Quickest Change-point Research Articles

Bayesian Quickest Change-Point Detection With an Energy Harvesting Sensor and Asymptotic Analysis

Bayesian Quickest Change-Point Detection With an Energy Harvesting Sensor and Asymptotic Analysis

Ultimate Limits for Quickest Quantum Change-Point Detection.

Detecting abrupt changes in data streams is crucial because they are often triggered by events that have important consequences if left unattended. Quickest change-point detection has become a vital sequential analysis primitive that aims at designing procedures that minimize the expected detection delay of a change subject to a bounded expected false alarm time. We put forward the quantum counterpart of this fundamental primitive on streams of quantum data. We give a lower bound on the mean minimum delay when the expected time of a false alarm is asymptotically large, under the most general quantum detection strategy, which is given by a sequence of adaptive collective (potentially weak) measurements on the growing string of quantum data. In addition, we give particular strategies based on repeated measurements on independent blocks of samples that asymptotically attain the lower bound and thereby establish the ultimate quantum limit for quickest change-point detection. Finally, we discuss online change-point detection in quantum channels.

Bayesian quickest changepoint detection approach to partially observe Markov processes

Bayesian quickest changepoint detection approach to partially observe Markov processes

Quickest change point detection with multiple postchange models

We study the sequential quickest change point detection for systems with multiple possible postchange models. A change point is the time instant at which the distribution of a random process changes. In many practical applications, the prechange model can be easily obtained, yet the postchange distribution is unknown due to the unexpected nature of the change. In this article, we consider the case that the postchange model is from a finite set of possible models. The objective is to minimize the average detection delay (ADD), subject to upper bounds on the probability of false alarm (PFA). Two different quickest change detection algorithms are proposed under Bayesian and non-Bayesian settings. Under the Bayesian setting, the prior probabilities of the change point and prior probabilities of possible postchange models are assumed to be known, yet this information is not available under the non-Bayesian setting. Theoretical analysis is performed to quantify the analytical performance of the proposed algorithms in terms of exact or asymptotic bounds on PFA and ADD. It is shown through theoretical analysis that when PFA is small, both algorithms are asymptotically optimal in terms of ADD minimization for a given PFA upper bound. Numerical results demonstrate that the proposed algorithms outperform existing algorithms in the literature.

On the evaluation of an integral involving the Whittaker [formula omitted] function

The topic of interest is the evaluation of the integral I(α;β)≔∫α+∞W1,β2(x)dxx2withα∈(0,∞)andβ∈ℂsuch thatW1,β(α)=0,where Wa,b(z) denotes the Whittaker W function. The question is brought about by a certain Sturm–Liouville problem: the (discrete) spectrum of the respective Sturm–Liouville operator is captured by β=βk, k∈N, and I(α;β) gives the (positive but finite) “length” of the kth non-normalized eigenfunction. The Sturm–Liouville problem is of importance in financial mathematics, mathematical physics, stochastic processes, and quickest change-point detection. We obtain a new analytic expression for I(α;β). The expression is in the form of a (convergent) series, which can be computed numerically via successive iteration of a homogeneous linear second-order difference equation. We also offer an exact closed-form solution to the difference equation. We conclude with a curious connection of our result to the latest developments in the theory of hypergeometric functions.

On the problems of sequential statistical inference for Wiener processes with delayed observations

We study the sequential hypothesis testing and quickest change-point (or disorder) detection problems with linear delay penalty costs for observable Wiener processes under (constantly) delayed detection times. The method of proof consists of the reduction of the associated delayed optimal stopping problems for one-dimensional diffusion processes to the equivalent free-boundary problems and solution of the latter problems by means of the smooth-fit conditions. We derive closed-form expressions for the Bayesian risk functions and optimal stopping boundaries for the associated weighted likelihood ratio processes in the original problems of sequential analysis.

