Guangming Wang

Jonathan J. Wylie

Qiang Zhang

Samir Ben Hariz

Wang, Guangming

Ben Hariz, Samir

Wylie, Jonathan J.

Zhang, Qiang

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 346 (2008) 467–470http://france.elsevier.com/direct/CRASS1/StatisticsChange-point detection for continuous processeswith high-frequency samplingGuangming Wang a, Samir Ben Hariz b, Jonathan J. Wylie c,1, Qiang Zhang ca School of Mathematics and Statistics, Wuhan University, Hubei, 430072, People’s Republic of Chinab Laboratoire de statistique et processus, département de mathématiques, université du Maine, avenue Olivier-Messiaen,72085 Le Mans cedex 9, Francec Department of Mathematics, City University of Hong Kong, 83, Tat Chee Avenue, Kowloon, Hong KongReceived 29 November 2007; accepted 4 December 2007Available online 3 March 2008Presented by Philippe G. CiarletAbstractWe consider the detection of a jump in a continuous process over a fixed time interval. We aim to locate the jump position viadiscrete observations and consider how increasing the frequency of the observations affects the accuracy of the detection process.We show that the classical cumulative-sum estimator fails, and propose a new estimator based on local information that we proveconverges exponentially fast. To cite this article: G. Wang et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).© 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméDétection de rupture pour des processus continus avec un échantillonnage à haute fréquence. Nous considérons la détectionde saut dans un processus en temps continu sur un intervalle de temps fini et fixé. On cherche à localiser dans le temps la positionde rupture via des observations discrètes. Nous étudions l’effet de croissance de la fréquence d’échantillonnage sur la précisionde l’estimation. Nous montrons que la procédure classique avec les estimateurs à sommes cumulatives échoue et nous proposonsune alternative basée sur une localisation de l’information. Nous prouvons que le nouvel estimateur converge avec une vitesseexponentielle. Pour citer cet article : G. Wang et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).© 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionSudden localized changes in data occur in a very broad range of applications that span image processing, signalprocessing, economics and the physical sciences. Given a finite set of observations, one typically needs to estimatethe location at which changes occur. In this paper, we consider a continuous-time process with both dependent andE-mail addresses: G.M.Wang@student.cityu.edu.hk (G. Wang), Samir.Ben_Hariz@univ-lemans.fr (S. Ben Hariz), mawylie@cityu.edu.hk(J.J. Wylie), mazq@cityu.edu.hk (Q. Zhang).1 The work of J.J. Wylie was supported by a grant from CityU (7001752).1631-073X/$ – see front matter © 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2008.02.005468 G. Wang et al. / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 467–470independent increments over a fixed time interval with a single jump at an unknown location. Correlations are inherentin continuous processes and these correlations cause a number of challenging practical and theoretical difficulties.The change-point problem has a wide literature that spans a number of fields. There are also a wide range oftechniques including linear-model based approaches and nonparametric approaches. For a comprehensive review ofthe subject we refer the reader to Basseville and Nikiforov [1].For uncorrelated processes or for independent data a number of important results have been obtained. Carlstein [4]introduced a class of nonparametric estimators based on cumulative sums for problems in which the distribution isstationary on both sides of the jump. Dumbgen [5] considered a more general class of estimators and proved anOp(n−1) rate of convergence, where n is the number of observations.Wang [8] proposed a method based on wavelets that considered a white-noise model in which the mean variescontinuously on both sides of the jump. Wong et al. [9] considered jump detection in a heteroscedastic autoregressivemodel and proved the consistency of an estimator that used wavelets.Nikiforov [6] considered generalized change detection problems for stochastic models. He provided a statisticalmethod for both detecting and isolating changes in stochastic systems. He also considered the optimality of the ap-proach in the sense of minimizing the worst mean detection delay. Nikiforov [7] made a detailed comparison betweentwo strategies for both change detection and isolation. He compared optimal sequential and fixed-size sample strate-gies and showed that a simple nonsequential method is almost as efficient as an optimal sequential algorithm, but withless costly computation.When there are heavy correlations in the noise the problem is significantly more challenging. Ben Hariz et al. [3]considered data with long-range dependence where correlations decay to zero algebraically or faster. They showedthat the classical cumulative-sum (CUSUM) estimators also achieve the Op(n−1) rate of convergence in this case.To our best knowledge, previous works have focussed on processes with independent increments, weakly dependentstructure, or at least with correlations structures that tend to zero. They hence do not apply to situations in which thereis a finite interval and the sampling frequency tends to infinity.There are a number of important applications related to high-frequency sampling in which correlations betweenneighboring points do not decay to zero. For this class of problems we show that the traditional cumulative-sum andwavelet-based estimators may not yield consistent estimation. We propose an alternative estimator that is based onlocal averages and show that the estimator performs best when it uses localized information. For Gaussian data wederive a bound for the probability of the estimator selecting an incorrect change-point location and show that thisprobability tends to zero exponentially fast as the sampling rate tends to infinity. Detailed proofs of these results canbe found in Ben Hariz et al. [2].2. Framework and methodologyWe consider the following modelYt = δ1{t>θ} + Xt, t ∈ [0, T ], (1)where δ is the size of the jump and θ ∈ (0, T ) is the location of the jump. We assume that (Xt )0tT is a ran-dom process with a constant mean, which, without loss of generality, is taken to be zero. We also assume that theprocess (Xt ) satisfies the following regularity condition∃α > 0,C > 0, such that for s, t ∈ [0, T ], E(Xs − Xt)2  C|s − t |α. (2)This is a typical assumption to ensure the continuity of the process.Observe that here the main concern is the local fluctuations of X rather than the long-range correlations of X. Thisis in sharp contrast with the common assumption that the random part of the data is either a white noise process,a weakly dependent process or a process with correlations that tend to zero as the distance between points tends toinfinity.Clearly, the process can only be observed at a finite number of grid points. For the sake of simplicity we considerthe case T = 1 with uniformly distributed grid points at t1 = T/n, . . . , tk = kT /n, . . . , tn = T . Given the observations(Yti )1in , we aim to estimate within which interval [θ̂ , θ̂ + 1/n[ the change-point lies. Here θ̂ is determined by alocalized version of the cumulative-sum estimatorθ̂ = 1 min(arg max{|Uk|}), (3)n 1k<nG. Wang et al. / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 467–470 469withUk = 1k − kl + 1k∑i=klYi − 1ku − kku∑i=k+1Yi, k = 1, . . . , n − 1, (4)where kl = max(1, k − L + 1) and ku = min(k + L,n).The implementation of these estimators is straight-forward, namely for a given continuous process, we sample thedata uniformly at the points ti = i/n, i = 1,2, . . . , n, and consider each interval [ti , ti+1[, i = 1,2, . . . , n − 1, as acandidate for the segment of the process within which the change-point lies. We take a window with 2L observationswith an obvious modification near the end points of the data set (see Eq. (4)), and determine the difference betweenthe mean of Y based on the data from the first L points and that from the second L points. The algorithm identifiesthe interval which gives the maximum of the absolute value of the difference as the segment where the change pointlies. The classical CUSUM estimator takes the whole set as the window size, i.e., L is equal to n.3. Main resultsWe have performed numerical tests of processes with correlation functions that satisfy (2). The simulations showthat the classical CUSUM change-point estimator that one obtains by setting L = n in (4) does not yield consistentresults. The results also show that smaller values of L yield more accurate estimators. This suggests that choosingL = 1 will be best choice since it gives the smallest estimation error. Therefore, in the rest of this Note, we will onlyconsider the case of L = 1 and develop