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The fractal energy measurement and the singularity energy spectrum analysis

Listed author(s):
  • Xiong, Gang
  • Zhang, Shuning
  • Yang, Xiaoniu
Registered author(s):

    The singularity exponent (SE) is the characteristic parameter of fractal and multifractal signals. Based on SE, the fractal dimension reflecting the global self-similar character, the instantaneous SE reflecting the local self-similar character, the multifractal spectrum (MFS) reflecting the distribution of SE, and the time-varying MFS reflecting pointwise multifractal spectrum were proposed. However, all the studies were based on the depiction of spatial or differentiability characters of fractal signals. Taking the SE as the independent dimension, this paper investigates the fractal energy measurement (FEM) and the singularity energy spectrum (SES) theory. Firstly, we study the energy measurement and the energy spectrum of a fractal signal in the singularity domain, propose the conception of FEM and SES of multifractal signals, and investigate the Hausdorff measure and the local direction angle of the fractal energy element. Then, we prove the compatibility between FEM and traditional energy, and point out that SES can be measured in the fractal space. Finally, we study the algorithm of SES under the condition of a continuous signal and a discrete signal, and give the approximation algorithm of the latter, and the estimations of FEM and SES of the Gaussian white noise, Fractal Brownian motion and the multifractal Brownian motion show the theoretical significance and application value of FEM and SES.

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    Article provided by Elsevier in its journal Physica A: Statistical Mechanics and its Applications.

    Volume (Year): 391 (2012)
    Issue (Month): 24 ()
    Pages: 6347-6361

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    Handle: RePEc:eee:phsmap:v:391:y:2012:i:24:p:6347-6361
    DOI: 10.1016/j.physa.2012.07.056
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    1. Stanley, H.E. & Afanasyev, V. & Amaral, L.A.N. & Buldyrev, S.V. & Goldberger, A.L. & Havlin, S. & Leschhorn, H. & Maass, P. & Mantegna, R.N. & Peng, C.-K. & Prince, P.A. & Salinger, M.A. & Stanley, M., 1996. "Anomalous fluctuations in the dynamics of complex systems: from DNA and physiology to econophysics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 224(1), pages 302-321.
    2. Wei-Xing Zhou, 2009. "The components of empirical multifractality in financial returns," Papers 0908.1089,, revised Oct 2009.
    3. Alvarez-Ramirez, Jose & Rodriguez, Eduardo & Carlos Echeverría, Juan, 2005. "Detrending fluctuation analysis based on moving average filtering," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 354(C), pages 199-219.
    4. Schumann, Aicko Y. & Kantelhardt, Jan W., 2011. "Multifractal moving average analysis and test of multifractal model with tuned correlations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(14), pages 2637-2654.
    5. Zhi-Qiang Jiang & Wei-Xing Zhou, 2011. "Multifractal detrending moving average cross-correlation analysis," Papers 1103.2577,, revised Mar 2011.
    6. B. Lashermes & S. G. Roux & P. Abry & S. Jaffard, 2008. "Comprehensive multifractal analysis of turbulent velocity using the wavelet leaders," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 61(2), pages 201-215, January.
    7. Castro e Silva, A. & Moreira, J.G., 1997. "Roughness exponents to calculate multi-affine fractal exponents," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 235(3), pages 327-333.
    8. Kantelhardt, Jan W. & Zschiegner, Stephan A. & Koscielny-Bunde, Eva & Havlin, Shlomo & Bunde, Armin & Stanley, H.Eugene, 2002. "Multifractal detrended fluctuation analysis of nonstationary time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 316(1), pages 87-114.
    9. Ayache, Antoine & Lévy Véhel, Jacques, 2004. "On the identification of the pointwise Hölder exponent of the generalized multifractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 111(1), pages 119-156, May.
    10. Gao-Feng Gu & Wei-Xing Zhou, 2010. "Detrending moving average algorithm for multifractals," Papers 1005.0877,, revised Jun 2010.
    11. Peng, C.-K. & Buldyrev, S.V. & Goldberger, A.L. & Havlin, S. & Mantegna, R.N. & Simons, M. & Stanley, H.E., 1995. "Statistical properties of DNA sequences," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 221(1), pages 180-192.
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