Wavelet Based Multifractal Formalism: Applications to DNA Sequences, Satellite Images of the Cloud Structure, and Stock
We elaborate on a unified thermodynamic description of multifractal distributions including measures and functions. This new approach relies on the computation of partition functions from the wavelet transform skeleton defined by the wavelet transform mod
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A. Bunde et al., The Science of Disasters © Springer-Verlag Berlin Heidelberg 2002
2. Wavelet Based Multifractal Formalism: Applications to DNA Sequences, Satellite Images of the Cloud Structure, and Stock Market Data Alain Arneodo, Benjamin Audit, Nicolas Decoster, Jean-Francois Muzy, and Cedric Vaillant
We elaborate on a unified thermodynamic description of multifractal distributions including measures and functions. This new approach relies on the computation of partition functions from the wavelet transform skeleton defined by the wavelet transform modulus maxima (WTMM). This skeleton provides an adaptive space-scale partition of the fractal distribution under study, from which one can extract the D(h) singularity spectrum as the equivalent of a thermodynamic function. With some appropriate choice of the analyzing wavelet, we show that the WTMM method provides a natural generalization of the classical box-counting and structure function techniques. We then extend this method to multifractal image analysis, with the specific goal to characterize statistically the roughness fluctuations of fractal surfaces. As a very promising perspective, we demonstrate that one can go even deeper in the multifractal analysis by studying correlation functions in both space and scales. Actually, in the arborescent structure of the WT skeleton is somehow uncoded the multiplicative cascade process that underlies the multifractal properties of the considered deterministic or random function. To illustrate our purpose, we report on the most significant results obtained when applying our concepts and methodology to three experimental situations, namely the statistical analysis of DNA sequences, of high resolution satellite images of the cloud structure, and of stock market data.
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Fig. 2.0. Local estimate of the r.m.s. a(a, x) of the WT coefficients of the A DNA walk, computed with the Mexican hat analyzing wavelet g(2). a( a) is computed over a window of width l = 2000, sliding along the first 106 bp of the yeast chromosome IV (a), Escherichia coli (b), and a human contig (c). loglO a(a) - 2/3log lO a is coded using 128 colors from black (min) to red (max). In this space-scale wavelet like representation, x and a are expressed in nucleotide units. The horizontal white dashed lines mark the scale a* where some minimum is observed consistently along the entire genomes: a* = 200 bp for Saccharomyces cerevisiae, a* = 200 bp for Escherichia coli, and a* = 100bp for the human contig
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Alain Arneodo et al.
2.1 Introduction In the real world, it is often the case that a wide range of scales is needed to characterize physical properties. Actually, multiscale phenomena seem to be ubiquitous in nature. A paradigmatic illustration of such situation are fractals which are complex mathematical objects that have no minimal natural length scale. The relevance of fractals to physics and many other fields was pointed out by Mandelbrot [2.1] who demonstrated the richness of fractal
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