some surprising theoretical results for this apparently simple choice of L.Before progressing to the next section in which we present our main theoretical results, we provide some simpleintuitive explanation about why the conventional approach based on whole data set, that is L = n, fails and why L = 1is a better choice.The classical change-point problem corresponds to the situation in which the data is zero correlated. In this case,choosing larger value of L will reduce the statistical errors in the estimates for the empirical means in (4). If oneconsiders correlated processes in which the correlations decay to zero a similar argument holds. These processescorrespond to situations in which the frequency is fixed, but the time over which the process is recorded tends toinfinity. For the class of problems considered in this Note, the time over which the process is recorded is fixed, but thesampling frequency increases. As one increases the frequency the correlations among all sampled data will not tendto zero. These correlations imply that the statistical errors in (4) do not decay. In this situation, as n tends to infinity,the data between the neighboring points becomes almost perfectly correlated. This is due to the fact the noise betweenneighboring points becomes almost perfectly correlated. Therefore the difference between data from the neighboringpoints will be very small except across the change point where the difference will be dominated by the jump. ThereforeL = 1 will be the best choice when we consider high frequency correlated data. As one increases L, the noise of thedata within the window containing the 2L observations becomes less correlated, and consequently the performanceof the estimator deteriorates.In the following theorem, we state an exponential bound for the probability of missing the change-point locationand give some examples:Theorem 3.1. Assume that (Xt ) is a Gaussian process. Then we haveP(θ /∈[θ̂ , θ̂ + 1n]) 2(n − 1)P(N(0,1)  δ√εn), (5)where εn := maxi =k(Var(Zi − Zk),Var(Zi + Zk)), Zi := Xi/n − X(i+1)/n and k := [nθ ].In particular,(i) If there exist constants α > 0,C > 0, such that for s, t ∈ [0, T ], E(Xs − Xt)2  C|s − t |α , then εn  4Cn−α .(ii) If E(Xs − Xt)2  C|s − t |α and for s1  s2  t1  t2, |E(Xs2 − Xs1)(Xt2 − Xt1)|  C1|s2 − s1|α1 |t2 − t1|α2 forα1 + α2 > α, then εn  2Cn−α + o(n−α).Remark. If X is weak stationary and |corr(Xs,Xt ) − 1|  C|s − t |α , then (i) is satisfied. Condition (ii) is satisfiedwhen (Xt ) has independent increments.470 G. Wang et al. / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 467–470In summary, we have considered the detection of change-points for continuous processes over a fixed time interval.We have shown that as the frequency of observations increases the classical CUSUM estimator fails. We have proposedan alternative, extremely simple, estimator. For Gaussian processes with natural conditions to control the regularityof the process, we have derived a bound for the probability of missing the change-point and shown that our estimatorhas exponentially fast convergence.References[1] M. Basseville, I.V. Nikiforov, Detection of Abrupt Changes: Theory and Application, Prentice-Hall, Englewood Cliffs, NJ, 1993.[2] S. Ben Hariz, G.M. Wang, J.J. Wylie, Q. Zhang, Jump detection for discretely observed continuous processes, Preprint, 2007.[3] S. Ben Hariz, J.J. Wylie, Q. Zhang, Optimal rate of convergence for nonparametric change-point estimators for non-stationary sequences, Ann.Statist. 35 (2007) 1802–1826.[4] E. Carlstein, Nonparametric change-point estimation, Ann. Statist. 16 (1988) 188–197.[5] L. Dumbgen, The asymptotic behavior of some nonparametric change-point estimators, Ann. Statist. 19 (1991) 1471–1495.[6] I.V. Nikiforov, A generalized change detection problem, IEEE Trans. Inform. Theory 41 (1) (1995) 171–187.[7] I.V. Nikiforov, Two strategies in the problem of change detection and isolation, IEEE Trans. Inform. Theory 43 (2) (1997) 770–776.[8] Y. Wang, Jump and sharp cusp detection by wavelets, Biometrika 82 (2) (1995) 385–397.[9] H. Wong, W. Ip, Y. Li, Detection of jumps by wavelets in a heteroscedastic autoregressive model, Statist. Probab. Lett. 52 (4) (2001) 365–372.

Change-point detection for continuous processes with high-frequency sampling

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Change-point detection for continuous processes with high-frequency sampling

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