Automatic algorithm to decompose discrete paths of fractional Brownian motion into self-similar intrinsic components

Regular Article


Fractional Brownian motion (fBm) is a nonstationary self-similar continuous stochastic process used to model many natural phenomena. A realization of the fBm can be numerically approximated by discrete paths which do not entirely preserve the self-similarity. We investigate the self-similarity at different time scales by decomposing the discrete paths of fBm into intrinsic components. The decomposition is realized by an automatic numerical algorithm based on successive smoothings stopped when the maximum monotonic variation of the averaged time series is reached. The spectral properties of the intrinsic components are analyzed through the monotony spectrum defined as the graph of the amplitudes of the monotonic segments with respect to their lengths (characteristic times). We show that, at intermediate time scales, the mean amplitude of the intrinsic components of discrete fBms scales with the mean characteristic time as a power law identical to that of the corresponding continuous fBm. As an application we consider hydrological time series of the transverse component of the transport process generated as a superposition of diffusive movements on advective transport in random velocity fields. We found that the transverse component has a rich structure of scales, which is not revealed by the analysis of the global variance, and that its intrinsic components may be self-similar only in particular cases.


Computational Methods 


  1. 1.
    J. Gao, Y. Cao, W. Tung, J. Hu, Multiscale Analysis of Complex Time Series (Wiley, Hoboken, 2007)Google Scholar
  2. 2.
    D.B. Percival, A.T. Walden, Wavelet Methods for Time Series Analysis (Cambridge University Press, Cambridge, 2000) Google Scholar
  3. 3.
    N.E. Huang, Z. Shen, S.R. Long, M.C. Wu, H.H. Shih, Q. Zheng, N.C. Yen, C.C. Tung, H.H. Liu, Proc. R. Soc. Lond. A 454, 903 (1998) MathSciNetCrossRefADSMATHGoogle Scholar
  4. 4.
    L. Lin, Y. Wang, H. Zhou, Adv. Adapt. Data Anal. 1, 543 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    I. Daubechies, J. Lu, H.-T. Wu, Appl. Comput. Harmon. Anal. 30, 243 (2011)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    T.Y. Hou, Z. Shi, Adv. Adapt. Data Anal. 3, 1 (2011)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    B. Boashash, Proc. IEEE 80, 520 (1992)CrossRefADSGoogle Scholar
  8. 8.
    C. Vamoş, M. Crăciun, Eur. Phys. J. B 87, 301 (2014)CrossRefADSGoogle Scholar
  9. 9.
    B.B. Mandelbrot, J.W. Van Ness, SIAM Rev. 10, 422 (1968)MathSciNetCrossRefADSMATHGoogle Scholar
  10. 10.
    A. Carbone, G. Castelli, H.E. Stanley, Phys. Rev. E 69, 026105 (2004) CrossRefADSGoogle Scholar
  11. 11.
    A. Carbone, Phys. Rev. E 76, 056703 (2007) MathSciNetCrossRefADSGoogle Scholar
  12. 12.
    B.B. Mandelbrot, J.R. Wallis Noah, Water Resour. Res. 4, 909 (1968)CrossRefADSGoogle Scholar
  13. 13.
    T. Graves, R.B. Gramscy, N.Watkins, C.L.E. Franzke, arXiv:1406.6018 [stat.OT] (2014)Google Scholar
  14. 14.
    N. Suciu, Phys. Rev. E 81, 056301 (2010) CrossRefADSGoogle Scholar
  15. 15.
    A.N. Shiryaev, Essentials of Stochastic Finance. Facts, Models, Theory (World Scientific, Singapore, 1999) Google Scholar
  16. 16.
    R. Cont, in Proceedings of the Fractals in Engineering, edited by J. Lévy Véhel, E. Lutton (Springer, London, 2005), p. 159Google Scholar
  17. 17.
    W.E. Leland, M.S. Taqqu, W. Willinger, D.V. Wilson, IEEE/ACM Trans. Networking 2, 1 (1994)CrossRefGoogle Scholar
  18. 18.
    S. Stoev, M.S. Taqqu, C. Park, J.S. Marron, Computer Networking 48, 423 (2005)CrossRefGoogle Scholar
  19. 19.
    D.L. Turcotte, Fractals and Chaos in Geology and Geophysics, 2nd edn. (Cambridge University Press, Cambridge, 1997) Google Scholar
  20. 20.
    P.S. Addison, Fractals and Chaos. An Illustrated Course, (Institute of Physics Publishing, London, 1997) Google Scholar
  21. 21.
    M.G. Trefry, F.P. Ruan, D. McLaughlin, Water Resour. Res. 39, 1063 (2003) ADSGoogle Scholar
  22. 22.
    A. Fiori, I. Jankovic, G. Dagan, Water Resour. Res. 42, W06D13 (2006)Google Scholar
  23. 23.
    C. Vamoş, M. Crăciun, Phys. Rev. E 78, 036707 (2008) CrossRefADSGoogle Scholar
  24. 24.
    C. Vamoş, M. Crăciun, Automatic Trend Estimation (Springer, Dordrecht, 2012)Google Scholar
  25. 25.
    J.-M. Bardet, G. Lang, G. Oppenheim, A. Philippe, S. Stoev, M.S. Taqqu, in Theory and applications of long-range dependence, edited by P. Doukhan, G. Oppenheim, M. Taqqu, (Birkhäuser, Boston, 2003), p. 579Google Scholar
  26. 26.
    Y. Meyer, F. Sellan, M.S. Taqqu, J. Fourier Anal. Appl. 5, 465 (2000)MathSciNetCrossRefGoogle Scholar
  27. 27.
    P. Abry, F. Sellan, Appl. Comp. Harmonic Anal., 3, 377 (1996) MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    N. Suciu, Adv. Water Resour. 69, 114 (2014)CrossRefADSGoogle Scholar
  29. 29.
    H. Schwarze, U. Jaekel, H. Vereecken, Transport Porous Med. 43, 265 (2001)CrossRefGoogle Scholar
  30. 30.
    R.H. Kraichnan, Phys. Fluids 13, 22 (1970)CrossRefADSMATHGoogle Scholar
  31. 31.
    J. Eberhard, N. Suciu, C. Vamos, J. Phys. A 40, 597 (2007)MathSciNetCrossRefADSMATHGoogle Scholar
  32. 32.
    N. Suciu, S. Attinger, F.A. Radu, C. Vamoş, J. Vanderborght, H. Vereecken, P. Knabner, An. St. Univ. Ovidius Constanta 23, 167 (2015)Google Scholar
  33. 33.
    Y. Huang, F.G. Schmitt, J.-P. Hermand, Y. Gagne, Z. Lu, Y. Liu, Phys. Rev. E 84, 016208 (2011) CrossRefADSGoogle Scholar
  34. 34.
    L. Calvet, A. Fisher, B. Mandelbrot, Large deviations and the distribution of price changes (Cowles Foundation Discussion Paper, 1997)Google Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.T. Popoviciu Institute of Numerical AnalysisRomanian AcademyCluj-NapocaRomania
  2. 2.Mathematics DepartmentFriedrich Alexander University of Erlangen-NurembergErlangenGermany

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