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The fractal energy measurement and the singularity energy spectrum analysis

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  • Xiong, Gang
  • Zhang, Shuning
  • Yang, Xiaoniu

Abstract

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.

Suggested Citation

  • Xiong, Gang & Zhang, Shuning & Yang, Xiaoniu, 2012. "The fractal energy measurement and the singularity energy spectrum analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(24), pages 6347-6361.
  • 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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    Cited by:

    1. Xiong, Gang & Yu, Wenxian & Zhang, Shuning, 2015. "Singularity power spectrum distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 431(C), pages 63-73.
    2. 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.

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