Multivariate scale-free temporal dynamics: From spectral (Fourier) to fractal (wavelet) analysis

Patrice Abry; Herwig Wendt; Stéphane Jaffard; Gustavo Didier

doi:10.1016/j.crhy.2019.08.005

Fourier and the science of today / Fourier et la science d'aujourd'hui

Multivariate scale-free temporal dynamics: From spectral (Fourier) to fractal (wavelet) analysis
[Dynamique temporelle multivariée invariante d'échelle : de l'analyse spectrale (Fourier) à l'analyse fractale (ondelette)]

Patrice Abry ¹ ; Herwig Wendt ² ; Stéphane Jaffard ³ ; Gustavo Didier ⁴

¹ Université de Lyon, ENS de Lyon, CNRS, Laboratoire de physique, Lyon, France
² IRIT, CNRS (UMR 5505), Université de Toulouse, France
³ Université Paris-Est, LAMA (UMR 8050), UPEM, UPEC, CNRS, Créteil, France
⁴ Department of Mathematics, Tulane University, New Orleans, LA, USA

Comptes Rendus. Physique, Volume 20 (2019) no. 5, pp. 489-501.

Résumé
Abstract

La transformée de Fourier (ou analyse spectrale) est aujourd'hui devenue un outil universel pour l'analyse de données issues de nombreuses applications réelles de natures très différentes, particulièrement pertinent pour la caractérisation de la dynamique temporelle ou spatiale. La transformée en ondelettes (ou analyse multéchelle) peut être vue comme une analyse spectrale adaptée à des classes de signaux ou fonctions dont la dynamique est invariante d'échelle. La présente contribution propose d'abord de faire un état de l'art des relations formelles entre ces deux analyses dans le cadre des processus aléatoires stationaires multivariés, puis de montrer la capacité de la transformée en ondelettes à étendre l'analyse de l'invariance d'échelle multivariée au-delà des statistiques de second ordre (fonction de covariance et spectre de Fourier), à l'auto-similarité multivariée et à la multifractalité multivariée. Quelques illustrations et éléments de discussion sur la pertinence de ces concepts et outils pour l'analyse de l'activité cérébrale macroscopique sont proposés.

The Fourier transform (or spectral analysis) has become a universal tool for data analysis in many different real-world applications, notably for the characterization of temporal/spatial dynamics in data. The wavelet transform (or multiscale analysis) can be regarded as tailoring spectral estimation to classes of signals or functions defined by scale-free dynamics. The present contribution first formally reviews these connections in the context of multivariate stationary processes, and second details the ability of the wavelet transform to extend multivariate scale-free temporal dynamics analysis beyond second-order statistics (Fourier spectrum and autocovariance function) to multivariate self-similarity and multivariate multifractality. Illustrations and qualitative discussions of the relevance of scale-free dynamics for macroscopic brain activity description using MEG data are proposed.

Publié le : 2019-07-01

DOI : 10.1016/j.crhy.2019.08.005

Keywords: Fourier transform, wavelet transform, Multivariate signals, Scale-free dynamics, Self-similarity, Multifractality
Mot clés : Transformée de Fourier, transformée en ondelettes, Signaux multivariés, Dynamique invariante d'échelle, Auto-similarité, Multifractalité

Affiliations des auteurs :

Patrice Abry ¹ ; Herwig Wendt ² ; Stéphane Jaffard ³ ; Gustavo Didier ⁴

¹ Université de Lyon, ENS de Lyon, CNRS, Laboratoire de physique, Lyon, France
² IRIT, CNRS (UMR 5505), Université de Toulouse, France
³ Université Paris-Est, LAMA (UMR 8050), UPEM, UPEC, CNRS, Créteil, France
⁴ Department of Mathematics, Tulane University, New Orleans, LA, USA

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     title = {Multivariate scale-free temporal dynamics: {From} spectral {(Fourier)} to fractal (wavelet) analysis},
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Patrice Abry; Herwig Wendt; Stéphane Jaffard; Gustavo Didier. Multivariate scale-free temporal dynamics: From spectral (Fourier) to fractal (wavelet) analysis. Comptes Rendus. Physique, Volume 20 (2019) no. 5, pp. 489-501. doi : 10.1016/j.crhy.2019.08.005. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2019.08.005/

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