The Experts below are selected from a list of 32019 Experts worldwide ranked by ideXlab platform
Patrice Abry - One of the best experts on this subject based on the ideXlab platform.
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Multivariate Multifractal analysis
Applied and Computational Harmonic Analysis, 2019Co-Authors: Stéphane Jaffard, Stéphane Seuret, Roberto Leonarduzzi, Herwig Wendt, Stéphane Roux, Patrice AbryAbstract:We show how a joint Multifractal analysis of a collection of signals unravels correlations between the locations of their pointwise singularities. The multivariate Multifractal formalism, reformulated in the general setting supplied by multireso-lution quantities, provides a framework which allows to estimate joint Multifractal spectra. General results on joint Multifractal spectra are derived, and illustrated by the theoretical derivation and practical estimation of the joint Multifractal spectra of simple mathematical models, including correlated binomial cascades.
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A Generalized Multifractal Formalism for the Estimation of Nonconcave Multifractal Spectra
IEEE Transactions on Signal Processing, 2019Co-Authors: Roberto Leonarduzzi, Patrice Abry, Herwig Wendt, Hugo Touchette, Stéphane JaffardAbstract:Multifractal analysis has become a powerful signal processing tool that characterizes signals or images via the fluctuations of their pointwise regularity, quantified theoretically by the so-called Multifractal spectrum. The practical estimation of the Multifractal spectrum fundamentally relies on exploiting the scale dependence of statistical properties of appropriate multiscale quantities, such as wavelet leaders, that can be robustly computed from discrete data. Despite successes of Multifractal analysis in various real-world applications, current estimation procedures remain essentially limited to providing concave upper-bound estimates, while there is a priori no reason for the Multifractal spectrum to be a concave function. This paper addresses this severe practical limitation and proposes a novel formalism for Multifractal analysis that enables nonconcave Multifractal spectra to be estimated in a stable way. The key contributions reside in the development and theoretical study of a generalized Multifractal formalism to assess the multiscale statistics of wavelet leaders, and in devising a practical algorithm that permits this formalism to be applied to real-world data, allowing for the estimation of nonconcave Multifractal spectra. Numerical experiments are conducted on several synthetic Multifractal processes as well as on a real-world remote-sensing image and demonstrate the benefits of the proposed Multifractal formalism over the state of the art. Index Terms-Multifractal analysis, nonconcave multifrac-tal spectrum, wavelet leaders, Legendre transform, generalized canonical ensemble.
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A Generalized Multifractal Formalism for the Estimation of Nonconcave Multifractal Spectra
IEEE Transactions on Signal Processing, 2019Co-Authors: Roberto Leonarduzzi, Patrice Abry, Stéphane Jaffard, Herwig Wendt, Hugo TouchetteAbstract:Multifractal analysis has become a powerful signal processing tool that characterizes signals or images via the fluctuations of their pointwise regularity, quantified theoretically by the so-called Multifractal spectrum . The practical estimation of the Multifractal spectrum fundamentally relies on exploiting the scale dependence of statistical properties of appropriate multiscale quantities, such as wavelet leaders, that can be robustly computed from discrete data. Despite successes of Multifractal analysis in various real-world applications, current estimation procedures remain essentially limited to providing concave upper-bound estimates, while there is a priori no reason for the Multifractal spectrum to be a concave function. This paper addresses this severe practical limitation and proposes a novel formalism for Multifractal analysis that enables nonconcave Multifractal spectra to be estimated in a stable way. The key contributions reside in the development and theoretical study of a generalized Multifractal formalism to assess the multiscale statistics of wavelet leaders, and in devising a practical algorithm that permits this formalism to be applied to real-world data, allowing for the estimation of nonconcave Multifractal spectra. Numerical experiments are conducted on several synthetic Multifractal processes as well as on a real-world remote-sensing image and demonstrate the benefits of the proposed Multifractal formalism over the state of the art.
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Hyperspectral image analysis using Multifractal attributes
2015Co-Authors: Sébastien Combrexelle, Herwig Wendt, Jean-yves Tourneret, Stephen Mclaughlin, Patrice AbryAbstract:The increasing spatial resolution of hyperspectral remote sensors requires the development of new processing methods capable of combining both spectral and spatial information.In this article, we focus on the spatial component and propose the use of novel Multifractal attributes, which extract spatial information in terms of the fluctuations of the local regularity of image amplitudes. The novelty of the proposed approach is twofold. First, unlike previous attempts, we study attributes that efficiently summarize Multifractal information in a few parameters. Second, we make use of the most recent developments in Multifractal analysis: wavelet leader Multifractal formalism, the current theoretical and practical benchmark in Multifractal analysis, and a novel Bayesian estimation procedure for one of the central Multifractal parameters. Attributes provided by these state-of-the-art Multifractal analysis procedures are studied on two sets of hyperspectral images. The experiments suggest that Multifractal analysis can provide relevant spatial/textural attributes which can in turn be employed in tasks such as classification or segmentation.
