Multidecision Quickest Change-Point Detection: Previous Achievements and Open Problems

Abstract

The following multidecision quickest detection problem, which is of importance for a variety of applications, is considered. There are N populations that are either statistically identical or where the change occurs in one of them at an unknown point in time. Alternatively, there may be N “isolated” points/hypotheses associated with a change. It is necessary to detect the change in distribution as soon as possible and indicate which population is “corrupted” or which hypothesis is true after a change occurs. Both the false alarm rate and misidentification rate should be controlled by given (usually low) levels. We discuss performance of natural multihypothesis/multipopulation generalizations of the Page and Shiryaev-Roberts procedures, including certain asymptotically optimal properties of these tests when both the false alarm and the misidentification rates are low. Specifically, we show that under certain conditions the proposed multihypothesis detection-identification procedures asymptotically minimize the trade-off between any positive moment of the detection lag and the false alarm/misclassification rates in the worst-case scenario. At the same time, the corresponding sequential detection-identification procedures are computationally simple and can be easily implemented online in a variety of applications such as rapid detection of intrusions in large-scale distributed computer networks, target detection in cluttered environment, and detection of terrorist' malicious activity. Limitations of the existing and proposed solutions to this challenging problem are also discussed.     

Monitoring a Poisson process subject to gradual changes in the arrival rates

Abstract

We look at a Poisson process where the arrival rates change from a known λ₁ to a known λ₂. Whereas in most of the literature the change-point is abrupt, we model the more realistic assumption that states that the change happens gradually over a period of time η where η is known. We calculate the probability that the change has started and completed. We also look at optimal stopping rules assuming that there is a cost for a false alarm and a cost per time unit to stop early. We conclude with some numerical results.     

Data-Efficient Quickest Change Detection with On–Off Observation Control

Abstract

In this article we extend Shiryaev's quickest change detection formulation by also accounting for the cost of observations used before the change point. The observation cost is captured through the average number of observations used in the detection process before the change occurs. The objective is to select an on–off observation control policy that decides whether or not to take a given observation, along with the stopping time at which the change is declared, to minimize the average detection delay, subject to constraints on both the probability of false alarm and the observation cost. By considering a Lagrangian relaxation of the constraint problem and using dynamic programming arguments, we obtain an a posteriori probability-based two-threshold algorithm that is a generalized version of the classical Shiryaev algorithm. We provide an asymptotic analysis of the two-threshold algorithm and show that the algorithm is asymptotically optimal—that is, the performance of the two-threshold algorithm approaches that of the Shiryaev algorithm—for a fixed observation cost, as the probability of false alarm goes to zero. We also show, using simulations, that the two-threshold algorithm has good observation cost-delay trade-off curves and provides significant reduction in observation cost compared to the naïve approach of fractional sampling, where samples are skipped randomly. Our analysis reveals that, for practical choices of constraints, the two thresholds can be set independent of each other: one based on the constraint of false alarm and another based on the observation cost constraint alone.     

Is Average Run Length to False Alarm Always an Informative Criterion?

Abstract

Apart from Bayesian approaches, the average run length (ARL) to false alarm has always been seen as the natural performance criterion for quantifying the propensity of a detection scheme to make false alarms, and no researchers seem to have questioned this on grounds that it does not always apply. In this article, we show that in the change-point problem with mixture prechange models, detection schemes with finite detection delays can have infinite ARLs to false alarm. We also discuss the implication of our results on the change-point problem with either exchangeable prechange models or hidden Markov models. Alternative minimax formulations with different false alarm criteria are proposed.     

