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Fitting piecewise linear threshold autoregressive models by means of genetic algorithms. (English) Zbl 1429.62383

Summary: A nonlinear version of the threshold autoregressive model for time series is introduced. A peculiar requirement on parameters, except possibly for the constant term, is the continuity, that seems a natural and useful assumption. This model is a special case of the general state-dependent models, where the moving-average term is dropped and a particular form for the dependence on the state is specified. Such model meets also the functional autoregressive model formulation, but the “least demanding” functional form is assumed. Further restrictive assumptions are not needed. Both identification and estimation problems will be taken into account. The proposed approach brings together the genetic algorithm, in its simplest binary form, and some basic features from spline theory. It results in a powerful flexible tool which is shown to be able to approximate a wide class of nonlinear time series models. This method is found to compare favorably with existing procedures in modeling some well-known real-time series, which often are taken as a benchmark for testing and comparing modeling procedures.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62-08 Computational methods for problems pertaining to statistics
90C59 Approximation methods and heuristics in mathematical programming

Software:

AS 183; TSDL; TDSL; Genocop
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References:

[1] Akaike, H., An entropy maximisation principle, (Krishnaiah, P. R., Proceedings of the Symposium on Applied Statistics (1977), North-Holland: North-Holland Amsterdam), 27-41
[2] Applied Statistics Algorithms, web site http://lib.stat.cmu.edu/apstat/; Applied Statistics Algorithms, web site http://lib.stat.cmu.edu/apstat/
[3] Cai, Z.; Fan, J.; Yao, Q., Functional-coefficient regression models for nonlinear time series, J. Amer. Statist. Assoc, 95, 941-956 (2000) · Zbl 0996.62078
[4] Chan, W.-S.; Cheung, S.-H., On robust estimation of threshold autoregressions, J. Forecasting, 13, 37-49 (1994)
[5] Chan, K. S.; Tong, H., On estimating thresholds in autoregressive models, J. Time Series Anal, 7, 179-190 (1986) · Zbl 0596.62085
[6] Chatterjee, S.; Laudato, M., Genetic algorithms in statisticsprocedures and applications, Comm. Statist. Theory Methods, 26, 4, 1617-1630 (1997) · Zbl 0925.62011
[7] Chatterjee, S.; Laudato, M.; Lynch, L. A., Genetic algorithms and their statistical applicationsan introduction, Comput. Statist. Data Anal, 22, 633-651 (1996) · Zbl 0900.62336
[8] Chen, R.; Tsay, R. S., Functional-coefficient autoregressive models, J. Amer. Statist. Assoc, 88, 298-308 (1993) · Zbl 0776.62066
[9] De Jong, K.A., 1975. An analysis of the behavior of a class of genetic adaptive systems. Ph.D. Thesis, Department of Computer and Communication Sciences, University of Michigan, Ann Arbor, MI.; De Jong, K.A., 1975. An analysis of the behavior of a class of genetic adaptive systems. Ph.D. Thesis, Department of Computer and Communication Sciences, University of Michigan, Ann Arbor, MI.
[10] Fuller, W. A., Introduction to Statistical Time Series (1996), Wiley: Wiley New York · Zbl 0851.62057
[11] Ghaddar, D. K.; Tong, H., Data transformation and self-exciting threshold autoregression, Appl. Statist, 30, 238-248 (1981)
[12] Goldberg, D. E., Genetic Algorithms in Search, Optimization and Machine Learning (1989), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0721.68056
[13] Haggan, V.; Ozaki, T., Modelling nonlinear random vibrations using an amplitude-dependent autoregressive time series model, Biometrika, 68, 189-196 (1981) · Zbl 0462.62070