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Bayesian estimation for the Multifractality parameter
2013Co-Authors: Herwig Wendt, Jean-yves Tourneret, Nicolas Dobigeon, Patrice AbryAbstract:Multifractal analysis has matured into a widely used signal and image processing tool. Due to the statistical nature of Multifractal processes (strongly non-Gaussian and intricate dependence) the accurate estimation of Multifractal parameters is very challenging in situations where the sample size is small (notably including a range of biomedical applications) and currently available estimators need to be improved. To overcome such limitations, the present contribution proposes a Bayesian estimation procedure for the Multifractality (or intermittence) parameter. Its originality is threefold: First, the use of wavelet leaders, a recently introduced multiresolution quantity that has been shown to yield significant benefits for Multifractal analysis; Second, the construction of a simple yet generic semi-parametric model for the marginals and covariance structure of wavelet leaders for the large class of multiplicative cascade based Multifractal processes; Third, the construction of original Bayesian estimators associated with the model and the constraints imposed by Multifractal theory. Performance are numerically assessed and illustrated for synthetic Multifractal processes for a range of Multifractal parameter values. The proposed procedure yields significantly improved estimation performance for small sample sizes.
Stéphane Jaffard - One of the best experts on this subject based on the ideXlab platform.
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Multivariate Multifractal analysis
Applied and Computational Harmonic Analysis, 2019Co-Authors: Stéphane Jaffard, Stéphane Seuret, Roberto Leonarduzzi, Herwig Wendt, Stéphane Roux, Patrice AbryAbstract:We show how a joint Multifractal analysis of a collection of signals unravels correlations between the locations of their pointwise singularities. The multivariate Multifractal formalism, reformulated in the general setting supplied by multireso-lution quantities, provides a framework which allows to estimate joint Multifractal spectra. General results on joint Multifractal spectra are derived, and illustrated by the theoretical derivation and practical estimation of the joint Multifractal spectra of simple mathematical models, including correlated binomial cascades.
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A Generalized Multifractal Formalism for the Estimation of Nonconcave Multifractal Spectra
IEEE Transactions on Signal Processing, 2019Co-Authors: Roberto Leonarduzzi, Patrice Abry, Stéphane Jaffard, Herwig Wendt, Hugo TouchetteAbstract:Multifractal analysis has become a powerful signal processing tool that characterizes signals or images via the fluctuations of their pointwise regularity, quantified theoretically by the so-called Multifractal spectrum . The practical estimation of the Multifractal spectrum fundamentally relies on exploiting the scale dependence of statistical properties of appropriate multiscale quantities, such as wavelet leaders, that can be robustly computed from discrete data. Despite successes of Multifractal analysis in various real-world applications, current estimation procedures remain essentially limited to providing concave upper-bound estimates, while there is a priori no reason for the Multifractal spectrum to be a concave function. This paper addresses this severe practical limitation and proposes a novel formalism for Multifractal analysis that enables nonconcave Multifractal spectra to be estimated in a stable way. The key contributions reside in the development and theoretical study of a generalized Multifractal formalism to assess the multiscale statistics of wavelet leaders, and in devising a practical algorithm that permits this formalism to be applied to real-world data, allowing for the estimation of nonconcave Multifractal spectra. Numerical experiments are conducted on several synthetic Multifractal processes as well as on a real-world remote-sensing image and demonstrate the benefits of the proposed Multifractal formalism over the state of the art.
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A Generalized Multifractal Formalism for the Estimation of Nonconcave Multifractal Spectra
IEEE Transactions on Signal Processing, 2019Co-Authors: Roberto Leonarduzzi, Patrice Abry, Herwig Wendt, Hugo Touchette, Stéphane JaffardAbstract:Multifractal analysis has become a powerful signal processing tool that characterizes signals or images via the fluctuations of their pointwise regularity, quantified theoretically by the so-called Multifractal spectrum. The practical estimation of the Multifractal spectrum fundamentally relies on exploiting the scale dependence of statistical properties of appropriate multiscale quantities, such as wavelet leaders, that can be robustly computed from discrete data. Despite successes of Multifractal analysis in various real-world applications, current estimation procedures remain essentially limited to providing concave upper-bound estimates, while there is a priori no reason for the Multifractal spectrum to be a concave function. This paper addresses this severe practical limitation and proposes a novel formalism for Multifractal analysis that enables nonconcave Multifractal spectra to be estimated in a stable way. The key contributions reside in the development and theoretical study of a generalized Multifractal formalism to assess the multiscale statistics of wavelet leaders, and in devising a practical algorithm that permits this formalism to be applied to real-world data, allowing for the estimation of nonconcave Multifractal spectra. Numerical experiments are conducted on several synthetic Multifractal processes as well as on a real-world remote-sensing image and demonstrate the benefits of the proposed Multifractal formalism over the state of the art. Index Terms-Multifractal analysis, nonconcave multifrac-tal spectrum, wavelet leaders, Legendre transform, generalized canonical ensemble.
