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p-exponent and p-leaders, Part II: Multifractal analysis. Relations to detrended fluctuation analysis

Author

Listed:
  • Leonarduzzi, R.
  • Wendt, H.
  • Abry, P.
  • Jaffard, S.
  • Melot, C.
  • Roux, S.G.
  • Torres, M.E.

Abstract

Multifractal analysis studies signals, functions, images or fields via the fluctuations of their local regularity along time or space, which capture crucial features of their temporal/spatial dynamics. It has become a standard signal and image processing tool and is commonly used in numerous applications of different natures. In its common formulation, it relies on the Hölder exponent as a measure of local regularity, which is by nature restricted to positive values and can hence be used for locally bounded functions only. In this contribution, it is proposed to replace the Hölder exponent with a collection of novel exponents for measuring local regularity, the p-exponents. One of the major virtues of p-exponents is that they can potentially take negative values. The corresponding wavelet-based multiscale quantities, the p-leaders, are constructed and shown to permit the definition of a new multifractal formalism, yielding an accurate practical estimation of the multifractal properties of real-world data. Moreover, theoretical and practical connections to and comparisons against another multifractal formalism, referred to as multifractal detrended fluctuation analysis, are achieved. The performance of the proposed p-leader multifractal formalism is studied and compared to previous formalisms using synthetic multifractal signals and images, illustrating its theoretical and practical benefits. The present contribution is complemented by a companion article studying in depth the theoretical properties of p-exponents and the rich classification of local singularities it permits.

Suggested Citation

  • Leonarduzzi, R. & Wendt, H. & Abry, P. & Jaffard, S. & Melot, C. & Roux, S.G. & Torres, M.E., 2016. "p-exponent and p-leaders, Part II: Multifractal analysis. Relations to detrended fluctuation analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 448(C), pages 319-339.
  • Handle: RePEc:eee:phsmap:v:448:y:2016:i:c:p:319-339
    DOI: 10.1016/j.physa.2015.12.035
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    References listed on IDEAS

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    Cited by:

    1. Patrice Abry & Yannick Malevergne & Herwig Wendt & Stéphane Jaffard & Marc Senneret & Laurent Jaffrès, 2022. "Foreign Exchange Multivariate Multifractal Analysis," Post-Print hal-03735497, HAL.
    2. Yun Chen & Huaizhong Li & Liang Hou & Xiangjian Bu & Shaogan Ye & Ding Chen, 2022. "Chatter detection for milling using novel p-leader multifractal features," Journal of Intelligent Manufacturing, Springer, vol. 33(1), pages 121-135, January.

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