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Time-scale analyses and self-similar stochastic processes

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Wavelets and Their Applications

Part of the book series: NATO ASI Series ((ASIC,volume 442))

Abstract

A number of different physical situations (e.g.,“1 /f noise,” turbulence, texture analysis,… ) give rise to fractal or fractal-like signals, modeled as samples of self-similar processes. This motivates the development of specific methods for characterizing self-similarity structures in signals and for efficiently estimating the corresponding scaling laws. The recently introduced techniques of time-scale analysis (wavelet transforms and bilinear generalizations) offer such a possibility, especially in the case of locally self-similar processes, i.e., those for which scaling laws are time-dependent. In this respect, basics of linear and bilinear time-scale theories will be reviewed in connection with more classical methods. Starting from idealizations such as white noise, Poisson process, or (fractional) Brownian motion and variations thereof, we will show what advantages can be gained from time-scale approaches, either for obtaining almost Karhunen-Loève (doubly orthogonal) representations via orthogonal wavelet bases, or for defining general classes of estimators (aimed at scaling exponents) via bilinear time-scale representations which generalize the usual Wigner-Ville distribution.

Part of the material reported here is based upon joint works with P. Abry and P. Gonçalvès, who are gratefully acknowledged for stimulating and fruitful discussions.

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Author information

Authors and Affiliations

  1. Ecole Normale Supérieure de Lyon, Laboratoire de Physique (URA 1325 CNRS), 46 allée d’Italie, 69364, Lyon Cedex 07, France

    Patrick Flandrin

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  1. Patrick Flandrin
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Editor information

Editors and Affiliations

  1. Prometheus Inc., Newport, Rhode Island, USA

    J. S. Byrnes

  2. University of Massachusetts, Boston, Massachusetts, USA

    J. S. Byrnes , Kathryn A. Hargreaves  & Karl Berry ,  & 

  3. Harvard University, Cambridge, Massachusetts, USA

    Jennifer L. Byrnes

  4. Interleaf, Cambridge, Massachusetts, USA

    Karl Berry

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© 1994 Springer Science+Business Media Dordrecht

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Flandrin, P. (1994). Time-scale analyses and self-similar stochastic processes. In: Byrnes, J.S., Byrnes, J.L., Hargreaves, K.A., Berry, K. (eds) Wavelets and Their Applications. NATO ASI Series, vol 442. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-1028-0_7

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  • DOI: https://doi.org/10.1007/978-94-011-1028-0_7

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-4448-6

  • Online ISBN: 978-94-011-1028-0

  • eBook Packages: Springer Book Archive

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