Bootstrap procedures for detecting multiple persistence shifts in heteroskedastic time series

This article proposes new bootstrap procedures for detecting multiple persistence shifts in a time series driven by non-stationary volatility. The assumed volatility process can accommodate discrete breaks, smooth transition variation as well as trending volatility. We develop wild bootstrap sup-Wald tests of the null hypothesis that the process is either stationary [I(0)] or has a unit root [I(1)] throughout the sample. We also propose a sequential procedure to estimate the number of persistence breaks based on ordering the regime-specific bootstrap p-values. The asymptotic validity of the advocated procedures is established both under the null of stability and a variety of persistence change alternatives. A comparison with existing tests that assume homoskedasticity illustrates the finite sample improvements offered by our methods. An application to OECD inflation rates highlights the empirical relevance of the proposed approach and weakens the case for persistence change relative to existing procedures.     

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.     

Some Nonparametric Tests for Change-Point Detection Based on the ℙ-ℙ and ℚ-ℚ Plot Processes

Abstract

We propose nonparametric procedures for testing change-point by using the ?-? and ?-? plots processes. The limiting distributions of the proposed statistics are characterized under the null hypothesis of no change and also under contiguous alternatives. We give an estimator of the change-point coefficient and obtain its strong consistency. We introduce the bootstrapped version of ?-? and ?-? processes, requiring the estimation of quantile density, and obtain their limiting laws. Finally, we propose and investigate the exchangeable bootstrap of the empirical ?-? plot and ?-? plot processes which avoids the problem of the estimation of quantile density, which is of its own interest. These results are used for calculating p-values of the proposed test statistics. Emphasis is placed on the explanation of the strong approximation methodology.     

Estimating a change point in the long memory parameter

We propose an estimator of change point in the long memory parameter d of an ARFIMA(p, d, q) process using the sup Wald test. We derive the consistency and the rate of convergence of the estimator for the time of change. The convergence rate of our change point estimator depends on the magnitude of a shift. Furthermore, we obtain the limiting distribution of our change point estimator without depending on the distribution of the process. Therefore, we can construct confidence intervals for the change point. Simulations show the validity of the asymptotic theory of our estimator if the sample size is large enough. We apply our change point estimator to the yearly Nile river minimum water level.     

Online Change Detection for a Poisson Process with a Phase-Type Change-Time Prior Distribution

Abstract

We consider a change detection problem in which the arrival rate of a Poisson process changes suddenly at some unknown and unobservable disorder time. It is assumed that the prior distribution of the disorder time is known. The objective is to detect the disorder time with an online detection rule (a stopping time) in a way that balances the frequency of false alarms and detection delay. So far in the study of this problem, the prior distribution of the disorder time is taken to be exponential distribution for analytical tractability. Here, we will take the prior distribution to be a phase-type distribution, which is the distribution of the absorption time of a continuous time Markov chain with a finite state space. We find an optimal stopping rule for this general case. We illustrate our findings on two numerical examples.     

Difference Equations for the Higher Order Moments and Cumulants of the INAR(p) Model

Abstract. Here we obtain difference equations for the higher order moments and cumulants of a time series {X_t} satisfying an INAR(p) model. These equations are similar to the difference equations for the higher order moments and cumulants of the bilinear time series model. We obtain the spectral and bispectral density functions for the INAR(p) process in state–space form, thus characterizing it in the frequency domain. We consider a frequency domain method – the Whittle criterion – to estimate the parameters of the INAR(p) model and illustrate it with the series of the number of epilepsy seizures of a patient.     

Detection of Stationary Errors in Multiple Regressions with Integrated Regressors and Cointegration

Abstract

This article studies the problem of detecting sequentially stationary error terms in a multiple regression model with a difference-stationary multivariate I(1)-regressor. The detection of cointegration is covered as a special case. We provide the asymptotic distribution theory for a monitoring procedure that is related to a well-known nonparametric unit root test statistic calculated from sequentially updated least squares residuals. Functional limit theorems for the corresponding sequential processes and central limit theorems for the detectors used to raise an alarm are established under the no-change null hypothesis as well as under change-point models covering a change to I(0)-errors and a change of the regression coefficients as well. We also discuss extensions to the case that continuous time processes are discretely sampled to obtain the data allowing to apply the procedures to high-frequency data as well. Our results show that we can handle the infill asymptotics assuming that nonstationary continuous-time processes such as semimartingales are discretely observed, by virtue of the general assumptions that we impose. The finite sample properties are investigated by a simulation study.     