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Comprehensive Multifractal analysis of turbulent velocity using the wavelet leaders
The European Physical Journal B, 2008Co-Authors: Bruno Lashermes, Patrice Abry, Stéphane G. Roux, Stéphane JaffardAbstract:The Multifractal framework relates the scaling properties of turbulence to its local regularity properties through a statistical description as a collection of local singularities. The Multifractal properties are moreover linked to the multiplicative cascade process that creates the peculiar properties of turbulence such as intermittency. A comprehensive estimation of the Multifractal properties of turbulence from data analysis, using a tool valid for all kind of singularities (including oscillating singularities) and mathematically well-founded, is thus of first importance in order to extract a reliable information on the underlying physical processes. The wavelet leaders yield a new Multifractal formalism which meets all these requests. This paper aims at describing it and at applying it to experimental turbulent velocity data. After a detailed discussion of the practical use of the wavelet leader based Multifractal formalism, the following questions are carefully investigated: (1) What is the dependence of Multifractal properties on the Reynolds number? (2) Are oscillating singularities present in turbulent velocity data? (3) Which Multifractal model does correctly account for the observed Multifractal properties? Results from several data set analysis are used to discuss the dependence of the computed Multifractal properties on the Reynolds number but also to assess their common or universal component. An exact though partial answer (no oscillating singularities are detected) to the issue of the presence of oscillating singularities is provided for the first time. Eventually an accurate parameterization with cumulant exponents up to order 4 confirms that the log-normal model (with c2 = -0.025±0.002) correctly accounts for the universal Multifractal properties of turbulent velocity.
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Wavelet Leaders in Multifractal Analysis
Wavelet Analysis and Applications, 1Co-Authors: Stéphane Jaffard, Bruno Lashermes, Patrice AbryAbstract:The properties of several Multifractal formalisms based on wavelet coefficients are compared from both mathematical and numerical points of view. When it is based directly on wavelet coefficients, the Multifractal formalism is shown to yield, at best, the increasing part of the weak scaling exponent spectrum. The formalism has to be based on new multiresolution quantities, the wavelet leaders, in order to yield the entire and correct spectrum of Holder singularities. The properties of this new Multifractal formalism and of the alternative weak scaling exponent Multifractal formalism are investigated. Examples based on known synthetic Multifractal processes are illustrating its numerical implementation and abilities.
Toshihico Arimitsu - One of the best experts on this subject based on the ideXlab platform.
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the world according to renyi thermodynamics of Multifractal systems
Annals of Physics, 2004Co-Authors: Petr Jizba, Toshihico ArimitsuAbstract:Abstract We discuss basic statistical properties of systems with Multifractal structure. This is possible by extending the notion of the usual Gibbs–Shannon entropy into more general framework—Renyi’s information entropy. We address the renormalization issue for Renyi’s entropy on (multi)fractal sets and consequently show how Renyi’s parameter is connected with Multifractal singularity spectrum. The maximal entropy approach then provides a passage between Renyi’s information entropy and thermodynamics of Multifractals. Important issues such as Renyi’s entropy versus Tsallis–Havrda–Charvat entropy and PDF reconstruction theorem are also studied. Finally, some further speculations on a possible relevance of our approach to cosmology are discussed.
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the world according to renyi thermodynamics of Multifractal systems
arXiv: Statistical Mechanics, 2002Co-Authors: Petr Jizba, Toshihico ArimitsuAbstract:We discuss basic statistical properties of systems with Multifractal structure. This is possible by extending the notion of the usual Gibbs--Shannon entropy into more general framework - Renyi's information entropy. We address the renormalization issue for Renyi's entropy on (multi)fractal sets and consequently show how Renyi's parameter is connected with Multifractal singularity spectrum. The maximal entropy approach then provides a passage between Renyi's information entropy and thermodynamics on Multifractals. Important issues as, for instance, Renyi's entropy versus Tsallis--Havrda--Charvat entropy and PDF reconstruction theorem are also studied. Finally, some further speculations on a possible relevance of our approach to cosmology are discussed.
