Estimation of parameters in one-dimensional maps from noisy chaotic time series. (English) Zbl 1136.37361

Summary: The problem of parameter estimation in model maps from noisy time series is addressed. We suggest a new technique for a special case of one-dimensional maps and chaotic signals. It is based on the maximum likelihood (ML) principle and evaluation of the cost function via backward iterations of a model map. We demonstrate in numerical experiments and, in part, justify theoretically that this “backward ML technique” gives more accurate estimates than previously known techniques for low and moderate noise levels. In particular, global optimisation of the cost function becomes much easier; biases in the estimates vanish as the time series length \(N\) increases; variances of the estimates decrease as fast as \(N - \alpha \) where \(\alpha \) depends on the original system, typical values being about \(\alpha =2.0\) under mild conditions on the original systems.


37M10 Time series analysis of dynamical systems
Full Text: DOI


[1] Hegger, R.; Kantz, H.; Schmuser, F., Chaos, 8, 727 (1998)
[2] Swameye, I.; Muller, T. G.; Timmer, J., Proc. Natl. Acad. Sci. USA, 100, 1028 (2003)
[3] Ljung, L., System Identification. Theory for the User (1991), Moscow
[4] Kantz, H.; Schreiber, T., Nonlinear Time Series Analysis (1997), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0873.62085
[5] (Gouesbet, G.; Meunier-Guttin-Cluzel, S.; Menard, O., Chaos and its Reconstructions (2003), Nova Science Publishers: Nova Science Publishers New York)
[6] Baake, E.; Baake, M.; Bock, H. J.; Briggs, K. M., Phys. Rev. A, 45, 5524 (1992)
[7] Brown, R.; Rulkov, N. F.; Tracy, E. R., Phys. Lett. A, 194, 71 (1994)
[8] Anishchenko, V. S.; Pavlov, A. N.; Janson, N. B., Chaos Solitons Fractals, 8, 1267 (1998)
[9] Timmer, J.; Rust, H.; Horbelt, W.; Voss, H. U., Phys. Lett. A, 274, 123 (2000) · Zbl 1055.37586
[10] Bezruchko, B. P.; Smirnov, D. A., Phys. Rev. E, 63, 016207 (2001)
[11] Sitz, A.; Schwartz, U.; Kurths, J.; Voss, H. U., Phys. Rev. E, 66, 016210 (2002)
[12] Casdagli, M., Physica D, 35, 335 (1989)
[13] Smith, L. A., Physica D, 58, 50 (1992)
[14] Judd, K.; Mees, A. I., Physica D, 82, 426 (1995)
[15] Aguirre, L. A.; Mendes, E. M.A. M., Int. J. Bifur. Chaos, 6, 279 (1996)
[16] Parlitz, U.; Merkwirth, C., Phys. Rev. Lett., 84, 1890 (2000)
[17] Feigin, A. M.; Mol’kov, Ya. I.; Mukhin, D. N.; Loskutov, Ye. M., Izv. Vyssh. Uchebn. Zaved. Radiofiz., 44, 5-6, 376 (2001)
[18] Smirnov, D.; Bezruchko, B., Phys. Rev. E, 68, 046209 (2003)
[19] (Gouesbet, G.; Meunier-Guttin-Cluzel, S.; Ménard, O., Chaos and its Reconstruction (2003), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 1-160
[20] Jaeger, L.; Kantz, H., Chaos, 6, 440 (1996)
[21] McSharry, P. E.; Smith, L. A., Phys. Rev. Lett., 83, 4285 (1999)
[22] Judd, K., Phys. Rev. E, 67, 026212 (2003)
[23] Horbelt, W.; Timmer, J., Phys. Lett. A, 310, 269 (2003)
[24] Pisarenko, V. F.; Sornette, D., Phys. Rev. E, 69, 036122 (2004)
[25] Farmer, J. D.; Sidorowich, J. J., Physica D, 47, 373 (1991)
[26] Andreyev, Yu. V.; Dmitriev, A. S.; Efremova, E. V., Phys. Rev. E, 65, 046220 (2002)
[27] Dennis, J. E.; Schnabel, R. B., Numerical Methods for Unconstrained Optimization and Nonlinear Equations (1983), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0579.65058
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.