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The time-singularity multifractal spectrum distribution

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  • Xiong, Gang
  • Zhang, Shuning
  • Liu, Qiang

Abstract

Although the multifractal singularity spectrum revealed the distribution of singularity exponent, it failed to consider the temporal information, therefore it is hard to describe the dynamic evolving process of non-stationary and nonlinear systems. In this paper, we aim for a multifractal analysis and propose a time-singularity multifractal spectrum distribution (TS-MFSD), which will hopefully reveal the spatial dynamic character of fractal systems. Similar to the Wigner–Ville time-frequency distribution, the time-delayed conjugation of fractal signals is selected as the windows function. Furthermore, the time-varying Holder exponent and the time-varying wavelet singularity exponent are deduced based on the instantaneous self-correlation fractal signal. The time-singularity exponent distribution i.e. TS-MFSD is proposed, which involves time-varying Hausdorff singularity spectrum distribution, time-varying large deviation multifractal spectrum and time-varying Legendre spectrum distribution, which exhibit the singularity exponent distribution of fractal signal at arbitrary time. Finally, we studied the algorithm of the TS-MFSD based on the wavelet transform module maxima method, analyzed and discussed the characteristic of TS-MFSD based on Devil Staircase signal, stochastic fractional motion and real sea clutter.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:phsmap:v:391:y:2012:i:20:p:4727-4739
    DOI: 10.1016/j.physa.2012.05.026
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    1. Lahmiri, Salim, 2017. "Multifractal analysis of Moroccan family business stock returns," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 486(C), pages 183-191.
    2. 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.
    3. Xiong, Gang & Zhang, Shuning & Zhao, Huichang, 2014. "Multifractal spectrum distribution based on detrending moving average," Chaos, Solitons & Fractals, Elsevier, vol. 65(C), pages 97-110.
    4. Liu, Jie & Li, Qiuping & Wang, Xiaoran & Wang, Zaiquan & Lu, Shouqing & Sa, Zhanyou & Wang, Hao, 2022. "Dynamic multifractal characteristics of acoustic emission about composite coal-rock samples with different strength rock," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    5. Xi, Caiping & Zhang, Shunning & Xiong, Gang & Zhao, Huichang, 2016. "A comparative study of two-dimensional multifractal detrended fluctuation analysis and two-dimensional multifractal detrended moving average algorithm to estimate the multifractal spectrum," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 454(C), pages 34-50.
    6. Shen, Na & Chen, Jiayi, 2023. "Asymmetric multifractal spectrum distribution based on detrending moving average cross-correlation analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 615(C).

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