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

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 1. From global to local characterization of the regularity of fractal functions In many situations in physics as well as in some applied sciences, one is faced to the problem of characterizing very irregular functions [1–11]. The examples range from plots of various kinds of random walks, e.g. Brownian signals [12,13], to nancial timeseries [14–16], 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 quali ed as fractal functions [1,13,23–25] whenever their graphs are fractal sets in R2 (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 : ∀x0 ∈R; ∃H ∈R such that for any ¿0, one has f(x0 + x)− f(x0)' H (f(x0 + x)− f(x0)) : (1) ∗ Corresponding author. Tel.: +33 556845665; fax: +33 556845600; e-mail: [email protected]. 0378-4371/98/$19.00 Copyright c © 1998 Elsevier Science B.V. All rights reserved PII S 0378-4371(98)00002 -8 A. Arneodo et al. / Physica A 254 (1998) 24–45 25 If f is a stochastic process, this identity holds in law for xed and x0. The exponent H is called the roughness or Hurst exponent [1,3,6]. The graph of f is self-similar if it is invariant under some isotropic dilations, i.e., when the Hurst exponent H =1. Let us note that if H¡1, then f is not di erentiable and the smaller the exponent H , the more singular f. Thus the Hurst exponent provides indication of how globally irregular the function f is. Di erent methods [6] (e.g., height-height correlation function, variance and power spectral methods) have been used to estimate the roughness exponent which is supposed to be related to the fractal dimension DF = 2 − H of the graph of the considered function. But beyond some practical algorithmic limitations, there exists a more fundamental intrinsic insu ciency in the fractal dimension and=or roughness exponent measurement in the sense that DF as well as H are global quantities that do not account for possible uctuations (from point to point) in the local regularity of f. To describe these nonhomogeneous functions, one thus needs to change slightly the de nition of the Hurst regularity so that it becomes a local quantity [26–29]: |f(x0 + l)− f(x0)| ∼ l0 : (2) This “local roughness exponent” h(x0) corresponds to the H older exponent of f at the point x0 when h(x0)¡1. The H older exponent characterizes the strength of the singularity of f at the point x0 and is de ned as the largest exponent such that there exists a polynomial Pn(x − x0) of order n satisfying [30–33] |f(x)− Pn(x − x0)|6C|x − x0| ; (3) for x in the neighborhood of x0. If h(x0)∈ ]n; n + 1[, one can easily prove that f is n times, but not n + 1 times di erentiable at the point x0. The polynomial Pn(x − x0) corresponds to the Taylor series of f around x= x0 up to the order n. Thus the higher the exponent h(x0), the more regular the function f at the point x0. 2. Statistical analysis of the regularity of fractal functions: the multifractal formalism The multifractal formalism [34–43] has been originally established to account for the statistical scaling properties of singular measures arising in various situations in physics, chemistry, geology or biology. Notable examples include the invariant probability distribution on a strange attractor [36,38,43], the distribution of voltage drops across a random resistor network [2,3,44], the distribution of growth probabilities on the boundary of a di usion-limited aggregate [3,4,45] and the spatial distribution of dissipative regions in a turbulent ow [41,46–48]. This formalism lies upon the determination of the so-called f( ) singularity spectrum [36] which characterizes the relative contribution of each singularity of the measure: let S be the subset of points x where the measure of an -box Bx( ), centered at x, scales like (Bx( ))∼ in the limit → 0+, then by de nition, f( ) is the Hausdor dimension of S : f( )= dimH(S ). 26 A. Arneodo et al. / Physica A 254 (1998) 24–45 Actually, there exists a deep analogy that links the multifractal formalism with that of statistical thermodynamics [49–51]. This analogy provides a natural connection between the f( ) spectrum and a directly observable spectrum (q) de ned from the power-law behavior, in the limit → 0+, of the partition function [36]: