Abstract
Parameter estimation is an important research topic in nonlinear dynamics. Based on the evolutionary algorithm (EA), Wang et al. (2014) present a new scheme for nonlinear parameter estimation and numerical tests indicate that the estimation precision is satisfactory. However, the convergence rate of the EA is relatively slow when multiple unknown parameters in a multidimensional dynamical system are estimated simultaneously. To solve this problem, an improved method for parameter estimation of nonlinear dynamical equations is provided in the present paper. The main idea of the improved scheme is to use all of the known time series for all of the components in some dynamical equations to estimate the parameters in single component one by one, instead of estimating all of the parameters in all of the components simultaneously. Thus, we can estimate all of the parameters stage by stage. The performance of the improved method was tested using a classic chaotic system—Rössler model. The numerical tests show that the amended parameter estimation scheme can greatly improve the searching efficiency and that there is a significant increase in the convergence rate of the EA, particularly for multiparameter estimation in multidimensional dynamical equations. Moreover, the results indicate that the accuracy of parameter estimation and the CPU time consumed by the presented method have no obvious dependence on the sample size.
Similar content being viewed by others
References
Aksoy A, Zhang F, Nielsen-Gammon JW (2006a) Ensemble-based simultaneous state and parameter estimation in a two-dimensional sea-breeze model. Mon Weather Rev 134:2951–2970
Aksoy A, Zhang F, Nielsen-Gammon JW, Epifanio CC (2006b) Ensemble-based simultaneous state and parameter estimation with MM5. Geophys Res Lett 33, L12801
Annan JD, Hargreaves JC (2004) Efficient parameter estimation for a highly chaotic system. Tellus A 56:520–526
Back T, Hammel U, Schwefel HP (1997) Evolutionary computation: comments on the history and current state. IEEE Trans Evol Comput 1:3–17
Cao HQ, Kang LS, Chen Y, Yu J (2000) EM of systems of ordinary differential equations with genetic programming. Genet Program Evolvable Mach 1:309–337
Evensen G (1994) Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte-Carlo methods to forecast error statistics. J Geophys Res 99:10143–10162
Feng GL, Cao HX, Dong WJ, Chou JF (2001) On temporal evolution of precipitation probability of the Yangtze River delta in the last 50 years. Chin Phys 10:1004–1010
Feng GL, Dong WJ, Jia XJ (2003) Evaluation of the applicability of a retrospective scheme based on comparison with several difference schemes. Chin Phys 12:1076–1086
Feng GL, Dong WJ, Li JP (2004) On temporal evolution of precipitation probability of the Yangtze River delta in the last 50 years. Chin Phys 13:1582–1587
He WP, Wang L, Wan SQ, Liao LJ, He T (2012) Evolutionary modeling for dryness and wetness prediction. Acta Phys Sin 61:119201 (In Chinese)
Huang JP, Yi YH (1991) Nonlinear dynamical model inversion with observational data. Sci China B 21:331–336 (In Chinese)
Kivman GA (2003) Sequential parameter estimation for stochastic systems. Nonlinear Processes Geophys 10:253–259
Lea DJ, Allen MR, Haine TWN (2000) Sensitivity analysis of the climate of a chaotic system. Tellus A 52:523–532
Lea DJ, Haine TWN, Allen MR, Hansen JA (2002) Sensitivity analysis of the climate of a chaotic ocean circulation model. Q J R Meteorol Soc 128:2587–2605
Li LX, Yang YX, Peng HP, Wang XD (2006a) Parameters identification of chaotic systems via chaotic ant swarm. Int J Bifurcation Chaos 16:1204–1211
Li LX, Peng HP, Wang XD, Yang YX (2006b) An optimization method inspired by ‘chaotic’ ant behaviour. Int J Bifurcation Chaos 16:2351–2364
Mu M, Duan W, Wang Q, Zhang R (2010) An extension of conditional nonlinear optimal perturbation approach and its applications. Nonlinear Processes Geophys 17:211–220
Rössler OE (1976) An equation for continuous chaos. Phys Lett A 57:397–398
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
Schirber S, Klocke D, Pincus R, Quaas J, Anderson J (2013) Parameter estimation using data assimilation in an atmospheric general circulation model: from a perfect toward the real world. J Adv Model Earth Syst 5:58–70
Siebesma AP, Cuijpers JWM (1995) Evaluation of parametric assumptions for shallow cumulus convection. J Atmos Sci 52:650–666
Song JQ, Cao XQ, Zhang WM, Zhu XQ (2012) Estimating parameters for coupled air-sea model with variational method. Acta Phys Sin 61:110401 (In Chinese)
Tong M, Xue M (2008) Simultaneous estimation of microphysical parameters and atmospheric state with simulated radar data and ensemble square root Kalman filter. Part II: Parameter estimation experiments. Mon Weather Rev 136:1649–1668
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
Wang L, He WP, Wan SQ, Liao LJ, He T (2014) Evolutionary modeling for parameter estimation for chaotic system. Acta Phys Sin 63:019203 (In Chinese)
Wang L, He WP, Liao LJ, Wan SQ, He T (2015) A new method for parameter estimation in nonlinear dynamical equations. Theor Appl Climatol 119:193–202. doi:10.1007/s00704-014-1113-3
Acknowledgments
The authors would like to thank the anonymous reviewers and editors for the beneficial and helpful suggestions for this manuscript. This research was jointly supported by the National Natural Science Foundation of China (Grant No. 41475073), the National Basic Research Program of China (973 Program) (2012CB955902), and the National Natural Science Foundation of China (Grant Nos. 41275074 and 41475064).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
He, WP., Wang, L., Jiang, YD. et al. An improved method for nonlinear parameter estimation: a case study of the Rössler model. Theor Appl Climatol 125, 521–528 (2016). https://doi.org/10.1007/s00704-015-1528-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00704-015-1528-5