Numerical comparison of popular quickest changepoint detection procedures

Numerical comparison of popular quickest changepoint detection procedures

Comparative performance analysis of the Cumulative Sum chart and the Shiryaev‐Roberts procedure for detecting changes in autocorrelated data

AbstractWe consider the problem of quickest changepoint detection where the observations form a first‐order autoregressive (AR(1)) process driven by temporally independent standard white Gaussian noise. Subject to possible change are both the drift of the AR(1) process (μ) and its correlation coefficient (λ), which are both known. The change is abrupt and persistent, and of known magnitude, with |λ|<1 throughout. For this scenario, we carry out a comparative performance analysis of the popular cumulative sum (CUSUM) chart and its less well‐known but worthy competitor, ie, the Shiryaev‐Roberts (SR) procedure. Specifically, the performance is measured through Pollak's supremum (conditional) average delay to detection (SADD) constrained to a pre‐specified level of the average run length (ARL) to false alarm. Particular attention is drawn to the sensitivity of each procedure's SADD and ARL with respect to the value of λ before and after the change. The performance is studied through the solution of the respective integral renewal equations obtained via Monte Carlo simulations. The simulations are designed to estimate the sought performance metrics in an unbiased and consistent manner, and within a prescribed proportional closeness (also asymptotically). Our extensive numerical studies suggest that both the CUSUM chart and the SR procedure are asymptotically second‐order optimal, even though the CUSUM chart is found to be slightly better than the SR procedure, irrespective of the model parameters. Moreover, the existence of a worst‐case post‐change correlation parameter corresponding to the poorest detectability of the change for a given ARL to false alarm is established as well. To the best of our knowledge, this is the first time that the performance of the SR procedure is studied for autocorrelated data.

Asymptotic Near-Minimaxity of the Randomized Shiryaev--Roberts--Pollak Change-Point Detection Procedure in Continuous Time

For the classical continuous-time quickest change-point detection problem it is shown that the randomized Shiryaev--Roberts--Pollak procedure is asymptotically nearly minimax-optimal (in the sense of Pollak [Ann. Statist., 13 (1985), pp. 206--227]) in the class of randomized procedures with vanishingly small false alarm risk. The proof is explicit in that all of the relevant performance characteristics are found analytically and in a closed form. The rate of convergence to the (unknown) optimum is elucidated as well. The obtained optimality result is a one-order improvement of that previously obtained by Burnaev, Feinberg, and Shiryaev [Theory Probab. Appl., 53 (2009), pp. 519--536] for the very same problem.

Asymptotic exponentiality of the first exit time of the Shiryaev–Roberts diffusion with constant positive drift

ABSTRACTWe consider the first exit time of a Shiryaev–Roberts diffusion with constant positive drift from the interval [0,A] with A>0 fixed. We show that the moment generating function (Laplace transform) of a suitably standardized version of the first exit time converges to that of the unit-mean exponential distribution as A→+∞. The proof is explicit in that the moment generating function of the first exit time is first expressed analytically and in a closed form and then the desired limit as A→+∞ is evaluated directly. The result is of importance in the area of quickest change-point detection, and its discrete-time counterpart has been previously established—although in a different manner—by Pollak and Tartakovsky (2009a).

On the sequential testing and quickest change-point detection problems for Gaussian processes

We study the sequential hypothesis testing and quickest change-point (disorder) detection problems with linear delay penalty costs for certain observable time-inhomogeneous Gaussian diffusions and fractional Brownian motions. The method of proof consists of the reduction of the initial problems into the associated optimal stopping problems for one-dimensional time-inhomogeneous diffusion processes and the analysis of the associated free boundary problems for partial differential operators. We derive explicit estimates for the Bayesian risk functions and optimal stopping boundaries for the associated weighted likelihood ratios and obtain their exact asymptotic growth rates under large time values.

Asymptotically optimal pointwise and minimax quickest change-point detection for dependent data

