Thermodynamics of fractal signals based on wavelet analysis: application to fully developed turbulence data and DNA sequences

Abstract

We use the continuous wavelet transform to generalize the multifractal formalism to fractal functions. We report the results of recent applications of the so-called wavelet transform modulus maxima (WTMM) method to fully developed turbulence data and DNA sequences. We conclude by brie y describing some works currently under progress, which are likely to be the guidelines for future research. c 1998 Elsevier Science B.V. All rights reserved From global to local characterization of the regularity of fractal functionsIn many situations in physics as well as in some applied sciences, one is faced to the problem of characterizing very irregular functions [1][2][3][4][5][6][7][8][9][10][11]. The examples range from plots of various kinds of random walks, e.g. Brownian signals [12,13], to ÿnancial timeseries [14][15][16], to geological shapes [1,9,17], to medical time-series [18], to interfaces developing in far from equilibrium growth processes [4,6,11], to turbulent velocity signals [10,19,20] and to "DNA walks" coding nucleotide sequences [21,22]. These functions can be qualiÿed as fractal functions [1,13,[23][24][25] whenever their graphs are fractal sets in R 2 (for our purpose here we will only consider functions from R to R). They are commonly called self-a ne functions since their graphs are similar to themselves when transformed by anisotropic dilations, i.e., when shrinking along the x-axis by a factor followed by a rescaling of the increments of the function by a di erent factor −H

Wavelet to predict bacterial ori and ter: a tendency towards a physical balance

“…We used wavelet transform methods [27,28]. in the analysis of these bacterial genomes to detect the genomic features that might be associated with the locations of ori and ter , including AT and GC skews [16,20].…”

“…These features of wavelet analysis are ideal for genomic analysis. As one application, wavelet analysis has been used on the G+C patterns occurring in genomes [27-29]. More recently, this technique has also been shown to be very successful in extracting quantitative information on the structure and dynamics of the nucleosomes [30].…”

Multifractality in the stock market: price increments versus waiting times

“…The concept of multifractality was developed in order to describe the scaling properties of singular measures and functions which exhibit the presence of various distinct scaling exponents in their different parts [19,20]. Soon the related formalism was successfully applied to characterize empirical data in many distant fields like turbulence [21,22], earth science [23], genetics [24,25,26], physiology [27,28,29] and, as already mentioned, in finance. The problem of detecting multifractality in real data is delicate, however.…”

A novel iterative approach for mapping local singularities from geochemical data

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