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On the existence of proper stochastic Markov models for statistical reconstruction and prediction of chaotic time series. (English) Zbl 1448.37099

Summary: In this paper, the problem of statistical reconstruction and prediction of chaotic systems with unknown governing equations using stochastic Markov models is investigated. Using the time series of only one measurable state, an algorithm is proposed to design any orders of Markov models and the approach is state transition matrix extraction. Using this modeling, two goals are followed: first, using the time series, statistical reconstruction is performed through which the probability density and conditional probability density functions are reconstructed; and second, prediction is performed. For this problem, some estimators are required and here the maximum likelihood and the conditional expected value are used. The efficiency of this algorithm has been investigated by applying it to some typical data. To illustrate the advantages of this model, a deterministic model based on the nearest neighbors in the delayed phase space is used. It will be seen that using the appropriate Markov model order and sufficient number of meshes, the proposed algorithm, unlike the deterministic model, is capable of reconstructing and predicting the chaotic data in a many steps.

MSC:

37M10 Time series analysis of dynamical systems
62M05 Markov processes: estimation; hidden Markov models
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[1] Yin, G. G.; Zhang, Q., Discrete-time Markov chains: two-time-scale methods and applications, 55 (2006), Springer Science & Business Media
[2] Ching, W.-K.; Ng, M. K., Markov chains, Models, algorithms and applications (2006) · Zbl 1089.60003
[3] Fink, G. A., Markov models for pattern recognition: from theory to applications (2014), Springer Science & Business Media · Zbl 1307.68001
[4] Tongal, H.; Berndtsson, R., Impact of complexity on daily and multi-step forecasting of streamflow with chaotic, stochastic, and black-box models, Stochastic Environ Res Risk Assess, 31, 3, 661-682 (2017)
[5] Box, G. E., Time series analysis: forecasting and control (2015), John Wiley & Sons · Zbl 1348.91001
[6] Vangelov, B, Barahona, M. Modelling the dynamics of biological systems with the geometric hidden Markov model. bioRxiv, 2017: p. 224063.; Vangelov, B, Barahona, M. Modelling the dynamics of biological systems with the geometric hidden Markov model. bioRxiv, 2017: p. 224063.
[7] Xie, N.-M.; Yuan, C.-Q.; Yang, Y.-J., Forecasting China’s energy demand and self-sufficiency rate by grey forecasting model and Markov model, Int J Electr Power Energy Syst, 66, 1-8 (2015)
[8] Ye, N., Vehicle trajectory prediction based on hidden Markov model, KSII Trans Internet Inform Syst, 10, 7 (2016)
[9] Ye, N., A method for driving route predictions based on hidden Markov model, Math Probl Eng, 2015 (2015)
[10] Krumm, J., A Markov model for driver turn prediction (2016)
[11] Myers, C., Modeling chaotic systems with hidden Markov models, (Acoustics, Speech, and signal processing, 1992. ICASSP-92., 1992 IEEE international conference on (1992), IEEE)
[12] Wu, Q.; Cao, Y., An equivalent stochastic system model for control of chaotic dynamics, (Decision and control, 1995., proceedings of the 34th IEEE conference on (1995), IEEE)
[13] Stamp, D.; Wu, Q., Prediction of chaotic time series using hidden Markov models, (Control’98. UKACC international conference on (Conf. Publ. No. 455) (1998), IET)