We consider the quickest change-point detection problem in pointwise and minimax settings for general dependent data models. Two new classes of sequential detection procedures associated with the maximal “local” probability of a false alarm within a period of some fixed length are introduced. For these classes of detection procedures, we consider two popular risks: the expected positive part of the delay to detection and the conditional delay to detection. Under very general conditions for the observations, we show that the popular Shiryaev–Roberts procedure is asymptotically optimal, as the local probability of false alarm goes to zero, with respect to both these risks pointwise (uniformly for every possible point of change) and in the minimax sense (with respect to maximal over point of change expected detection delays). The conditions are formulated in terms of the rate of convergence in the strong law of large numbers for the log-likelihood ratios between the “change” and “no-change” hypotheses, specifically as a uniform complete convergence of the normalized log-likelihood ratio to a positive and finite number. We also develop tools and a set of sufficient conditions for verification of the uniform complete convergence for a large class of Markov processes. These tools are based on concentration inequalities for functions of Markov processes and the Meyn–Tweedie geometric ergodic theory. Finally, we check these sufficient conditions for a number of challenging examples (time series) frequently arising in applications, such as autoregression, autoregressive GARCH, etc.

Sequential detection/isolation of abrupt changes

ABSTRACTThe quickest multidecision change detection/isolation problem is the generalization of the quickest changepoint detection problem to the case of M post-change hypotheses. It is necessary to detect the change in distribution as soon as possible and to indicate which hypothesis is true after a change occurs. Both the rate of false alarms and the misidentification (false isolation) rate should be controlled by given levels. Several detection/isolation procedures that asymptotically minimize the tradeoff between the expected detection delay and the false alarm/false isolation rates in the worst-case scenario are discussed.

An Analytic Expression for the Distribution of the Generalized Shiryaev–Roberts Diffusion

We consider the quickest change-point detection problem where the aim is to detect the onset of a pre-specified drift in "live"-monitored standard Brownian motion; the change-point is assumed unknown (nonrandom). The topic of interest is the distribution of the Generalized Shryaev-Roberts (GSR) detection statistic set up to "sense" the presence of the drift. Specifically, we derive a closed-form formula for the transition probability density function (pdf) of the time-homogeneous Markov diffusion process generated by the GSR statistic when the Brownian motion under surveillance is "drift-free", i.e., in the pre-change regime; the GSR statistic's (deterministic) nonnegative headstart is assumed arbitrarily given. The transition pdf formula is found analytically, through direct solution of the respective Kolmogorov forward equation via the Fourier spectral method to achieve separation of the spacial and temporal variables. The obtained result generalizes the well-known formula for the (pre-change) stationary distribution of the GSR statistic: the latter's stationary distribution is the temporal limit of the distribution sought in this work. To conclude, we exploit the obtained formula numerically and briefly study the pre-change behavior of the GSR statistic versus three factors: (a) drift-shift magnitude, (b) time, and (c) the GSR statistic's headstart.

Exact distribution of the Generalized Shiryaev–Roberts stopping time under the minimax Brownian motion setup

ABSTRACTWe consider the quickest change-point detection problem where the aim is to detect the onset of a prespecified drift in “live”-monitored standard Brownian motion; the change-point is assumed unknown (nonrandom). The object of interest is the distribution of the stopping time associated with the Generalized Shryaev–Roberts (GSR) detection procedure set up to “sense” the presence of the drift in the Brownian motion under surveillance. Specifically, we seek the GSR stopping time's survival function (the tail probability that no alarm is triggered by the GSR procedure prior to a given point in time), and distinguish two scenarios: (a) when the drift never sets in (prechange regime) and (b) when the drift is in effect ab initio (postchange regime). Under each scenario, we obtain a closed-form formula for the respective survival function, with the GSR statistic's (deterministic) nonnegative headstart assumed arbitrarily given. The two formulae are found analytically, through direct solution of the respective Kolmogorov forward equation via the Fourier spectral method to achieve separation of the spacial and temporal variables. We then exploit the obtained formulae numerically and characterize the pre- and postchange distributions of the GSR stopping time depending on three factors: (1) magnitude of the drift, (2) detection threshold, and (3) the GSR statistic's headstart.

