Change detection for uncertain autoregressive dynamic models through nonparametric estimation. (English) Zbl 1487.62096
Summary: A new statistical approach for on-line change detection in uncertain dynamic system is proposed. In change detection problem, the distribution of a sequence of observations can change at some unknown instant. The goal is to detect this change, for example a parameter change, as quickly as possible with a minimal risk of false detection. In this paper, the observations come from an uncertain system modeled by an autoregressive model containing an unknown functional component. The popular Page’s CUSUM rule is not applicable anymore since it requires the full knowledge of the model. A new detection CUSUM-like scheme is proposed, which is based on the nonparametric estimation of the unknown component from a learning sample. Moreover, the estimation procedure can be updated on-line which ensures a better detection, especially at the beginning of the monitoring procedure. Simulation trials were performed on a model describing a water treatment process and show the interest of this new procedure with respect to the classic CUSUM rule.
MSC:
| 62L10 | Sequential statistical analysis |
| 62G05 | Nonparametric estimation |
| 62L15 | Optimal stopping in statistics |
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 62P30 | Applications of statistics in engineering and industry; control charts |
References:
| [1] | Baillo, A.; Cuevas, A., Parametric versus nonparametric tolerance regions in detection problems, Comput. Statist., 21, 3-4, 523-536 (2006) · Zbl 1164.62441 |
| [2] | Bansal, R. K.; Papantoni-Kazakos, P., An algorithm for detecting a change in a stochastic process, IEEE Trans. Inf. Theory, IT-32, 227-235 (1986) · Zbl 0596.60040 |
| [3] | Basseville, M.; Nikiforov, I., Detection of Abrupt Changes. Theory and Application (1993), Prentice-Hall · Zbl 1407.62012 |
| [4] | Bastin, G.; Dochain, D., On-line Estimation and Adaptive Control of Bioreactors (1990), Elsevier |
| [5] | Bernard, O.; Hadj-Sadok, Z.; Dochain, D.; Genovesi, A.; Steyer, J.-P., Dynamical model development and parameter identification for an anaerobic wastewater treatment process, Biotechnol. Bioeng., 75, 4, 424-438 (2001) |
| [6] | Bosq, D., (Nonparametric Statistics for Stochastic Processes. Nonparametric Statistics for Stochastic Processes, Lecture Notes in Statistics, vol. 110 (1996), Springer) · Zbl 0857.62081 |
| [7] | Duflo, M., Random Iterative Models (1997), Springer-Verlag · Zbl 0868.62069 |
| [8] | Fouladirad, M.; Nikiforov, I., Optimal statistical fault detection with nuisance parameters, Automatica, 41, 1157-1171 (2005) · Zbl 1116.62111 |
| [9] | Fuh, C.-D., SPRT and CUSUM in hidden markov models, Ann. Statist., 31, 942-977 (2003) · Zbl 1036.60005 |
| [10] | Genest, C.; Rémillard, B., Tests of independence and randomness based on the empirical copula process, TEST, 13, 2, 335-369 (2004) · Zbl 1069.62039 |
| [11] | Goel, A. L.; Wu, S. M., Determination of ARL and a contour nomogram for CUSUM charts to control normal mean, Technometrics, 13, 2, 221-230 (1971) · Zbl 0217.51801 |
| [12] | Gombay, E.; Serban, D., Monitoring parameter change in AR \((p)\) time series models, J. Multivariate Anal., 100, 715-725 (2009) · Zbl 1163.62063 |
| [13] | Harrou, F.; Fillatre, L.; Nikiforov, I., Anomaly detection/detectability for a linear model with a bounded nuisance parameter, Annu. Rev. Control, 38, 32-44 (2014) |
| [14] | Herkenrath, U., On the uniform ergodicity of Markov processes of order 2, J. Appl. Probab., 40, 455-472 (2003) · Zbl 1031.60065 |