Asymptotically Optimal Quickest Change Detection in Distributed Sensor Systems

Abstract

In the standard formulation of the quickest change-point detection problem, a sequence of observations, whose distribution changes at some unknown point in time, is available to a decision maker, and the goal is to detect this change as quickly as possible, subject to false alarm constraints. In this paper, we study the quickest change detection problem in the setting where the information available for decision-making is distributed across a set of geographically separated sensors, and only a compressed version of observations in sensors may be used for final decision-making due to communication bandwidth constraints. We consider the minimax, uniform, and Bayesian versions of the optimization problem, and we present asymptotically optimal decentralized quickest change detection procedures for two scenarios. In the first scenario, the sensors send quantized versions of their observations to a fusion center where the change detection is performed based on all the sensor messages. In the second scenario, the sensors perform local change detection and send their final decisions to the fusion center for combining. We show that our decentralized procedures for the latter scenario have the same first-order asymptotic performance as the corresponding centralized procedures that have access to all of the sensor observations. We also present simulation results for examples involving Gaussian and Poisson observations. These examples show that although the procedures with local decisions are globally asymptotically optimal as the false alarm rate (or probability) goes to zero, they perform worse than the corresponding decentralized procedures with binary quantization at the sensors, unless the false alarm rate (or probability) is unreasonably small.     

Alternate poly(amide-imide)s based on 1,4-and 1,3-bis(4-trimellitimido)benzene and their mixtures

Several series of alternate poly(amide-imide)s [P(A-alt-I)s] were synthesized by aromatic dicarboxylic acid (I- p or I- m), which was prepared by the condensation of p-phenylenediamine (or m-phenylenediamine), trimellitic anhydride, and various aromatic diamines by means of direct polycondensation. A diimide-diacid (I- p) with a p-phenylene group was used to synthesize P(A-alt-I)s III, and P(A-alt-I)s IV were synthesized by a diimide-diacid (I- m) prepared from m-phenylenediamine. Another series of P(A-alt-I)s V was synthesized from both I- p and I- m (1/1 mole) with various diamines. Polymers of series III have low inherent viscosities and limited solubility, but polymers of series IV have high degrees of polymerization. Series V copolycondensated from I- p and I- m has improved solubility and degrees of polymerization relative to series III. The degree of crystallinity was found to be III > V > IV. Glass transition temperatures for most of series III were not observed below 400 °C, and those of series IV and V were in the range of 238–325 °C and 262–328 °C, respectively. The 10% weight loss temperatures in nitrogen or in air of these three series are all in the range of 482–582 °C. Because series V has limited solubility for casting into films from DMAc solutions, two diamines were selected to synthesize series VI by changing the I- p/I- m ratio. Solubility was improved when the content of I- p in diimide-diacid was less than 15%, and the degree of crystallinity reduced as the content of I- p in diimide-diacid decreased. Polymers containing a few I- p showed an increase in the initial modulus.     

pKa-Values of Guaiacyl and Syringyl Phenols Related to Lignin

Abstract

The literature relating to the pK_a -values of guaiacyl- and syringyl-derived phenols has been thoroughly surveyed and summarized. In addition, the pK_a -values of a number of guaiacyl, syringyl and other phenols related to lignin have been determined using a spectrophotometric method combined with multivariate evaluation. Differences and similarities between the acidities of a number of substances are extensively discussed. The pK_a -value strongly affects the delignification during pulping, bleaching and leaching of lignin during pulp washing.     

