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Nonparametric reconstruction of a multifractal function from noisy data
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  • Published: 08 November 2008

Nonparametric reconstruction of a multifractal function from noisy data

  • Arnaud Gloter1 &
  • Marc Hoffmann1 

Probability Theory and Related Fields volume 146, Article number: 155 (2010) Cite this article

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Abstract

We estimate a real-valued function f of d variables, subject to additive Gaussian perturbation at noise level \({\varepsilon > 0}\), under L π-loss, for π ≥ 1. The main novelty is that f can have an extremely varying local smoothness, exhibiting a so-called multifractal behaviour. The results of Jaffard on the Frisch–Parisi conjecture suggest to link the singularity spectrum of f to Besov properties of the signal that can be handled by wavelet thresholding for denoising purposes. We prove that the optimal (minimax) rate of estimation of multifractal functions with singularity spectrum d(H) has explicit representation \({\varepsilon^{2v(d({\bullet}),\pi)}}\) , with

$$ v(d({\bullet}),\pi)=\min_{H}\frac{H+\left(d-d(H)\right)/\pi}{2H+d}.$$

The minimum is taken over a specific domain and the rate is corrected by logarithmic factors in some cases. In particular, the usual rate \({\varepsilon^{2s/(2s+d)}}\) is retrieved for monofractal functions (with spectrum reduced to a single value s) irrespectively of π. More interestingly, the sparse case of estimation over single Besov balls has a new interpretation in terms of multifractal analysis.

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Authors and Affiliations

  1. Laboratoire d’Analyse et de Mathématiques Appliquées, Université Paris-Est and CNRS (UMR 8050), 5 boulevard Descartes, 77454, Marne-la-Vallée, Cedex 2, France

    Arnaud Gloter & Marc Hoffmann

Authors
  1. Arnaud Gloter
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  2. Marc Hoffmann
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Correspondence to Arnaud Gloter.

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Cite this article

Gloter, A., Hoffmann, M. Nonparametric reconstruction of a multifractal function from noisy data. Probab. Theory Relat. Fields 146, 155 (2010). https://doi.org/10.1007/s00440-008-0187-1

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  • Received: 21 August 2007

  • Revised: 21 July 2008

  • Published: 08 November 2008

  • DOI: https://doi.org/10.1007/s00440-008-0187-1

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Keywords

  • Multifractal analysis
  • Frisch–Parisi conjecture
  • Wavelet threshold algorithm
  • Minimax rate of convergence

Mathematics Subject Classification (2000)

  • 62G05
  • 62G20
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