Journal : Acta Geophysica
Article : On the reconstruction of Ito models on the basis of time series with long-tail distributions

Authors :
Sobotka, J.
University of Wrocław, Institute of Geological Sciences, Department of Structural Geology, Wrocław, Poland,,
Sedighi, M.
K.N. Toosi University of Technology, Faculty of Geodesy and Geomatics Engineering, Tehran, Iran,,
Rezaei, K.
LMU University, Munich, Germany,,
Narayan, J.
Dept. of Earthquake Engineering, Indian Institute of Technology, Roorkee, India,,
Rozmarynowska, A.
Institute of Geophysics, Polish Academy of Sciences, Warszawa, Poland,,
Abstract : Two methods of reconstruction of Ito equations on the basis of time series were analysed. The Sequin method appeared to be completely inadequate in cases of considerable noise. The histogram method required some improvements; therefore, the procedure of smoothing of joint distribution function was proposed and verified. Ito equations may constitute some macroscopic models of phenomena in which microscopic interactions are averaged in an adequate way. Analysis of two geophysical phenomena is presented. For these two examples, Ito equations are constructed and some physical causes of resulting forms of the deterministic and stochastic force are deduced.

Keywords : Ito equation, stochastic process, nonlinear time series analysis, histogram method, Sequin method,
Publishing house : Instytut Geofizyki PAN
Publication date : 2009
Number : Vol. 57, no. 2
Page : 311 – 329

: 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, 1, 95-103, DOI: 10.1007/BF01573843.
2. Borland, L., and H. Haken (1992b), Unbiased determination of forces from measured correlation funtions, including the case of strong multiplicative noise, Ann. Phys. 1.
3. Borland, L., and H. Haken (1993), On the constraints necessary for macroscopic prediction of stochastic time-dependent processes, Rep. Math. Phys. 33, 1-2, 35-42, DOI: 10.1016/0034-4877(93)90038-G.
4. Borland, L. (1996), Simultaneous modeling of nonlinear deterministic and stochastic dynamics, Physica D 99, 2-3, 175-190, DOI: 10.1016/S0167-2789(96)00143-1.
5. Czechowski, Z. (2002), On linking of two approaches which lead to the power behaviour: The privilege concept and the non-linear models, Acta Geophys. Pol. 50, 3, 321-335.
6. 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.
7. 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.
8. Czechowski, Z., and A. Rozmarynowska (2008), The importance of the privilege for appearance of inverse-power solution in Ito equations, Physica A 387, 22, 5403-5416.
9. Gaume, E., and R. Gosset (2003), Over-parameterisation, a major obstacle to the use of artificial neural networks in hydrology, Hydrol. Earth Syst. Sc. 7, 5, 693-706.
10. Grohens, J., and J. Progen (1999), Time Series, Wolfram Research Inc., Champaign, IL.
11. Haken, H. (1988), Information and Self-Organization, Springer-Verlag, Berlin.
12. Jarosławski, J., J.W. Krzyścin, S. Puchalski, and P. Sobolewski (2003), On the optical thickness in the UV range: Analysis of the ground-based data taken at Belsk, Poland, J. Geophys. Res. 108, D23, 4722-4730, DOI: 10.1029/2003JD003571.
13. Jaynes, E.T. (1957), Information theory and statistical mechanics, Phys. Rev. 106, 4, 620-630, DOI: 10.1103/PhysRev.106.620.
14. Risken, H. (1996), The Fokker-Planck Equation. Methods of Solution and Applications, Springer-Verlag, Berlin, 472 pp.
15. Rozmarynowska, A. (2008), Construction of Ito models on the basis of time series with long-tail distributions, Ph.D. Dissertation, Institute of Geophysics, Polish Academy of Sciences (in Polish).
16. Siegert, S., R. Friedrich, and J. Peinke (1998), Analysis of data sets of stochastic systems, Physics Letters A 243, 5-6, 275, DOI: 10.1016/S0375-9601(98) 00283-7.
Qute : Sobotka, J. ,Sedighi, M. ,Rezaei, K. ,Narayan, J. ,Rozmarynowska, A. ,Rozmarynowska, A. , On the reconstruction of Ito models on the basis of time series with long-tail distributions. Acta Geophysica Vol. 57, no. 2/2009