An Accurate Method for Determining the Pre-Change Run Length Distribution of the Generalized Shiryaev-Roberts Detection Procedure

Abstract

Change-of-measure is a powerful technique in wide use across statistics, probability, and analysis. Particularly known as Wald's likelihood ratio identity, the technique enabled the proof of a number of exact and asymptotic optimality results pertaining to the problem of quickest change-point detection. Within the latter problem's context we apply the technique to develop a numerical method to compute the generalized Shiryaev–Roberts (GSR) detection procedure's pre-change run length distribution. Specifically, the method is based on the integral equations approach and uses the collocation framework with the basis functions chosen to exploit a certain change-of-measure identity and a specific martingale property of the GSR procedure's detection statistic. As a result, the method's accuracy and robustness improve substantially, even though the method's theoretical rate of convergence is shown to be merely quadratic. A tight upper bound on the method's error is supplied as well. The method is not restricted to a particular data distribution or to a specific value of the GSR detection statistic's head start. To conclude, we offer a case study to demonstrate the proposed method at work, drawing particular attention to the method's accuracy and its robustness with respect to three factors: (1) partition size (rough vs. fine), (2) change magnitude (faint vs. contrast), and (3) average run length (ARL) to false alarm level (low vs. high). Specifically, assuming independent standard Gaussian observations undergoing a surge in the mean, we employ the method to study the GSR procedure's run length's pre-change distribution, its average (i.e., the usual ARL to false alarm), and its standard deviation. As expected from the theoretical analysis, the method's high accuracy and robustness with respect to the foregoing three factors are confirmed experimentally. We also comment on extending the method to handle other performance measures and other procedures.     

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.     

Quickest detection of an accumulated state-dependent change point

Abstract

Motivated by the practical investigation of a state-dependent quickest detection problem in continuous time, especially for Brownian observations, we propose an asymptotic scheme in discrete time called a quickest detection scheme of an accumulated state-dependent change point. Here the state-dependent means that the priori probability of the change point depends on the current state. We reduce the problem to finding an optimal stopping time of a vector-valued Markov process. We illustrate the scheme via a numerical example.     

State space models for time series with patches of unusual observations

Abstract. An alternative to leave‐k‐out diagnostics for detecting patches of outlying points in time series is developed. We propose that unusual behaviour should be modelled by the addition of shocks. By including shocks in the transition equation of a state space model, we admit the possibility of a persistent change associated with a patch of outliers. Persistent change may take the form of a level shift or a change in seasonal pattern. We provide an efficient mechanism for computing diagnostic statistics associated with the addition of k shocks using a simple adaptation of the Kalman filter. Statistics for detecting unspecified patterns of shocks and an interpretation of the output of the associated smoothing algorithm are derived. Illustrations using real series are given.     

Two-Stage Procedures for High-Dimensional Data

Abstract

In this article, we consider a variety of inference problems for high-dimensional data. The purpose of this article is to suggest directions for future research and possible solutions about p ? n problems by using new types of two-stage estimation methodologies. This is the first attempt to apply sequential analysis to high-dimensional statistical inference ensuring prespecified accuracy. We offer the sample size determination for inference problems by creating new types of multivariate two-stage procedures. To develop theory and methodologies, the most important and basic idea is the asymptotic normality when p → ∞. By developing asymptotic normality when p → ∞, we first give (a) a given-bandwidth confidence region for the square loss. In addition, we give (b) a two-sample test to assure prespecified size and power simultaneously together with (c) an equality-test procedure for two covariance matrices. We also give (d) a two-stage discriminant procedure that controls misclassification rates being no more than a prespecified value. Moreover, we propose (e) a two-stage variable selection procedure that provides screening of variables in the first stage and selects a significant set of associated variables from among a set of candidate variables in the second stage. Following the variable selection procedure, we consider (f) variable selection for high-dimensional regression to compare favorably with the lasso in terms of the assurance of accuracy and the computational cost. Further, we consider variable selection for classification and propose (g) a two-stage discriminant procedure after screening some variables. Finally, we consider (h) pathway analysis for high-dimensional data by constructing a multiple test of correlation coefficients.