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A multifractal formalism for non-concave and non-increasing spectra: the leaders profile method Esser, Céline ; Kleyntssens, Thomas ; Nicolay, Samuel in Applied & Computational Harmonic Analysis (in press) We present an implementation of a multifractal formalism based on the types of histogram of wavelet leaders. This method yields non-concave spectra and is not limited to their increasing part. We show ... [more ▼] We present an implementation of a multifractal formalism based on the types of histogram of wavelet leaders. This method yields non-concave spectra and is not limited to their increasing part. We show both from the theoretical and from the applied points of view that this approach is more e cient than the wavelet-based multifractal formalisms previously introduced. [less ▲] Detailed reference viewed: 28 (11 ULg)A Refined Method for Estimating the Global Hölder Exponent Kleyntssens, Thomas ; ; Nicolay, Samuel Conference (2016, April 12) We give a wavelet characterization of the generalized Hölder spaces and show how this result can be applied to detect logarithmic corrections appearing in Brownian processes. Detailed reference viewed: 28 (7 ULg)Mars Topography Investigated Through the Wavelet Leaders Method: a Multidimensional Study of its Fractal Structure Deliège, Adrien ; Kleyntssens, Thomas ; Nicolay, Samuel Poster (2016, April) This work examines the scaling properties of Mars topography through a wavelet-based formalism. We conduct exhaustive one-dimensional (both longitudinal and latitudinal) and two-dimensional studies based ... [more ▼] This work examines the scaling properties of Mars topography through a wavelet-based formalism. We conduct exhaustive one-dimensional (both longitudinal and latitudinal) and two-dimensional studies based on Mars Orbiter Laser Altimeter (MOLA) data using the multifractal formalism called Wavelet Leaders Method (WLM). This approach shows that a scale break occurs at approximately 15 km, giving two scaling regimes in both 1D and 2D cases. At small scales, these topographic profiles mostly display a monofractal behavior while a switch to multifractality is observed in several areas at larger scales. The scaling exponents extracted from this framework tend to be greater at small scales. In the 1D context, these observations are in agreement with previous works and thus suggest that the WLM is well-suited for examining scaling properties of topographic fields. Moreover, the 2D analysis is the first such complete study to our knowledge. It gives both a local and global insight on the scaling regimes of the surface of Mars and allows to exhibit the link between the scaling exponents and several famous features of the Martian topography. These results may be used as a solid basis for further investigations of the scaling laws of the Red planet and show that the WLM could be used to perform systematic analyses of the surface roughness of other celestial bodies. [less ▲] Detailed reference viewed: 13 (3 ULg)The Fractal Nature of Mars Topography Analyzed via the Wavelet Leaders Method Kleyntssens, Thomas ; Deliège, Adrien ; Nicolay, Samuel in Information Technology: New Generations (2016, April) This paper studies the scaling properties of Mars topography based on Mars Orbiter Laser Altimeter (MOLA) data through the wavelet leaders method (WLM). This approach shows a scale break at 15 km. At ... [more ▼] This paper studies the scaling properties of Mars topography based on Mars Orbiter Laser Altimeter (MOLA) data through the wavelet leaders method (WLM). This approach shows a scale break at 15 km. At small scales, these topographic profiles display a monofractal behavior while a multifractal nature is observed at large scales. The scaling exponents are greater at small scales. They also seem to be influenced by latitude and may indicate a slight anisotropy in topography. [less ▲] Detailed reference viewed: 26 (16 ULg)Wavelet-based Methods to Study the Regularity of a Signal: from Theory to Practice Kleyntssens, Thomas ; Nicolay, Samuel Conference (2016, March 23) In this talk, I use the notion of wavelet to design multifractal formalisms. I present the theoritical results obtained on the generalized Snu spaces and I show the utility of these generalization ... [more ▼] In this talk, I use the notion of wavelet to design multifractal formalisms. I present the theoritical results obtained on the generalized Snu spaces and I show the utility of these generalization. Besides, I also apply these formalisms on a practical example: the Mars topography. [less ▲] Detailed reference viewed: 17 (3 ULg)The Fractal Nature of Mars Topography Analyzed via the Wavelet Leaders Method Kleyntssens, Thomas ; Deliège, Adrien ; Nicolay, Samuel Poster (2016) This work studies the scaling properties of Mars topography based on Mars Orbiter Laser Altimeter (MOLA) data through the