location parameter for the change‐point in the mean based on Wilcoxon statistic and establish its consistency for L₁ near‐epoch dependent processes. It is shown that the consistency rate depends on the magnitude of the change. A simulation study is performed to evaluate the finite sample properties of the Wilcoxon‐type estimator under Gaussianity as well as under heavy‐tailed distributions and disturbances by outliers, and to compare it with a CUSUM‐type estimator. It shows that the Wilcoxon‐type estimator is equivalent to the CUSUM‐type estimator under Gaussianity but outperforms it in the presence of heavy tails or outliers in the data.     

An online change detection test for parametric discrete-time stochastic processes

Detecting a change as fast as possible in an observed stochastic process is an important task. In this article, an online procedure is presented to detect changes in the parameter of general discrete-time parametric stochastic processes. As examples, regression models, autoregressive processes, and Galton–Watson processes are investigated. The test is called cumulative sum (CUSUM) type because it is based on the cumulated sums of the estimates of certain martingale difference sequences belonging to the process. In case of a single change alternative hypothesis, the procedure is examined in terms of consistency. Due to the online manner, the time of change can also be estimated.     

Quasi-stationary biases of change point and change magnitude estimation after sequential cusum test

Assuming the observation process is a Brownian motion with the drift parameter subject to sudden change, estimations of the change point and change magnitude after the sequential CUSUM test are proposed and investigated. By assuming that the change occures far away from 0, the biases of the estimations conditioning on that the change is detected are obtained as the control limit approaches infinity     

Sequential algorithms for moving anomaly detection in networks

Abstract

The problem of quickest moving anomaly detection in networks is studied. Initially, the observations are generated according to a prechange distribution. At some unknown but deterministic time, an anomaly emerges in the network. At each time instant, one node is affected by the anomaly and receives data from a post-change distribution. The anomaly moves across the network, and the node that it affects changes with time. However, the trajectory of the moving anomaly is assumed to be unknown. A discrete-time Markov chain is employed to model the unknown trajectory of the moving anomaly in the network. A windowed generalized likelihood ratio–based test is constructed and is shown to be asymptotically optimal. Other detection algorithms including the dynamic Shiryaev-Roberts test, a quickest change detection algorithm with recursive change point estimation, and a mixture cumulative sum (CUSUM) algorithm are also developed for this problem. Lower bounds on the mean time to false alarm are developed. Numerical results are further provided to compare their performances.     

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.