[14] Haupt, R. L.; Haupt, S. E., Practical Genetic Algorithms (1998), Wiley: Wiley New York · Zbl 0940.68107
[15] Holland, J. H., Adaptation in Natural and Artificial Systems (1975), University of Michigan Press: University of Michigan Press Ann Arbor, (2nd Edition: The MIT Press, Cambridge, 1992)
[16] Jennison, C.; Sheehan, N., Theoretical and empirical properties of the genetic algorithm as a numerical optimizer, J. Comput. Graphic. Statist, 4, 296-318 (1995)
[17] Man, K. F.; Tang, K. S.; Kwong, S., Genetic Algorithms: Concepts and Designs (1999), Springer: Springer London · Zbl 0926.68113
[18] Michalewicz, Z., Genetic Algorithms + Data Structures = Evolution Programs (1996), Springer: Springer Berlin · Zbl 0841.68047
[19] Mitchell, M., An Introduction to Genetic Algorithms (1996), The MIT Press: The MIT Press Cambridge, MA
[20] Nicholson, A. J., The self-adjustment of populations to change, Cold Spring Harbour Symp. Quant. Biol, 22, 153-173 (1957)
[21] Ozaki, T., Non-linear threshold autoregressive models for non-linear random vibrations, J. Appl. Probab, 18, 443-451 (1981) · Zbl 0459.73077
[22] Ozaki, T., The statistical analysis of perturbed limit cycle processes using nonlinear time series models, J. Time Series Anal, 3, 29-41 (1982) · Zbl 0499.62079
[23] Ozaki, T.; Oda, H., Non-linear time series model identification by Akaike’s information criterion, (Dubuisson, B., Information and Systems (1978), Pergamon Press: Pergamon Press Oxford), 83-91
[24] Pittman, J.; Murthy, C. A., Fitting optimal piecewise linear functions using genetic algorithms, IEEE Trans. Pattern Anal. Mach. Intell, 22, 7, 701-718 (2000)
[25] Priestley, M. B., Non-linear and Non-stationary Time Series Analysis (1988), Academic Press: Academic Press New York · Zbl 0687.62072
[26] RWC, Belgium World Data Center for the Sunspot Index. Web site is http://sidc.oma.be; RWC, Belgium World Data Center for the Sunspot Index. Web site is http://sidc.oma.be
[27] Tong, H., 1980. A view on non-linear time series model building. In: Anderson, O.D. (Ed.), Time Series: Proceedings of the International Conference, Nottingham University, March 1979. North-Holland, Amsterdam, pp. 41-56.; Tong, H., 1980. A view on non-linear time series model building. In: Anderson, O.D. (Ed.), Time Series: Proceedings of the International Conference, Nottingham University, March 1979. North-Holland, Amsterdam, pp. 41-56.
[28] Tong, H., Threshold Models in Non-linear Time Series Analysis (1983), Springer: Springer New York · Zbl 0527.62083
[29] Tong, H., Non-linear Time Series. A Dynamical System Approach (1990), Oxford Science Publications, Clarendon Press: Oxford Science Publications, Clarendon Press Oxford
[30] Tong, H.; Dabas, P., Cluster of time series modelsan example, J. Appl. Statist, 17, 2, 187-198 (1990)
[31] Tong, H.; Lim, K. S., Threshold autoregression, limit cycles and cyclical data, J. Roy. Statist. Soc. Ser. B, 42, 245-292 (1980) · Zbl 0473.62081
[32] Tsay, R. S., Non-linear time series analysis of blowfly population, J. Time Series Anal, 9, 247-263 (1988)
[33] TSDL, Time Series Data Library maintained by Rob Hyndman and Muhammad Akram. Web site is http://www-personal.buseco.monash.edu.au/ hyndman/TSDL; TSDL, Time Series Data Library maintained by Rob Hyndman and Muhammad Akram. Web site is http://www-personal.buseco.monash.edu.au/ hyndman/TSDL
[34] Wu, B.; Chang, C.-L., Using genetic algorithms to parameters \((d,r)\) estimation for threshold autoregressive models, Comput. Statist. Data Anal, 38, 315-330 (2002) · Zbl 1028.62072
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