The Multifractal Formalism Revisited With Wavelets

Muzy, Jean–François; Bacry, Emmanuel; Arnéodo, A.

doi:10.1142/s0218127494000204

Cited by 543 publications

(602 citation statements)

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“…We also explain how to quantify LRC from the measurement of the so-called Hurst-exponent H [119,126]. We discuss the necessity of using wavelet analysis to investigate LRC in real DNA sequences.…”

Section: Theoretical Concepts and Methodology: Wavelet Analysis Of Lomentioning

confidence: 99%

“…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%

“…to low-frequency trends that can mask the existence of scale-invariance properties. Hence, by considering analyzing wavelets ψ (n) that have n ψ vanishing moments [119,120]:…”

Section: Mastering Nonlinear Mosaic Structuresmentioning

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%

“…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]. In a previous work, by applying the so-called wavelet transform modulus maxima (WTMM) method [119,120,122] to the analysis of various genomic sequences mainly selected in the human genome, we have found that the fluctuations in the patchy landscapes of both coding and noncoding DNA walks are homogeneous with Gaussian statistics [102,115]. The main consequence of this result is the justification of using a single exponent, namely the Hurst or roughness exponent H , to characterize the underlying fractal organization of DNA sequences.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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: Theoretical Concepts and Methodology: Wavelet Analysis Of Lomentioning

confidence: 99%

Section: A Solution: the Continuous Wavelet Transformmentioning

confidence: 99%

“…to low-frequency trends that can mask the existence of scale-invariance properties. Hence, by considering analyzing wavelets ψ (n) that have n ψ vanishing moments [119,120]:…”

Section: Mastering Nonlinear Mosaic Structuresmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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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On the local Hurst exponent of geomagnetic field fluctuations: Spatial distribution for different geomagnetic activity levels

Michelis

Consolini

2015

JGR Space Physics

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This study attempts to characterize the spatial distribution of the scaling features of the short time scale magnetic field fluctuations obtained from 45 ground-based geomagnetic observatories distributed in the Northern Hemisphere. We investigate the changes of the scaling properties of the geomagnetic field fluctuations by evaluating the local Hurst exponent and reconstruct maps of this index as a function of the geomagnetic activity level. These maps permit us to localize the different latitudinal structures responsible for disturbances and related to the ionospheric current systems. We find that the geomagnetic field fluctuations associated with the different ionospheric current systems have different scaling features, which can be evidenced by the local Hurst exponent. We also find that in general, the local Hurst exponent for quiet magnetospheric periods is higher than that for more active periods suggesting that the dynamical processes that are activated during disturbed times are responsible for changes in the nature of the geomagnetic field fluctuations.

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River basin organization around the main stem: Scale invariance in tributary branching and the incremental area function

2014

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The incremental increase in contributing area along a main stem river, called here the incremental area function (IAF), has direct relevance to the spatial heterogeneity of environmental fluxes (water, sediment, nutrients, etc.) entering the stream from hillslopes and side tributaries. It also dictates, to a large extent, possible ecohydrologic discontinuities or transitions resulting from large tributary contributions. Mathematically, the IAF directly reflects the topological and geometrical structure of the river network and maps the two-dimensional landscape organization into a one-dimensional function. In this paper, we use two approaches to investigate the spatial heterogeneity of the IAF. First, we implement a multithreshold decomposition on IAF to study the distribution of distances between tributaries as a function of the imposed threshold contributing area and verify the presence of a simple power law scaling relationship between the threshold and the average distance between tributaries. Second, we use a wavelet-based multiscale approach and document the presence of statistical self-affinity (multifractality) in the IAF with a high intermittency coefficient, reflecting the complex arrangement of extreme contributions of different size tributaries. We propose a multiplicative cascade model, parameterized in terms of basin-specific properties, to statistically simulate the IAF along the main stem. Finally, we point out the relation between the IAF and the widely used width function of a basin and show how the latter can be constructed from the former via a convolution on the self-similar structure of a tree.

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The Multifractal Formalism Revisited With Wavelets

Abstract: International audienceno abstrac

Cited by 543 publications

References 0 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

On the local Hurst exponent of geomagnetic field fluctuations: Spatial distribution for different geomagnetic activity levels

River basin organization around the main stem: Scale invariance in tributary branching and the incremental area function

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