On selecting models for nonlinear time series. (English) Zbl 0888.58034

Summary: Constructing models from time series with nontrivial dynamics involves the problem of how to choose the best model from within a class of models, or to choose between competing classes. This paper discusses a method of building nonlinear models of possibly chaotic systems from data, while maintaining good robustness against noise. The models that are built are close to the simplest possible according to a description length criterion. The method will deliver a linear model if that has shorter description length than a nonlinear model. We show how our models can be used for prediction, smoothing and interpolation in the usual way. We also show how to apply the results to identification of chaos by detecting the presence of homoclinic orbits directly from time series.


37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37N99 Applications of dynamical systems
Full Text: DOI


[1] Abarbanel, H. D.I.; Kennel, M. B., Local false nearest neighbors and dynamical dimensions from observed chaotic data, (Technical report (1992), Department of Physics, University of California: Department of Physics, University of California San Diego) · Zbl 0802.58041
[2] Albano, A. M.; Muench, J.; Schwartz, C.; Mees, A. I.; Rapp, P. E., Singular value decomposition and the Grassberger-Procaccia algorithm, Phys. Rev. A, 38, 3017-3026 (1988)
[3] Barron, A. R., Universal approximation bounds for superpositions of a sigmoidal function, IEEE Transactions on Information Theory, 39, 930-945 (1993) · Zbl 0818.68126
[4] Chen, S.; Cowan, C. F.N.; Grant, P. M., Orthogonal least squares learning algorithm for radial basis function networks, IEEE Transactions on Neural Networks, 2, 302-309 (1991)
[5] Garey, M. R.; Johnson, D. S., Computers and Intractability: A Guide to the Theory of NP-Completeness (1979), Freeman: Freeman San Francisco · Zbl 0411.68039
[6] Ghaddar, D. K.; Tong, H., Data transformation and self-exciting threshold autoregression, J.R. Stat. Soc. C, 30, 238-248 (1981)
[7] Glendinning, P.; Sparrow, C. T., Local and global behavior near homoclinic orbits, J. Stat. Phys., 35, 645-697 (1983) · Zbl 0588.58041
[8] Glover, J.; Mees, A. I., Reconstructing the dynamics of Chua’s circuit, J. Circuits Syst. Comput., 3, 201-214 (1992)
[9] Goldie, C. M.; Pinch, R. G.E., (Communication Theory, Vol. 20 (1991), Cambridge Univ. Press: Cambridge Univ. Press Cambridge) · Zbl 0746.94001
[10] Kline, M., Mathematical Thought from Ancient to Modern Times (1972), Oxford Univ. Press: Oxford Univ. Press New York · Zbl 0277.01001
[11] LeBaron, B., Persistence of the Dow Jones index on rising volume, (Technical report (1991), Santa Fe Institute)
[12] Leonard, J. A.; Kramer, M. A., Radial basis function networks for classifying process faults, (IEEE Control Systems (1991)), 281-294
[13] Mees, A. I., Dynamical systems and tesselations: Detecting determinism in data, Int. J. Bifurcation and Chaos, 1, 777-794 (1991) · Zbl 0876.58045
[14] Mees, A. I., Parsimonious dynamical reconstruction, Int. J. Bifurcation and Chaos, 3, 669-675 (1993) · Zbl 0875.62426
[15] Mees, A. I., Reconstructing chaotic systems in the presence of noise, (Yamaguti, M., Towards the Harnessing of Chaos (1994), Elsevier: Elsevier Tokyo), 305-321
[16] Mees, A. I.; Smith, R. K., Estimation and reconstruction in noisy chaotic systems (1995), in preparation
[17] Mees, A. I.; Sparrow, C. T., Some tools for analyzing chaos, (Proceedings IEEE, 75 (1987)), 1058-1070
[18] Murty, K., A fundamental problem in linear inequalities with an application to tsp, Math. Prog., 2, 296-308 (1972) · Zbl 0249.90038
[19] Noakes, L., The Takens embedding theorem, Int. J. Bifurcation and Chaos, 1, 867-872 (1991) · Zbl 0874.58075
[20] Rissanen, J., (Stochastic Complexity in Statistical Inquiry, Vol. 15 (1989), World Scientific: World Scientific Singapore) · Zbl 0800.68508
[21] Sauer, T.; Yorke, J. A.; Casdagli, M., Embedology, J. Stat. Phys., 65, 579-616 (1992) · Zbl 0943.37506
[22] Schwarz, G., Estimating the dimension of a model, Ann. Stat., 6, 461-464 (1978) · Zbl 0379.62005
[23] Takens, F., Detecting strange attractors in turbulence, (Rand, D. A.; Young, L. S., Dynamical Systems and Turbulence, Vol. 898 (1981), Springer: Springer Berlin), 365-381 · Zbl 0513.58032
[24] Tong, H., Nonlinear Time Series: a Dynamical Systems Approach (1990), Oxford Univ. Press: Oxford Univ. Press Oxford
[25] Whittle, P., Optimization Under Constraints (1971), Wiley: Wiley Chichester · Zbl 0218.90041
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