Scaling is a mathematical transformation that enlarges or diminishes objects. The technique is used in a variety of areas, including finance and image processing. This book is organized around the notions of scaling phenomena and scale invariance. The various stochastic models commonly used to describe scaling — self-similarity, long-range dependence and multi-fractals — are introduced. These models are compared and related to one another. Next, fractional integration, a mathematical tool closely related to the notion of scale invariance, is discussed, and stochastic processes with prescribed scaling properties (self-similar processes, locally self-similar processes, fractionally filtered processes, iterated function systems) are defined. A number of applications where the scaling paradigm proved fruitful are detailed: image processing, financial and stock market fluctuations, geophysics, scale relativity, and fractal time-space.
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Patrice Abry is a Professor in the Laboratoire de Physique
at the Ecole Normale Superieure de Lyon, France. His current
research interests include wavelet-based analysis and modelling of
scaling phenomena and related topics, stable processes,
multi-fractal, long-range dependence, local regularity of
processes, infinitely divisible cascades and departures from exact
scale invariance.
Paulo Goncalves graduated from the Signal Processing
Department of ICPI, Lyon, France in 1993. He received the Masters
(DEA) and Ph.D. degrees in signal processing from the Institut
National Polytechnique, Grenoble, France, in 1990 and 1993
respectively. While working toward his Ph.D. degree, he was with
Ecole Normale Superieure, Lyon. In 1994-96, he was a Postdoctoral
Fellow at Rice University, Houston, TX. Since 1996, he is associate
researcher at INRIA, first with Fractales (1996-99), and then with
a research team at INRIA Rhone-Alpes (2000-2003). His research
interests are in multiscale signal and image analysis, in
wavelet-based statistical inference, with application to
cardiovascular research and to remote sensing for land cover
classification.
Jacques Levy Vehel graduated from Ecole Polytechnique in
1983 and from Ecole Nationale Superieure des Telecommuncations in
1985. He holds a Ph.D in Applied Mathematics from Universite
d’Orsay. He is currently a research director at INRIA,
Rocquencourt, where he created the Fractales team, a research group
devoted to the study of fractal analysis and its applications to
signal/image processing. He also leads a research team at IRCCYN,
Nantes, with the same scientific focus. His current research
interests include (multi)fractal processes, 2-microlocal analysis
and wavelets, with application to Internet traffic, image
processing and financial data modelling.
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