Abstract
Parameter estimation is an important scientific problem in various fields such as chaos control, chaos synchronization and other mathematical models. In this paper, a new method for parameter estimation in nonlinear dynamical equations is proposed based on evolutionary modelling (EM). This will be achieved by utilizing the following characteristics of EM which includes self-organizing, adaptive and self-learning features which are inspired by biological natural selection, and mutation and genetic inheritance. The performance of the new method is demonstrated by using various numerical tests on the classic chaos model—Lorenz equation (Lorenz 1963). The results indicate that the new method can be used for fast and effective parameter estimation irrespective of whether partial parameters or all parameters are unknown in the Lorenz equation. Moreover, the new method has a good convergence rate. Noises are inevitable in observational data. The influence of observational noises on the performance of the presented method has been investigated. The results indicate that the strong noises, such as signal noise ratio (SNR) of 10 dB, have a larger influence on parameter estimation than the relatively weak noises. However, it is found that the precision of the parameter estimation remains acceptable for the relatively weak noises, e.g. SNR is 20 or 30 dB. It indicates that the presented method also has some anti-noise performance.
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References
Back T, Hammel U, Schwefel HP (1997) Evolutionary computation: comments on the history and current state. Evol Comput IEEE Trans 1:3–17
Cao HX, Kang L, Chen Y, Yu J (2000) EM of systems of ordinary differential equations with genetic programming. Genet Program Evolvable Mach 1:309–337
Cao XQ, Song JQ, Zhang WM, Zhao J, Zhang LL (2011) Estimating parameters of chaotic system with variational method. Acta Phys Sin 60:070511 (In Chinese)
Chou JF (1974) The problem for making use of observational data in numerical weather forecast. Sci Chin 6:635–644 (In Chinese)
Chou JF, Xu M (2001) Advancement and prospect in short-term climate prediction. Chin Sci Bull 46:890–896 (in Chinese)
Chou JF (2007) An innovative road to numerical weather prediction-from initial value problem to inverse problem. Acta Meteorol Sin 65:673–682 (In Chinese)
Dai D, Ma XK, Li FC, You Y (2002) An approach of parameter estimation for a chaotic system based on genetic algorithm. Acta Phys Sin 51:2459–2462 (In Chinese)
Guan XP, Peng HP, Li LX, Wang YQ (2001) Parameters identification and control of Lorenz chaotic system. Acta Phys Sin 50:26–29 (In Chinese)
Hakkarainen J, Ilin A, Solonen A, Laine M, Haario H, Tamminen J, Oja E, Järvinen H (2012) On closure parameter estimation in chaotic systems. Nonlin Process Geophys 19:127–143
He WP, Feng GL, Wu Q, He T, Wan SQ, Chou JF (2012) A new method for abrupt dynamic change detection of correlated time series. Int J Climatol 32:1604–1614
He WP, Wu Q, Cheng HY, Zhang W (2011) Comparison of applications of different filter methods for de-noising detrended fluctuation analysis. Acta Phys Sin 2011(60):029203 (In Chinese)
Huang JP, Yi YH (1991) Nonlinear dynamical model inversion with observational data. Sci Chin B 21:331–336 (In Chinese)
Jackson CS, Sen MK, Huerta G, Deng Y, Bowman KP (2008) Error reduction and convergence in climate prediction. J Clim 21:6698–6709
Kalman R (1960) A new approach to linear filtering and prediction problems. Transactions of the ASME. J Basic Eng 82(Series D):35–42
Kivman GA (2003) Sequential parameter estimation for stochastic systems. Nonlinear Processes Geophys 10:253–259
Li LX, Yang YX, Peng HP, Wang XD (2006a) Parameters identification of chaotic systems via chaotic ant swarm. Int J Bifurcat Chaos 16:1204–1211
Li LX, Peng HP, Wang XD, Yang YX (2006b) An optimization method inspired by ‘chaotic’ ant behaviour. Int J Bifurcat Chaos 16:2351–2364
Li LX, Peng HP, Yang YX, Wang XD (2007) Parameter estimation for Lorenz chaotic systems based on chaotic ant swarm algorithm. Acta Phys Sin 56(1):51–55 (In Chinese)
Lorenz EN (1963) Deterministic nonperiodic flow. J Atmos Sci 20:130–141
Rubin DB (1988) Using the SIR algorithm to simulate posterior distributions. In: Bernardo JM, DeGroot MH, Lindley DV, Smith, AFM (eds) Bayesian Statistics 3. Oxford University Press, Oxford, pp 395–402
Sexton D, Murphy J, Collins M, Webb M (2011) Multivariate prediction using imperfect climate models part I: outline of methodology. Clim Dyn 1–30
Song JQ, Cao XQ, Zhang WM, Zhu XQ (2012) Estimating parameters for coupled air-sea model with variational method. Acta Phys Sin 61(11):110401 (In Chinese)
Vondrak J (1969) A contribution to the problem of smoothing observational data. Bull Astron Inst Czech 20:349–355
Vondrak J (1977) Problem of smoothing observational data. Bull Astron Inst Czech 28:84–82
Wan SQ, He WP, Wang L, Jiang W, Zhang W (2012) Evolutionary modeling-based approach for model errors correction. Nonlinear Processes Geophys 19:439–447
Zechman EM, Ranji Ranjithan S (2007) Evolutionary computation-based approach for model error correction and calibration. Adv Water Resour 30:1360–1370
Acknowledgments
The authors thank the two anonymous reviewers and editors for the beneficial and helpful suggestions for this manuscript. In addition, we acknowledge Master Michael (in UK) for improving the language of our manuscript. This research was jointly supported by the National Basic Research Program of China (973 Program) (2012CB955902), and the National Natural Science Foundation of China (Grant nos. 41275074 and 41005041).
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Wang, L., He, WP., Liao, LJ. et al. A new method for parameter estimation in nonlinear dynamical equations. Theor Appl Climatol 119, 193–202 (2015). https://doi.org/10.1007/s00704-014-1113-3
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DOI: https://doi.org/10.1007/s00704-014-1113-3