Abstract
Ito equations are derived for simple stochastic cellular automaton with parameters describing efficiencies for avalanche triggering and cell occupation. Analytical results are compared with the numerical one obtained from the histogram method. Good agreement for various parameters supports the wide applicability of the Ito equation as a macroscopic model of some cellular automata and complex natural phenomena which manifest random energy release. Also, the paper is an example of effectiveness of histogram procedure as an adequate method of nonlinear modeling of time series.
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Białecki, M., and Z. Czechowski (2010a), Analytic approach to stochastic cellular automata: exponential and inverse power distributions out of Random Domino Automaton, arXiv:1009.4609 [nlin.CG].
Białecki, M., and Z. Czechowski (2010b), On a simple stochastic cellular automaton with avalanches: simulation and analytical results. In: V. de Rubeis, Z. Czechowski, and R. Teisseyre (eds.), Synchronization and Triggering: from Fracture to Earthquake Processess, GeoPlanet: Earth and Planetary Sciences, Vol. 1, Springer, Berlin, 63–75, DOI: 10.1007/978-3-642-12300-9-5.
Bollerslev, T. (1986), Generalized autoregressive conditional heteroskedasticity, J. Econometrics 31, 307–327.
Borland, L. (1999), Simultaneous modelling of nonlinear deterministic and stochastic dynamics, Physica D 90,2–3, 175–190, DOI: 10.1016/S0167-2789(96)00143-1.
Borland, L., and H. Haken (1992a), Unbiased determination of forces causing observed processes. The case of additive and weak multiplicative noise, Z. Phys. B 88, 95–103, DOI: 10.1007/BF01573843.
Borland, L., and H. Haken (1992b), Unbiased estimate of forces from measured correlation functions, including the case of strong multiplicative noise, Ann. Physik 504,6, 452–459, DOI: 10.1002/andp.19925040607.
Borland, L., and H. Haken (1993), On the constraints necessary for macroscopic prediction of time-dependent stochastic processes, Rep. Math. Phys. 33,1–2, 35–42, DOI: 10.1016/0034-4877(93)90038-G.
Cryer, J.D., and K.S. Chan (2008), Time Series Analysis: With Applications in R, Springer, New York.
Czechowski, Z., and M. Białecki (2010), Ito equations as macroscopic stochastic models of geophysical phenomena — construction of the models on the basis of time series. In: V. de Rubeis, Z. Czechowski, and R. Teisseyre (eds.), Synchronization and Triggering: from Fracture to Earthquake Processess, Geo-Planet: Earth and Planetary Sciences, Vol. 1, Springer, Berlin, 77–96, DOI: 10.1007/978-3-642-12300-9-6.
Czechowski, Z., and M. Białecki (2012), Three-level description of the domino cellular automaton, J. Phys. A: Math. Theor. 45, 155101, DOI:10.1088/1751-8113/45/15/155101.
Engle, R. (1982), Autoregressive conditional heteroscedasticity with estimates of variance of United Kingdom inflation, Econometrica 50,4, 987–1008.
Fan, J., and Q. Yao (2005), Nonlinear Time Series: Nonparametric and Parametric Methods, Springer, New York.
Granger, C.W.J., and A.P. Andersen (1978), Introduction to Bilinear Time Series Models, Vandenhoeck and Puprecht, Gottingen.
Haken, H. (1988), Information and Self-Organization: A Macroscopic Approach to Complex Systems, Springer, Berlin.
Racca, E., F. Laio, D. Poggi, and L. Ridolfi (2007), Test to determine the Markov order of time series, Phys. Rev. E 75,1, 011126, DOI: 10.1103/PhysRevE.75.011126.
Risken, H. (1996), The Fokker-Planck Equation, Springer, Berlin.
Rozmarynowska, A. (2009), On the reconstruction of Ito models on the base of time series with long-tail distributions, Acta Geophys. 57,2, 311–329, DOI: 10.2478/s11600-008-0074-2.
Siegert, S., R. Friedrich, and J. Peinke (1998), Analysis of data sets of stochastic systems, Phys. Lett. A 243,5–6, 275–280, DOI: 10.1016/S0375-9601(98)00283-7.
Takens, F. (1981), Detecting strange attractors in turbulence. In: D.A. Rand and L.S. Young (eds.), Dynamical Systems and Turbulence, Lecture Notes in Mathematics Vol. 898, Springer, New York.
Tong, H. (1983), Threshold Models in Nonlinear Time Series Analysis, Lecture Notes in Statistics, Springer, New York.
Tong, H. (1990), Nonlinear Time Series Analysis: A Dynamical Systems Approach, Oxford University Press, Oxford.
Tong, H. (2011), Threshold models in time series analysis¶30 years on (with discussions by P. Whittle, M. Rosenblatt, B.E. Hansen, P. Brockwell, N.I. Samia and F. Battaglia), Statistics and Its Interface 4, 107–136.
Tong, H., and K.S. Lim (1980), Threshold autoregression, limit cycles and cyclical data, J. Roy. Statist. Soc. B 42,3, 245–292.
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Czechowski, Z., Białecki, M. Ito equations out of domino cellular automaton with efficiency parameters. Acta Geophys. 60, 846–857 (2012). https://doi.org/10.2478/s11600-012-0021-0
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DOI: https://doi.org/10.2478/s11600-012-0021-0