Multivariate polynomial regression for identification of chaotic time series. (English) Zbl 1117.62099

Summary: Multivariate polynomial regression was used to generate polynomial iterators for time series exhibiting autocorrelations. A stepwise technique was used to add and remove polynomial terms to ensure the model contained only those terms that produce a statistically significant contribution to the fit. An approach is described in which datasets are divided into three subsets for identification, estimation, and validation. This produces a parsimonious global model that is can greatly reduce the tendency towards undesirable behaviours such as overfitting or instability.
The technique was found to be able to identify the nonlinear dynamic behaviour of simulated time series, as reflected in the geometry of the attractor and calculation of multiple Lyapunov exponents, even in noisy systems. The technique was applied to times series data obtained from simulations of the Lorenz and Mackey-Glass equations with and without measurement noise. The model was also used to determine the embedding dimension of the Mackey-Glass equation.


62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
37M10 Time series analysis of dynamical systems
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
Full Text: DOI Link


[1] Box G. E.P., Time Series Analysis, Forecasting and Control (1976) · Zbl 0363.62069
[2] DOI: 10.1177/003754979105700508
[3] DOI: 10.1103/PhysRevLett.59.845
[4] DOI: 10.1038/344734a0
[5] DOI: 10.1126/science.1519060
[6] Wolf A., Physica 16-D pp 235– (1985)
[7] DOI: 10.1103/PhysRevA.43.2787
[8] DOI: 10.1016/0375-9601(90)90841-B
[9] DOI: 10.1016/0167-2789(89)90074-2 · Zbl 0671.62099
[10] Chen S., Int. J. Control 49 pp 1013– (1989)
[11] DOI: 10.1021/i160036a018
[12] Wu, X. and Çinar, A. An automated knowledge-based system for nonlinear system identification. Gensym User Society Meeting. May26 – 28, Cambridge, MA.
[13] Draper N. R., Applied Regression Analysis (1966) · Zbl 0825.62222
[14] DOI: 10.1080/00207178608933633 · Zbl 0597.93058
[15] Piroddi, L and Spinelli, W. Structure selection for polynomial NARX models based on simulation error minimization. Proceedings of the 13th IFAC symposium on system identification. Rotterdam, The Netherlands. Edited by: van der Hof, P., Wahlberg, B. and Weiland, S. pp.371–376.
[16] Press W. H., Numerical Recipes (1989) · Zbl 0698.65001
[17] Billings S. A., Int. J. Cont. 49 pp 2157– (1989)
[18] Granger C. W.J., Modelling Nonlinear Economic Relationships (1993) · Zbl 0893.90030
[19] DOI: 10.1016/j.automatica.2004.11.017 · Zbl 1061.93094
[20] Vaccari D. A., Trans. Instr. Soc. Am. 31 pp 97– (1992)
[21] TaylorFit Software, Simetrica, LLC. 2004. Available online at:http://simetrica-LLC.com(accessed day month year)
[22] Wolf A., Chaos (1986)
[23] DOI: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 · Zbl 1417.37129
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.