| [15] | Hilgert, N.; Portier, B., Strong uniform consistency and asymptotic normality of a Kernel based error density estimator in functional autoregressive models, Stat. Inference Stoch. Process., 15, 2, 105-125 (2011) · Zbl 1242.62027 |
| [16] | Lai, T. L., Information bounds and quick detection of parameter changes in stochastic systems, IEEE Trans. Inform. Theory, 44, 2917-2929 (1998) · Zbl 0955.62084 |
| [17] | Lai, T. L., Sequential analysis: Some classical problems and new challenges, Statist. Sinica, 11, 303-408 (2001) · Zbl 1037.62081 |
| [18] | Ljung, G. M.; Box, G. E., On a measure of lack of fit in time series models, Biometrika, 65, 2, 297-303 (1978) · Zbl 0386.62079 |
| [19] | Lorden, G., Procedures for reacting to a change in distribution, Ann. Math. Statist., 42, 1897-1908 (1971) · Zbl 0255.62067 |
| [20] | Lorden, G., Open-ended tests for Koopman-Darmois families, Ann. Statist., 1, 633-643 (1973) · Zbl 0282.62072 |
| [21] | Margavio, T. M.; Conerly, M. D.; Woodall, W. H.; Drake, L. G., Alarm rates for quality control charts, Statist. Probab. Lett., 24, 3, 219-224 (1995) · Zbl 0835.62095 |
| [22] | Mei, Y., Sequential change-point detection when unknown parameters are present in the pre-change distribution, Ann. Statist., 34, 92-122 (2006) · Zbl 1091.62064 |
| [23] | Montgomery, D., Introduction to Statistical Quality Control (1996), Wiley: Wiley New York · Zbl 0851.62057 |
| [24] | Moustakides, G., Optimal procedures for detecting change in distribution, Ann. Statist., 14, 1379-1387 (1986) · Zbl 0612.62116 |
| [25] | Moustakides, G.; Polunchenko, A. S.; Tartakovsky, A. G., Numerical comparison of cusum and shiryaev-roberts procedures for detecting changes in distributions, Comm. Statist. Theory Methods, 38, 16-17, 3225-3239 (2009) · Zbl 1175.62084 |
| [26] | Moustakides, G.; Polunchenko, A. S.; Tartakovsky, A. G., A numerical approach to performance analysis of quickest change-point detection procedures, Statist. Sinica, 21, 571-596 (2011) · Zbl 1214.62084 |
| [27] | Nadaraya, E. A., On estimating regression, Theory Probab. Appl., 9, 141-142 (1964) · Zbl 0136.40902 |
| [28] | Page, E. S., Continuous inspection schemes, Biometrika, 41, 100-115 (1954) · Zbl 0056.38002 |
| [29] | Parzen, E., On estimation of a probability density function and mode, Ann. Math. Statist., 33, 1065-1076 (1962) · Zbl 0116.11302 |
| [31] | Portier, B.; Oulidi, A., Nonparametric estimation and adaptive control of functional autoregressive models, SIAM J. Control Optim., 39, 411-432 (2000) · Zbl 0980.93085 |
| [32] | Qiu, P., Distribution-free multivariate process control based on log-linear modeling, IIE Trans., 40, 7, 664-677 (2008) |
| [33] | Rosenblatt, M., Remarks on some nonparametric estimates of a density function, Ann. Math. Statist., 27, 832-837 (1956) · Zbl 0073.14602 |
| [34] | Senoussi, R., Uniform iterated logarithm laws for martingales and their application to functional estimation in controlled Markov chains, Stochastic Process. Appl., 89, 2, 193-211 (2000) · Zbl 1045.60030 |
| [36] | Su, J. Y.; Gan, F. F.; Tang, X., Optimal cumulative sum charting procedures based on Kernel densities, (Frontiers in Statistical Quality Control, Vol. 11 (2015), Springer), 119-134 · Zbl 1331.62504 |
| [37] | Verdier, G.; Hilgert, N.; Vila, J.-P., Adaptive threshold computation for CUSUM-type procedures in change detection and isolation problems, Comput. Statist. Data Anal., 52, 9, 4161-4174 (2008) · Zbl 1452.62135 |
| [38] | Verdier, G.; Hilgert, N.; Vila, J.-P., Optimality of CUSUM rule approximations in change-point detection problems: Application to nonlinear state-space systems, IEEE Trans. Inform. Theory, 54, 11, 5102-5112 (2008) · Zbl 1247.94011 |
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