Multifractal Processes

Riedi, Rudolf H.

doi:10.21236/ada531331

Cited by 92 publications

(178 citation statements)

References 66 publications

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“…It has long been recognized that wavelet coefficients constitute ideal quantities to study the regularity of data (see e.g., [34,18,14,15,16]). The characterization of p-exponents proposed here relies on the use of the d−dimensional discrete wavelet transform (dDWT), whose definition and properties are briefly recalled.…”

Section: Discrete Wavelet Coefficients and P-leadersmentioning

confidence: 99%

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p-exponent and p-leaders, Part I: Negative pointwise regularity

Jaffard

Mélot

Leonarduzzi

et al. 2016

Physica A: Statistical Mechanics and its Applications

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Multifractal analysis aims to characterize signals, functions, images or fields, via the fluctuations of their local regularity along time or space, hence capturing crucial features of their temporal/spatial dynamics. Multifractal analysis is becoming a standard tool in signal and image processing, and is nowadays widely used in numerous applications of different natures. Its common formulation relies on the measure of local regularity via the Hölder exponent, by nature restricted to positive values, and thus to locally bounded functions or signals. It is here proposed to base the quantification of local regularity on p-exponents, a novel local regularity measure potentially taking negative values. First, the theoretical properties of p-exponents are studied in detail. Second, wavelet-based multiscale quantities, the p-leaders, are constructed and shown to permit accurate practical estimation of p-exponents. Exploiting the potential dependence with p, it is also shown how the collection of p-exponents enriches the classification of locally singular behaviors in functions, signals or images. The present contribution is complemented by a companion article developing the p-leader based multifractal formalism associated to p-exponents.

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Section: Discrete Wavelet Coefficients and P-leadersmentioning

confidence: 99%

“…In its commonly, if not exclusively, used formulation, multifractal analysis relies on the quantification of local regularity via the so-called Hölder exponent [13,14,15,16]. In practice, several multiscale quantities have been involved in multifractal formalisms, the practical counterpart of multifractal analysis.…”

Section: Introductionmentioning

confidence: 99%

p-exponent and p-leaders, Part I: Negative pointwise regularity

Jaffard

Mélot

Leonarduzzi

et al. 2016

Physica A: Statistical Mechanics and its Applications

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“…Also, the measure is more singular for a x 0 < 1. In other words, smaller values of a x 0 correspond to the more singular measures, that is, burst of events around x 0 , while regions where events occur sparsely are less singular with higher value of a around x 0 (Riedi, 1999).…”

Section: Large Deviation Multifractal Spectrum (Ldms)mentioning

confidence: 99%

On multifractality of high-latitude geomagnetic fluctuations

Vörös

2000

Ann. Geophys.

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Abstract. In order to contribute to the understanding of solar wind±magnetosphere interactions the multifractal scaling properties of high-latitude geomagnetic uctuations observed at the Thule observatory have been studied. Using the local observatory data and the present experimental knowledge only it seems hard to characterize directly the, presumably intermittent, mesoscale energy accumulation and dissipation processes taking place at the magnetotail, auroral region, etc. Instead a positive probability measure, describing the accumulated local geomagnetic signal energy content at the given time scales has been introduced and its scaling properties have been studied. There is evidence for the multifractal nature of the so de®ned intermittent ®eld , a result obtained by using the recently introduced technique of large deviation multifractal spectra. This technique allows us to describe the geomagnetic¯uctuations locally in time by means of singularity exponents a, which represent a generalization of the local degree of di erentiability and characterize the power-law scaling dependence of the introduced measure on resolution. A global description of the geomagnetic¯uctuations is insured by the spectrum of exponents f a which represents a rate function quantifying the deviations of the observed singularities a from the expected value. The results show that there exists a multifractal counterpart of the previously reported spectral break and di erent types of f a spectra describe thē uctuations in direct dissipation or loading-unloading regimes of the solar wind±magnetosphere interaction. On the time scale of substorms and storms the multifractal structure of the loading±unloading mode¯uc-tuations seems to be analogous to the simple multiplicative P-model, while the f a spectra in direct dissipation regime are close but not equal to the features of a uniform distribution. Larger deviations from the multiplicative model are observed when the in¯uence of the solar wind¯uctuations is examined. On this basis it is expected that an extended multifractal analysis of the singularity structure of nearEarth plasma system¯uctuations would lead to improved geomagnetic diagnosis of the magnetospheric dynamics.

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“…Future research should consider some other approaches, for example the multifractional Brownian motion model (for example, Peltier and LevyVehel, 1995;Coeurjolly, 2005). However, the small dataset problem remains and such an approach would probably still require the generation of a model catalog by methods such as those described by, for example, Chan and Wood (1998), Chan (1999), or Reidi (2003.…”

Section: Discussionmentioning

confidence: 99%

“…Both localities NM65 (2005)). However, the small dataset problem remains and such an ould probably still require the generation of a model catalog by methods such as those y, for example, Chan and Wood (1998), Chan (1999), or Reidi (2003.…”

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confidence: 99%

Multifractal model of magnetic susceptibility distributions in some igneous rocks

Gettings

2012

Nonlin. Processes Geophys.

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Abstract. Measurements of in-situ magnetic susceptibility were compiled from mainly Precambrian crystalline basement rocks beneath the Colorado Plateau and ranges in Arizona, Colorado, and New Mexico. The susceptibility meter used measures about 30 cm 3 of rock and measures variations in the modal distribution of magnetic minerals that form a minor component volumetrically in these coarsely crystalline granitic to granodioritic rocks. Recent measurements include 50-150 measurements on each outcrop, and show that the distribution of magnetic susceptibilities is highly variable, multimodal and strongly non-Gaussian. Although the distribution of magnetic susceptibility is well known to be multifractal, the small number of data points at an outcrop precludes calculation of the multifractal spectrum by conventional methods. Instead, a brute force approach was adopted using multiplicative cascade models to fit the outcrop scale variability of magnetic minerals. Model segment proportion and length parameters resulted in 26 676 models to span parameter space. Distributions at each outcrop were normalized to unity magnetic susceptibility and added to compare all data for a rock body accounting for variations in petrology and alteration. Once the best-fitting model was found, the equation relating the segment proportion and length parameters was solved numerically to yield the multifractal spectrum estimate. For the best fits, the relative density (the proportion divided by the segment length) of one segment tends to be dominant and the other two densities are smaller and nearly equal. No other consistent relationships between the best fit parameters were identified. The multifractal spectrum estimates appear to distinguish between metamorphic gneiss sites and sites on plutons, even if the plutons have been metamorphosed. In particular, rocks that have undergone multiple tectonic events tend to have a larger range of scaling exponents.

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Multifractal Processes

Cited by 92 publications

References 66 publications

p-exponent and p-leaders, Part I: Negative pointwise regularity

p-exponent and p-leaders, Part I: Negative pointwise regularity

On multifractality of high-latitude geomagnetic fluctuations

Multifractal model of magnetic susceptibility distributions in some igneous rocks

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