On robustness of the Shiryaev–Roberts change-point detection procedure under parameter misspecification in the post-change distribution

ABSTRACTThe gist of the quickest change-point detection problem is to detect the presence of a change in the statistical behavior of a series of sequentially made observations, and do so in an optimal detection-speed-versus-“false-positive”-risk manner. When optimality is understood either in the generalized Bayesian sense or as defined in Shiryaev's multi-cyclic setup, the so-called Shiryaev–Roberts (SR) detection procedure is known to be the “best one can do”, provided, however, that the observations’ pre- and post-change distributions are both fully specified. We consider a more realistic setup, viz. one where the post-change distribution is assumed known only up to a parameter, so that the latter may be misspecified. The question of interest is the sensitivity (or robustness) of the otherwise “best” SR procedure with respect to a possible misspecification of the post-change distribution parameter. To answer this question, we provide a case study where, in a specific Gaussian scenario, we allow the SR procedure to be “out of tune” in the way of the post-change distribution parameter, and numerically assess the effect of the “mistuning” on Shiryaev's (multi-cyclic) Stationary Average Detection Delay delivered by the SR procedure. The comprehensive quantitative robustness characterization of the SR procedure obtained in the study can be used to develop the respective theory as well as to provide a rational for practical design of the SR procedure. The overall qualitative conclusion of the study is an expected one: the SR procedure is less (more) robust for less (more) contrast changes and for lower (higher) levels of the false alarm risk.

Bayesian Quickest Change-Point Detection With Sampling Right Constraints

In this paper, Bayesian quickest change detection problems with sampling right constraints are considered. Specifically, there is a sequence of random vari ables whose probability density function will change at an unknown time. The goal is to detect such a change in a way such that a linear combination of the average detection delay and the false alarm probability is minimized. Two types of sampling right constrains are discussed. The first one is a li mited sampling right constraint, in which the observer can take at most N observations from this random sequence. Under this setup, we show that the cost function can be written as a set of iterative fun ctions, which can be solved by Markov optimal stopping theory. The optimal stopping rule is shown to be a threshold rule. An asymptotic upper bound of the average detection delay is developed as the false alarm probability goes to zero. This upper bound indicates that the performance of the limited sampling right problem is close to that of the classic Bayesian quickest detection for several scenar ios of practical interest. The second constraint discussed in this paper is a stochastic sampling right const raint, in which sampling rights are consumed by taking observations and are replenished randomly. The observer cannot take observations if there are no sampling rights left. We characterize the optimal solution, which has a very complex structure. For practical applications, we propose a low complexity algorithm, in which the sampling rule is to take observations as long as the observer has sampling rights left and the detection scheme is a threshold rule. We show that this low complexity scheme is first order as ymptotically optimal as the false alarm probability goes to zero.

Quickest Detection of Changes in the Generating Mechanism of a Time Series via the ε-Complexity of Continuous Functions

A novel methodology for the quickest detection of abrupt changes in the generating mechanisms (stochastic, deterministic, or mixed) of a time series, without any prior knowledge about them, is developed. This methodology has two components: the first is a novel concept of the ε-complexity and the second is a method for the quickest change point detection (Darkhovsky, 2013). The ε-complexity of a continuous function given on a compact segment is defined. The expression for the ε-complexity of functions with the same modulus of continuity is derived. It is found that, for the Hölder class of functions, there exists an effective characterization of the ε-complexity. The conjecture that the ε-complexity of an individual function from the Hölder class has a similar characterization is formulated. The algorithm for the estimation of the ε-complexity coefficients via finite samples of function values is described. The second conjecture that a change of the generating mechanism of a time series leads to a change in the mean of the complexity coefficients, is formulated. Simulations to support our conjectures and verify the efficiency of our quickest change point detection algorithm are performed.

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

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.

A numerical approach to performance analysis of quickest change-point detection procedures

For the most popular sequential change detection rules such as CUSUM, EWMA, and the Shiryaev-Roberts test, we develop integral equations and a concise numerical method to compute a number of performance metrics, including average detection delay and average time to false alarm. We pay special attention to the Shiryaev-Roberts procedure and evaluate its performance for various initialization strategies. Regarding the randomized initialization variant proposed by Pollak, known to be asymptotically optimal of order-3, we offer a means for numerically computing the quasi-stationary distribution of the Shiryaev-Roberts statistic, that is, the distribution of the initializing random variable, thus making this test appli- cable in practice. A significant side-product of our computational technique is the observation that deterministic initializations of the Shiryaev-Roberts procedure can also enjoy the same order-3 optimality property as Pollak's randomized test and, after careful selection, even uniformly outperform it.