Michel L. Lapidus - One of the best experts on this subject based on the ideXlab platform.
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Fractal Strings and Multifractal Zeta Functions
Letters in Mathematical Physics, 2009Co-Authors: Michel L. Lapidus, Jacques Lévy Véhel, John A. RockAbstract:International audienceFor a Borel measure on the unit interval and a sequence of scales that tend to zero, we define a one-parameter family of zeta functions called Multifractal zeta functions. These functions are a first attempt to associate a zeta function to certain Multifractal measures. However, we primarily show that they associate a new zeta function, the topological zeta function, to a fractal string in order to take into account the topology of its fractal boundary. This expands upon the geometric information garnered by the traditional geometric zeta function of a fractal string in the theory of complex dimensions. In particular, one can distinguish between a fractal string whose boundary is the classical Cantor set, and one whose boundary has a single limit point but has the same sequence of lengths as the complement of the Cantor set. Later work will address related, but somewhat different, approaches to Multifractals themselves, via zeta functions, partly motivated by the present paper
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Fractal Strings and Multifractal Zeta Functions
Letters in Mathematical Physics, 2009Co-Authors: Michel L. Lapidus, Jacques Lévy Véhel, John RockAbstract:For a Borel measure on the unit interval and a sequence of scales that tend to zero, we define a one-parameter family of zeta functions called Multifractal zeta functions. These functions are a first attempt to associate a zeta function to certain Multifractal measures. However, we primarily show that they associate a new zeta function, the topological zeta function, to a fractal string in order to take into account the topology of its fractal boundary. This expands upon the geometric information garnered by the traditional geometric zeta function of a fractal string in the theory of complex dimensions. In particular, one can distinguish between a fractal string whose boundary is the classical Cantor set, and one whose boundary has a single limit point but has the same sequence of lengths as the complement of the Cantor set. Later work will address related, but somewhat different, approaches to Multifractals themselves, via zeta functions, partly motivated by the present paper.
Espen A F Ihlen - One of the best experts on this subject based on the ideXlab platform.
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Multifractal analyses of response time series: A comparative study
Behavior Research Methods, 2013Co-Authors: Espen A F IhlenAbstract:Response time series with a non-Gaussian distribution and long-range dependent dynamics have been reported for several cognitive tasks. Conventional monofractal analyses numerically define a long-range dependency as a single scaling exponent, but they assume that the response times are Gaussian distributed. Ihlen and Vereijken (Journal of Experimental Psychology: General, 139, 436–463, 2010 ) suggested Multifractal extensions of the conventional monofractal analyses that are more suitable when the response time has a non-Gaussian distribution. Multifractal analyses estimate a Multifractal spectrum of scaling exponents that contain the single exponent estimated by the conventional monofractal analyses. However, a comparison of the performance of Multifractal analyses with behavioral variables has not yet been addressed. The present study compares the performance of seven Multifractal analyses. The Multifractal analyses were tested on multiplicative cascading noise that generates time series with a predefined Multifractal spectrum and with a structure of variation that mimics intermittent response time variation. Time series with 1,024 and 4,096 samples were generated with additive noise and multiharmonic trends of two different magnitudes (signal-to-noise/trend ratio; 0.33 and 1). The results indicate that all Multifractal analysis has individual pros and cons related to sample size, Multifractality, and the presence of additive noise and trends in the response time series. The summary of pros and cons of the seven Multifractal analyses provides a guideline for the choice of Multifractal analyses of response time series and other behavioral variables.
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introduction to Multifractal detrended fluctuation analysis in matlab
Frontiers in Physiology, 2012Co-Authors: Espen A F IhlenAbstract:Physiological and behavioural phenomena are often complex, characterized by variations in time series. Variations in time series reflect how these phenomena organize into coherent structures by interactions that span multiple scales in both time and space. The present tutorial is an introduction to Multifractal analyses that can identify these scale invariant interactions within time series by its Multifractal spectrum. The Multifractal spectrum can be estimated directly from scale-dependent measurements or from its q-order statistics. The tutorial emphasizes the most common scale-dependent measurements defined by the wavelet transforms and the detrended fluctuation analyses. The tutorial also emphasizes common features of all Multifractal analyses, like the choice of linear regression method, scaling range and elimination of spurious singularities, which are important for a robust estimation of the Multifractal spectrum. The tutorial ends with two brief examples where Multifractal analyses are employed to time series from Multifractal models and the complex phenomena of cognitive performance. References to available software for Multifractal analyses are included at the end of the tutorial. The main aim of the tutorial is to give the reader an introduction to Multifractal analyses without the extensive technicalities typically provided in mathematical journals.