[14] Dangelmayr, G., Time series prediction by estimating Markov probabilities through topology preserving maps, (Applications and Science of neural networks, fuzzy systems, and evolutionary computation II (1999), International Society for Optics and Photonics)
[15] Froyland, G., Extracting dynamical behavior via Markov models, (Nonlinear dynamics and statistics (2001), Springer), 281-321
[16] Ragwitz, M.; Kantz, H., Markov models from data by simple nonlinear time series predictors in delay embedding spaces, Phys Rev E, 65, 5, Article 056201 pp. (2002)
[17] Lorenz, E. N., The predictability of a flow which possesses many scales of motion, Tellus, 21, 3, 289-307 (1969)
[18] Piccardi, C., On parameter estimation of chaotic systems via symbolic time-series analysis, Chaos, 16, 4, Article 043115 pp. (2006) · Zbl 1146.37360
[19] Bhardwaj, S., Chaotic time series prediction using combination of hidden markov model and neural nets, (Computer information systems and industrial management applications (CISIM), 2010 international conference on (2010), IEEE)
[20] Takens, F., Detecting strange attractors in turbulence, (Dynamical systems and turbulence, Warwick 1980 (1981), Springer), 366-381 · Zbl 0513.58032
[21] Salarieh, H.; Alasty, A., Control of stochastic chaos using sliding mode method, J Comput Appl Math, 225, 1, 135-145 (2009) · Zbl 1162.65062
[22] Salarieh, H.; Alasty, A., Chaos synchronization of nonlinear gyros in presence of stochastic excitation via sliding mode control, J Sound Vib, 313, 3-5, 760-771 (2008)
[23] Verhulst, P., Mathematical researches into the law of population growth increase, Nouveaux Mémoires de l’Académie Royale des Sciences et Belles-Lettres de Bruxelles, 18, 1-42 (1845)
[24] Devaney, R. L., A first course in chaotic dynamical systems: theory and experiment, Comput Phys, 7, 4, 416-417 (1993)
[25] Gulick, D., Encounters with chaos (1992), McGraw-Hill
[26] Cao, L., Practical method for determining the minimum embedding dimension of a scalar time series, Physica D, 110, 1-2, 43-50 (1997) · Zbl 0925.62385
[27] Wolf, A., Determining Lyapunov exponents from a time series, Physica D, 16, 3, 285-317 (1985) · Zbl 0585.58037
[28] Kantz, H.; Schreiber, T., Nonlinear time series analysis, 7 (2004), Cambridge university press · Zbl 1050.62093
[29] Gutiérrez, J, Iglesias, A, Rodriguez, M.J.P.R.E. Logistic map driven by dichotomous noise. 1993.48(4): p. 2507.; Gutiérrez, J, Iglesias, A, Rodriguez, M.J.P.R.E. Logistic map driven by dichotomous noise. 1993.48(4): p. 2507.
[30] Munmuangsaen, B.; Srisuchinwong, B. J.C., Solitons, and Fractals, A hidden chaotic attractor in the classical Lorenz system, 107, 61-66 (2018) · Zbl 1380.34085
[31] us.spindices.com. 2019 [cited 2019 2/2/2019]; Available from: https://us.spindices.com/indices/commodities/sp-gsci-natural-gas; us.spindices.com. 2019 [cited 2019 2/2/2019]; Available from: https://us.spindices.com/indices/commodities/sp-gsci-natural-gas
[32] Rowlands, G.; Sprott, J., Extraction of dynamical equations from chaotic data, Physica D, 58, 1-4, 251-259 (1992) · Zbl 1194.37141
[33] Paduart, J., Identification of nonlinear systems using polynomial nonlinear state space models, Automatica, 46, 4, 647-656 (2010) · Zbl 1193.93089
[34] Hajiloo, R.; Salarieh, H.; Alasty, A., Chaos control in delayed phase space constructed by the Takens embedding theory, Commun Nonlinear Sci Numer Simul, 54, 453-465 (2018) · Zbl 1510.37121
[35] Kaveh, H.; Salarieh, H.; Hajiloo, R., On the control of unknown continuous time chaotic systems by applying Takens embedding theory, Chaos Solitons Fractals, 109, 53-57 (2018) · Zbl 1390.93387
[36] Judd, K.; Mees, A., On selecting models for nonlinear time series, Physica D, 82, 4, 426-444 (1995) · Zbl 0888.58034
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