Alain Arneodo

The thermodynamics of fractals revisited with wavelets

The multifractal formalism originally introduced to describe statistically the scaling properties of singular measures is revisited using the wavelet transform. This new approach is based on the definition of partition functions from the wavelet transform modulus maxima. We demonstrate that very much like thermodynamic functions, the generalized fractal dimensions Dq and the f(α) singularity spectrum can be readily determined from the scaling behavior of these partition functions. We show that this method provides a natural generalization of the classical box-counting techniques to fractal signals, the wavelets playing the role of “generalized boxes”. We illustrate our theoretical considerations on pedagogical examples, e.g., devils staircases and fractional Brownian motions. We also report the results of some recent application of the wavelet transform modulus maxima method to fully developed turbulence data. That we emphasize the wavelet transform as a mathematical microscope that can be further used to extract microscopic informations about the scaling properties of fractal objects. In particular, we show that a dynamical system which leaves invariant such an object can be uncovered form the space-scale arrangement of its wavelet transform modulus maxima. We elaborate on a wavelet based tree matching algorithm that provides a very promising tool for solving the inverse fractal problem. This step towards a statistical mechanics of fractals is illustrated on discrete period-doubling dynamical systems where the wavelet transform is shown to reveal the renormalization operation which is essential to the understanding of the universal properties of this transition to chaos. Finally, we apply our technique to analyze the fractal hierarchy of DLA azimuthal Cantor sets defined by intersecting the inner frozen region of large mass off-lattice diffusion-limited aggregates (DLA) wit a circle. This study clearly lets out the existence of an underlying multiplicative process that is likely to account for the Fibonacci structural ordering recently discovered in the apparently disordered arborescent DLA morphology.

Singularity Spectrum of Fractal Signals from Wavelet Analysis: Exact Results

The multifractal formalism for singular measures is revisited using the wavelet transform. For Bernoulli invariant measures of some expanding Markov maps, the generalized fractal dimensions are proved to be transition points for the scaling exponents of some partition functions defined from the wavelet transform modulus maxima. The generalization of this formalism to fractal signals is established for the class of distribution functions of these singular invariant measures. It is demonstrated that the Hausdorff dimensionD(h) of the set of singularities of Hölder exponenth can be directly determined from the wavelet transform modulus maxima. The singularity spectrum so obtained is shown to be not disturbed by the presence, in the signal, of a superimposed polynomial behavior of ordern, provided one uses an analyzing wavelet that possesses at leastN>n vanishing moments. However, it is shown that aC∞ behavior generally induces a phase transition in theD(h) singularity spectrum that somewhat masks the weakest singularities. This phase transition actually depends on the numberN of vanishing moments of the analyzing wavelet; its observation is emphasized as a reliable experimental test for the existence of nonsingular behavior in the considered signal. These theoretical results are illustrated with numerical examples. They are likely to be valid for a large class of fractal functions as suggested by recent applications to fractional Brownian motions and turbulent velocity signals.

Wavelet based fractal analysis of DNA sequences

Abstract The fractal scaling properties of DNA sequences are analyzed using the wavelet transform. Mapping nucleotide sequences onto a “DNA walk” produces fractal landscapes that can be studied quantitatively by applying the so-called wavelet transform modulus maxima method. This method provides a natural generalization of the classical box-counting techniques to fractal signals, the wavelets playing the role of “generalized oscillating boxes”. From the scaling behavior of partition functions that are defined from the wavelet transform modulus maxima, this method allows us to determine the singularity spectrum of the considered signal and thereby to achieve a complete multifractal analysis. Moreover, by considering analyzing wavelets that make the “wavelet transform microscope” blind to “patches” of different nucleotide composition that are observed in genomic sequences, we demonstrate and quantify the existence of long-range correlations in the noncoding regions. Although the fluctuations in the patchy landscape of the DNA walks reconstructed from both noncoding and (protein) coding regions are found homogeneous with Gaussian statistics, our wavelet-based analysis allows us to discriminate unambiguously between the fluctuations of the former which behave like fractional Brownian motions, from those of the latter which cannot be distinguished from uncorrelated random Brownian walks. We discuss the robustness of these results with respect to various legitimate codings of the DNA sequences. Finally, we comment about the possible understanding of the origin of the observed long-range correlations in noncoding DNA sequences in terms of the nonequilibrium dynamical processes that produce the “isochore structre of the genome”.

What can we learn with wavelets about DNA sequences

We use the wavelet transform to explore the complexity of DNA sequences. Long-range correlations are clearly identified and shown to be related to the sequence GC content. The significance of this observation to gene evolution is discussed.

Universal intermittent properties of particle trajectories in highly turbulent flows

We present a collection of eight data sets from state-of-the-art experiments and numerical simulations on turbulent velocity statistics along particle trajectories obtained in different flows with Reynolds numbers in the range R{lambda}in[120:740]. Lagrangian structure functions from all data sets are found to collapse onto each other on a wide range of time lags, pointing towards the existence of a universal behavior, within present statistical convergence, and calling for a unified theoretical description. Parisi-Frisch multifractal theory, suitably extended to the dissipative scales and to the Lagrangian domain, is found to capture the intermittency of velocity statistics over the whole three decades of temporal scales investigated here.

Wavelet Based Multifractal Formalism: Applications to DNA Sequences, Satellite Images of the Cloud Structure, and Stock Market Data

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 multiphcative cascade process that underhes 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.

Transcription-coupled TA and GC strand asymmetries in the human genome

Analysis of the whole set of human genes reveals that most of them present TA and GC skews, that these biases are correlated to each other and are specific to gene sequences, exhibiting sharp transitions between transcribed and non‐transcribed regions. The GC asymmetries cannot be explained solely by a model previously proposed for (G+T) skew based on transitions measured in a small set of human genes. We propose that the GC skew results from additional transcription‐coupled mutation process that would include transversions. During evolution, both processes acting on a large majority of genes in germline cells would have produced these transcription‐coupled strand asymmetries.

Long time correlations in lagrangian dynamics: a key to intermittency in turbulence.

Using a new experimental technique, based on the scattering of ultrasounds, we perform a direct measurement of particle velocities, in a fully turbulent flow. This allows us to approach intermittency in turbulence from a dynamical point of view and to analyze the Lagrangian velocity fluctuations in the framework of random walks. We find experimentally that the elementary steps in the walk have random uncorrelated directions but a magnitude that is extremely long range correlated in time. Theoretically, a Langevin equation is proposed and shown to account for the observed one- and two-point statistics. This approach connects intermittency to the dynamics of the flow.

Three-dimensional wavelet-based multifractal method: the need for revisiting the multifractal description of turbulence dissipation data.

We generalize the wavelet transform modulus maxima (WTMM) method to multifractal analysis of 3D random fields. This method is calibrated on synthetic 3D monofractal fractional Brownian fields and on 3D multifractal singular cascade measures as well as their random function counterpart obtained by fractional integration. Then we apply the 3D WTMM method to the dissipation field issued from 3D isotropic turbulence simulations. We comment on the need to revisit previous box-counting analyses which have failed to estimate correctly the corresponding multifractal spectra because of their intrinsic inability to master nonconservative singular cascade measures.

Wavelet transform of fractal aggregates

Abstract We apply the wavelet transform for characterizing the geometrical complexity of two-dimensional fractal aggregates. We illustrate the efficiency of this mathematical microscope to capture the local scaling properties of self-similar snowflake fractals. We report on the first unambiguous numerical evidence for the structural self-similarity of Witten and Sander diffusion-limited aggregates (DLA). We emphasize the wavelet transform to be readily applicable to experimental situations.