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Scale Dependencies and Self-Similar Models with Wavelet Scattering Spectra

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Listed:
  • Rudy Morel
  • Gaspar Rochette
  • Roberto Leonarduzzi
  • Jean-Philippe Bouchaud
  • St'ephane Mallat

Abstract

We introduce the wavelet scattering spectra which provide non-Gaussian models of time-series having stationary increments. A complex wavelet transform computes signal variations at each scale. Dependencies across scales are captured by the joint correlation across time and scales of wavelet coefficients and their modulus. This correlation matrix is nearly diagonalized by a second wavelet transform, which defines the scattering spectra. We show that this vector of moments characterizes a wide range of non-Gaussian properties of multi-scale processes. We prove that self-similar processes have scattering spectra which are scale invariant. This property can be tested statistically on a single realization and defines a class of wide-sense self-similar processes. We build maximum entropy models conditioned by scattering spectra coefficients, and generate new time-series with a microcanonical sampling algorithm. Applications are shown for highly non-Gaussian financial and turbulence time-series.

Suggested Citation

  • Rudy Morel & Gaspar Rochette & Roberto Leonarduzzi & Jean-Philippe Bouchaud & St'ephane Mallat, 2022. "Scale Dependencies and Self-Similar Models with Wavelet Scattering Spectra," Papers 2204.10177, arXiv.org, revised Jun 2023.
    • Handle: RePEc:arx:papers:2204.10177
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    References listed on IDEAS

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    Cited by:

    1. Marcus Haggbom & Morten Karlsmark & Joakim And'en, 2024. "Mean-Field Microcanonical Gradient Descent," Papers 2403.08362, arXiv.org.

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