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On reconstruction of the Ito-like equation from persistent time series

Abstract

The Langevin equation with finite-range persistence was introduced as a macroscopic model of various geophysical phenomena. The modified histogram procedure (MHP) of reconstruction of the equation from time series was proposed. An efficiency of MHP was tested on artificial persistent time series (with short and long-tail distributions) generated by different Ito-like equations. For an exemplary geophysical time series, the appropriate Ito-like equation was reconstructed.

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References

  1. Białecki, M. (2012a), Motzkin numbers out of Random Domino Automaton, Phys. Lett. A 376,45, 3098–3100, DOI: 10.1016/j.physleta.2012.09.022.

    Article  Google Scholar 

  2. Białecki, M. (2012b), An explanation of the shape of the universal curve of the Scaling Law for the Earthquake Recurrence Time Distributions, arXiv:1210.7142 [physics.geo-ph].

    Google Scholar 

  3. Białecki, M., and Z. Czechowski (2010), 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 Processes, GeoPlanet — Earth and Planetary Sciences, Vol. 1, Springer, Berlin Heidelberg, 63–75, DOI: 10.1007/978-3-642-12300-9_5.

    Chapter  Google Scholar 

  4. Białecki, M., and Z. Czechowski (2013), On one-to-one dependence of rebound parameters on statistics of clusters: exponential and inverse-power distributions out of Random Domino Automaton, J. Phys. Soc. Jpn. 82, 014003, DOI: 10.7566/JPSJ.82.014003.

    Article  Google Scholar 

  5. Box, G.E.P., and G.M. Jenkins (1970), Time Series Analysis: Forecasting and Control, Holden-Day, San Francisco.

    Google Scholar 

  6. Brockwell, P.J., and R.A. Davis (1987), Time Series: Theory and Methods, Springer, New York.

    Book  Google Scholar 

  7. Czechowski, Z. (1991), A kinetic model of crack fusion, Geophys. J. Int. 104,2, 419–422, DOI: 10.1111/j.1365-246X.1991.tb02521.x.

    Article  Google Scholar 

  8. Czechowski, Z. (1993), A kinetic model of nucleation, propagation and fusion of cracks, J. Phys. Earth 41,3, 127–137, DOI: 10.4294/jpe1952.41.127.

    Article  Google Scholar 

  9. Czechowski, Z. (2001), Transformation of random distributions into power-like distributions due to non-linearities: application to geophysical phenomena, Geophys. J. Int. 144,1, 197–205, DOI: 10.1046/j.1365-246x.2001.00318.x.

    Article  Google Scholar 

  10. Czechowski, Z. (2003), The privilege as the cause of power distributions in geophysics, Geophys. J. Int. 154,3, 754–766, DOI: 10.1046/j.1365-246X.2003.01994.x.

    Article  Google Scholar 

  11. Czechowski, Z. (2005), The importance of the privilege in resource redistribution models for appearance of inverse-power solutions, Physica A 345,1–2, 92–106, DOI: 10.1016/j.physa.2004.07.014.

    Google Scholar 

  12. Czechowski, Z. (2010), The importance of privilege for the appearance of long-tail distributions. In: V. de Rubeis, Z. Czechowski, and R. Teisseyre (eds.), Synchronization and Triggering: from Fracture to Earthquake Processes, GeoPlanet — Earth and Planetary Sciences, Vol. 1, Springer, Berlin Heidelberg, 97–119, DOI: 10.1007/978-3-642-12300-9_7.

    Chapter  Google Scholar 

  13. 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 Processes, GeoPlanet: Earth and Planetary Sciences, Vol. 1, Springer, Berlin Heidelberg, 77–96, DOI: 10.1007/978-3-642-12300-9_6.

    Chapter  Google Scholar 

  14. Czechowski, Z., and M. Białecki (2012a), Ito equations out of domino cellular automaton with efficiency parameters, Acta Geophys. 60,3, 846–857, DOI: 10.2478/s11600-012-0021-0.

    Article  Google Scholar 

  15. Czechowski, Z., and M. Białecki (2012b), Three-level description of the domino cellular automaton, J. Phys. A 45,15, 155101, DOI: 10.1088/1751-8113/45/15/155101.

    Article  Google Scholar 

  16. Czechowski, Z., and A. Rozmarynowska (2008), The importance of the privilege for appearance of inverse-power solutions in Ito equations, Physica A 387,22, 5403–5416, DOI: 10.1016/j.physa.2008.06.007.

    Article  Google Scholar 

  17. Czechowski, Z., and L. Telesca (2011), The construction of an Ito model for geoelectrical signals, Physica A 390,13, 2511–2519, DOI: 10.1016/j.physa.2011.02.049.

    Article  Google Scholar 

  18. Gardiner, C.W. (1985), Handbook of Stochastic Methods: For Physics, Chemistry and the Natural Sciences, Springer, Berlin.

    Google Scholar 

  19. Grasman, J., and O.A. van Herwaarden (1999), Asymptotic Methods for the Fokker-Planck Equation and the Exit Problem in Applications, Springer, Berlin Heidelberg, DOI: 10.1007/978-3-662-03857-4.

    Book  Google Scholar 

  20. Kantelhardt, J.W., E. Koscielny-Bunde, H.H.A. Rego, S. Havlin, and A. Bunde (2001), Detecting long-range correlations with detrended fluctuation analysis, Physica A 295,3-4, 441–454, DOI: 10.1016/S0378-4371(01)00144-3.

    Article  Google Scholar 

  21. Mandelbrot, B.B., and J.W. Van Ness (1968), Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10,4, 422–437, DOI: 10.1137/1010093.

    Article  Google Scholar 

  22. Øksendal, B. (1998), Stochastic Differential Equations: An Introduction with Applications, 5th ed., Springer, Berlin Heidelberg, 324 pp.

    Google Scholar 

  23. Risken, H. (1996), The Fokker-Planck Equation. Methods of Solution and Applications, 3rd ed., Springer, Berlin Heidelberg.

    Google Scholar 

  24. Rozmarynowska, A. (2009), On the reconstruction of Ito models on the basis of time series with long-tail distributions, Acta Geophys. 57,2, 311–329, DOI: 10.2478/s11600-008-0074-2.

    Article  Google Scholar 

  25. 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.

    Article  Google Scholar 

  26. Telesca, L., and Z. Czechowski (2012), Discriminating geoelectrical signals measured in seismic and aseismic areas by using Ito models, Physica A 391,3, 809–818, DOI: 10.1016/j.physa.2011.09.006.

    Article  Google Scholar 

  27. Tsallis, C. (2012), Nonadditive entropy Sq and nonextensive statistical mechanics: applications in geophysics and elsewhere, Acta Geophys. 60,3, 502–525, DOI: 10.2478/s11600-012-0005-0.

    Article  Google Scholar 

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Correspondence to Zbigniew Czechowski.

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Czechowski, Z. On reconstruction of the Ito-like equation from persistent time series. Acta Geophys. 61, 1504–1521 (2013). https://doi.org/10.2478/s11600-013-0117-1

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Key words

  • stochastic processes
  • nonlinear time series modeling
  • persistence
  • Ito equation
  • Fokker-Planck equation