Wavelet Transform of Multifractals

Arnéodo, A.; Grasseau, G.; Holschneider, Matthias

doi:10.1103/physrevlett.61.2281

Cited by 305 publications

(154 citation statements)

References 8 publications

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“…The WT consists in expanding signals in terms of wavelets which are constructed from a single function, the analyzing wavelet ψ, by mean of translations and dilations. The WT of a distribution µ is defined as [118,119]:…”

Section: A Solution: the Continuous Wavelet Transformmentioning

confidence: 99%

“…[116] where the WT is used as a tool for visualizing regular patterns in DNA sequences). As already experienced in various fields, the WT can be seen as a mathematical microscope that is well suited for characterizing the scaling properties of fractal objects and this even in the presence of some polynomial component [113,117,118,119,120]. By considering analyzing wavelets that make the microscope blind to low frequency trends, one can reveal and quantify the scale invariance properties of DNA walks [102,115,121].…”

Section: Introductionmentioning

confidence: 99%

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Wavelet Analysis of DNA Bending Profiles reveals Structural Constraints on the Evolution of Genomic Sequences

Audit

Vaillant

Arnéodo

et al. 2004

Journal of Biological Physics

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Abstract. Analyses of genomic DNA sequences have shown in previous works that base pairs are correlated at large distances with scale-invariant statistical properties. We show in the present study that these correlations between nucleotides (letters) result in fact from long-range correlations (LRC) between sequence-dependent DNA structural elements (words) involved in the packaging of DNA in chromatin. Using the wavelet transform technique, we perform a comparative analysis of the DNA text and of the corresponding bending profiles generated with curvature tables based on nucleosome positioning data. This exploration through the optics of the so-called 'wavelet transform microscope' reveals a characteristic scale of 100 − 200 bp that separates two regimes of different LRC. We focus here on the existence of LRC in the small-scale regime ( 200 bp). Analysis of genomes in the three kingdoms reveals that this regime is specifically associated to the presence of nucleosomes. Indeed, small scale LRC are observed in eukaryotic genomes and to a less extent in archaeal genomes, in contrast with their absence in eubacterial genomes. Similarly, this regime is observed in eukaryotic but not in bacterial viral DNA genomes. There is one exception for genomes of Poxviruses, the only animal DNA viruses that do not replicate in the cell nucleus and do not present small scale LRC. Furthermore, no small scale LRC are detected in the genomes of all examined RNA viruses, with one exception in the case of retroviruses. Altogether, these results strongly suggest that small-scale LRC are a signature of the nucleosomal structure. Finally, we discuss possible interpretations of these small-scale LRC in terms of the mechanisms that govern the positioning, the stability and the dynamics of the nucleosomes along the DNA chain. This paper is maily devoted to a pedagogical presentation of the theoretical concepts and physical methods which are well suited to perform a statistical analysis of genomic sequences. We review the results obtained with the so-called waveletbased multifractal analysis when investigating the DNA sequences of various organisms in the three kingdoms. Some of these results have been announced in B. Audit et al. [1,2].

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Section: A Solution: the Continuous Wavelet Transformmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Wavelet Analysis of DNA Bending Profiles reveals Structural Constraints on the Evolution of Genomic Sequences

Audit

Vaillant

Arnéodo

et al. 2004

Journal of Biological Physics

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“…Multiresolution wavelet analysis [14][15][16][17][18] has proved to be a mathematically clear and practical tool for analyzing signals at multiple scales, even in the presence of nonstationarities [19,20], which are obviously present in the center of pressure time series x and y, Fig. 1b.…”

Section: B Wavelet-variance Functionmentioning

confidence: 99%

Scaling-violation phenomena and fractality in the human posture control systems

et al. 2000

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By analyzing the movements of quiet standing persons by means of wavelet statistics, we observe multiple scaling regions in the underlying body dynamics. The use of the wavelet-variance function opens the possibility to relate scaling violations to different modes of posture control. We show that scaling behavior becomes close to perfect, when correctional movements are dominated by the vestibular system.

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“…As we have pointed out in our previous study of Laplacian growth processes [23,24,30], the application of the wavelet transform [39,41,42] may eventually result in a more complete understanding of the local scaling properties of fractal aggregates. In fact, as seen through the « wavelet microscope », the self-similarity of the branched geometry of DLA clusters and electrochemical deposits is likely to be the expression of a nonlinear chaotic recursive construction process which accounts for the proliferation of tip-splitting and side-branching instabilities observed during the growth [24].…”

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confidence: 99%

Experimental evidence for deterministic chaos in electrochemical deposition

Argoul

Arnéodo

1990

J. Phys. France

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We report an experimental study of the statics and dynamics of electrodeposition of zinc in an aqueous medium, in the intermediate regime between dendritic and DLA-like metallic clusters. During the growth process, periodic and nonperiodic oscillations of the voltage are recorded under constant applied current intensity. Period-doubling bifurcations are identified. Analysis of the data in terms of phase portraits, Poincaré maps and 1-D maps shows that the fractal geometry of these electrochemical deposits is the signature of a low-dimensional deterministic chaotic dynamics which displays sensitivity to initial conditions. The study of the apparently more complex dynamics recorded in the DLA limit is a very promising experimental challenge

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Wavelet Transform of Multifractals

Cited by 305 publications

References 8 publications

Wavelet Analysis of DNA Bending Profiles reveals Structural Constraints on the Evolution of Genomic Sequences

Wavelet Analysis of DNA Bending Profiles reveals Structural Constraints on the Evolution of Genomic Sequences

Scaling-violation phenomena and fractality in the human posture control systems

Experimental evidence for deterministic chaos in electrochemical deposition

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