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Time-singularity multifractal spectrum distribution based on detrended fluctuation analysis

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  • Xiong, Gang
  • Yu, Wenxian
  • Zhang, Shuning

Abstract

The time-singularity multifractal spectrum distribution (TS-MFSD) generalizes the singularity spectrum in a time-varying framework. In this paper, a new method to compute MFSD based on detrended fluctuation analysis (DFA-MFSD) is introduced. We relate DFA-MFSD method to the standard partition function based multifractal spectrum distribution formalism, and prove that both approaches are equivalent for fractal time series with compact support. Furthermore, we find that DFA-MFSD has equivalent results, better mathematic foundation, less computational cost and is more adapted for fractal time series with arbitrary length, compared with MFSD based on wavelet transform modulus maxima (WTMM-MFSD). By analyzing several examples, this paper shows that DFAm-MFSD with different polynomial fitting orders can reliably determine the time-varying multifractal scaling behavior of time series, including processes embodying chirp-type or oscillating singularities. To illustrate these results, simulations are executed using binomial multiplicative cascades, wavelet series and real sea clutter, and simulations indicate that DFAm-MFSD benefits from excellent theoretical and practical performances.

Suggested Citation

  • Xiong, Gang & Yu, Wenxian & Zhang, Shuning, 2015. "Time-singularity multifractal spectrum distribution based on detrended fluctuation analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 437(C), pages 351-366.
  • Handle: RePEc:eee:phsmap:v:437:y:2015:i:c:p:351-366
    DOI: 10.1016/j.physa.2015.05.049
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    References listed on IDEAS

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    1. Serrano, E. & Figliola, A., 2009. "Wavelet Leaders: A new method to estimate the multifractal singularity spectra," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(14), pages 2793-2805.
    2. Zhi-Qiang Jiang & Wei-Xing Zhou, 2011. "Multifractal detrending moving average cross-correlation analysis," Papers 1103.2577, arXiv.org, revised Mar 2011.
    3. Bashan, Amir & Bartsch, Ronny & Kantelhardt, Jan W. & Havlin, Shlomo, 2008. "Comparison of detrending methods for fluctuation analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(21), pages 5080-5090.
    4. Gulich, Damián & Zunino, Luciano, 2014. "A criterion for the determination of optimal scaling ranges in DFA and MF-DFA," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 397(C), pages 17-30.
    5. Amir Bashan & Ronny Bartsch & Jan W. Kantelhardt & Shlomo Havlin, 2008. "Comparison of detrending methods for fluctuation analysis," Papers 0804.4081, arXiv.org.
    6. 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.
    7. Gao-Feng Gu & Wei-Xing Zhou, 2010. "Detrending moving average algorithm for multifractals," Papers 1005.0877, arXiv.org, revised Jun 2010.
    8. Kantelhardt, Jan W & Koscielny-Bunde, Eva & Rego, Henio H.A & Havlin, Shlomo & Bunde, Armin, 2001. "Detecting long-range correlations with detrended fluctuation analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 295(3), pages 441-454.
    9. Wei-Xing Zhou, 2008. "Multifractal detrended cross-correlation analysis for two nonstationary signals," Papers 0803.2773, arXiv.org.
    10. Xiong, Gang & Zhang, Shuning & Liu, Qiang, 2012. "The time-singularity multifractal spectrum distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(20), pages 4727-4739.
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    Cited by:

    1. Zhao, Zhen-yu & Zhu, Jiang & Xia, Bo, 2016. "Multi-fractal fluctuation features of thermal power coal price in China," Energy, Elsevier, vol. 117(P1), pages 10-18.

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