wavelet leaders method (WLM). This approach shows a scale break at 15 km. At small ... [more ▼] This work studies the scaling properties of Mars topography based on Mars Orbiter Laser Altimeter (MOLA) data through the wavelet leaders method (WLM). This approach shows a scale break at 15 km. At small scales, these topographic profiles display a monofractal behavior while a multifractal nature is observed at large scales. The scaling exponents are greater at small scales. They also seem to be influenced by latitude and may indicate a slight anisotropy in topography. [less ▲] Detailed reference viewed: 25 (12 ULg)Use of the wavelet theory as a tool to investigate the l-abelian complexity of a sequence Kleyntssens, Thomas ; Nicolay, Samuel ; Vandomme, Elise et al Poster (2015, September 23) The concept of k-automatic sequences is at the intersection of number theory and formal language theory. It has been generalized by the notion of k-regularity that allows to study sequences with values in ... [more ▼] The concept of k-automatic sequences is at the intersection of number theory and formal language theory. It has been generalized by the notion of k-regularity that allows to study sequences with values in a (possibly infinite) ring. This concept provides us with structural information about how the different terms of the sequence are related to each other. They are many different notions related to the measure of complexity of an infinite sequence w. A classical approach is its factor complexity. In an abelian context, the analogue to the factor complexity is the abelian complexity where the number of distinct factors of length n is counted up to abelian equivalence. The notion of abelian complexity was extended to that of l-abelian complexity. In this talk, I propose to use tools from the wavelet theory to analyze the l-abelian complexity. For the numerical simulations, I apply the wavelet leaders method that allows to study the pointwise regularity of signals. [less ▲] Detailed reference viewed: 21 (8 ULg)The leaders profile method: detection of distinct processes in a signal Kleyntssens, Thomas ; Nicolay, Samuel Poster (2015, September 08) The leaders profile method is a multifractal formalism that allows to compute non-concave and non-increasing spectra. Our implementation can detect the presence of distinct processes in a signal. We ... [more ▼] The leaders profile method is a multifractal formalism that allows to compute non-concave and non-increasing spectra. Our implementation can detect the presence of distinct processes in a signal. We present here the first results obtained. [less ▲] Detailed reference viewed: 30 (10 ULg)Les algorithmes : entre quotidien et créativité Nicolay, Samuel ; Kleyntssens, Thomas ; Mainz, Isabelle Conference given outside the academic context (2015) Detailed reference viewed: 26 (4 ULg)A generalization of the Snu spaces: getting rid of dyadic scales Kleyntssens, Thomas ; Nicolay, Samuel Conference (2015, June 16) The Snu spaces have been introduced by S. Jaffard to develop a new multifractal formalism that allows to improve the study of irregular functions. This type of formalism is connected to Besov spaces. From ... [more ▼] The Snu spaces have been introduced by S. Jaffard to develop a new multifractal formalism that allows to improve the study of irregular functions. This type of formalism is connected to Besov spaces. From a theoretical point of view, the Snu spaces gave birth to counterexamples in functional analysis. In this talk, I present the first results on a generalization of these spaces. I also present some links between these new spaces and the generalized Besov spaces defined with wavelet coefficients. [less ▲] Detailed reference viewed: 30 (8 ULg)De l’importance des échelles dyadiques dans les espaces Snu Kleyntssens, Thomas ; Nicolay, Samuel Conference (2014, September 23) Le but de l’analyse multifractale est de fournir une méthode permettant d’approximer le spectre de singularités d’une fonction. En 1985, Frisch et Parisi ont proposé un premier formalisme. D'autres ... [more ▼] Le but de l’analyse multifractale est de fournir une méthode permettant d’approximer le spectre de singularités d’une fonction. En 1985, Frisch et Parisi ont proposé un premier formalisme. D'autres formalismes, basés sur les coefficients d'ondelettes, ont été introduits (ex WLM). Cependant, de part leurs natures, ces méthodes ne peuvent détecter que des spectres concaves. En 2004, Jaffard introduit les espaces Snu pour palier à ce problème. Ces espaces sont inclus dans une intersection d'espaces de Besov. Dans cet exposé, je présente une généralisation des espaces Snu. Ceux-ci sont mis en relation avec les espaces de Besov généralisés et une mise en pratique est présentée. [less ▲] Detailed reference viewed: 35 (9 ULg)Detection of non concave and non increasing multifractal spectra using wavelet leaders (Part I) Esser, Céline ; Kleyntssens, Thomas ; Bastin, Françoise et al Conference (2014, May 22) Multifractal analysis is concerned with the study of very irregular signals. For such functions, the pointwise regularity may change widely from a point to another. Therefore, it is more interesting to ... [more ▼] Multifractal analysis is concerned with the study of very irregular signals. For such functions, the pointwise regularity may change widely from a point to another. Therefore, it is more interesting to determine the spectrum of singularities of the signal, which is the Hausdor ff dimension of the set of points which have the same H ölder exponent. For real-life signals, the computation of the spectrum of singularities from its de finition is not feasible. Multifractal formalisms are used to approximate this spectrum. Currently, there exist several methods. In this talk, we present a new multifractal formalism based on the wavelet leaders of a signal which allows to detect non concave and non increasing spectra. [less ▲] Detailed reference viewed: 44 (7 ULg)Detection of non concave and non increasing multifractal spectra using wavelet leaders (Part II) Kleyntssens, Thomas ; Esser, Céline ; Nicolay, Samuel Conference (2014, May 22) This talk follows "Detection of non concave and non increasing multifractal spectra using wavelet leaders (Part I)" given by Céline Esser. A multifractal formalism is a numerically computable formula that ... [more ▼] This talk follows "Detection of non concave and non increasing multifractal spectra using wavelet leaders (Part I)" given by Céline Esser. A multifractal formalism is a numerically computable formula that approximates the spectrum of singularities of a function. A new multifractal formalism based on the wavelet leaders is presented as well as a comparison with other formalisms. Its main advantages are that it allows to detect non concave and non increasing spectra. An implementation is proposed. [less ▲] Detailed reference viewed: 50 (18 ULg)A new multifractal formalism based on wavelet leaders : detection of non concave and non increasing spectra (Part I) Esser, Céline ; Kleyntssens, Thomas ; Nicolay, Samuel et al Conference (2014, March 25) Multifractal analysis is concerned with the study of very irregular signals. For such functions, the pointwise regularity may change widely from a point to another. Therefore, it is more interesting to ... [more ▼] Multifractal analysis is concerned with the study of very irregular signals. For such functions, the pointwise regularity may change widely from a point to another. Therefore, it is more interesting to determine the spectrum of singularities of the signal, which is the Hausdorff dimension of the set of points which have the same Hölder exponent. The spectrum of singularities of many mathematical functions can be determined directly from its definition. However, for many real-life signals, the numerical determination of their Hölder regularity is not feasible. Therefore, one cannot expect to have a direct access to their spectrum of singularities and one has to find an indirect way to compute it. A multifractal formalism is a formula which is expected to yield the spectrum of singularities from quantities which are numerically computable. Several multifractal formalisms based on the wavelet coefficients of a signal have been proposed to estimate its spectrum. The most widespread of these formulas is the so-called thermodynamic multifractal formalism, based on the Frish-Parisi conjecture. This formalism presents two drawbacks: it can hold only for spectra that are concave and it can yield only the increasing part of the spectrum. This first problem can be avoided using Snu spaces. The second one can be avoided using a formalism based on wavelet leaders of the signal. In this talk, we propose a new multifractal formalism, based on a generalization of the Snu spaces using wavelet leaders. It allows to detect non concave and non increasing spectra. An implementation of this method is presented in the talk "A new multifractal formalism based on wavelet leaders: detection of non concave and non increasing spectra (Part II)" of T. Kleyntssens. [less ▲] Detailed reference viewed: 79 (13 ULg)A new multifractal formalism based on wavelet leaders: detection of non concave and non increasing spectra (Part II) Kleyntssens, Thomas ; Esser, Céline ; Nicolay, Samuel Conference (2014, March 25) This talk follows "A new multifractal formalism based on wavelet leaders: detection of non concave and non increasing spectra (Part I)" given by Céline Esser. For real-life signals, it is impossible to ... [more ▼] This talk follows "A new multifractal formalism based on wavelet leaders: detection of non concave and non increasing spectra (Part I)" given by Céline Esser. For real-life signals, it is impossible to compute the spectrum of singularities by using its definition. A multifractal formalism is used to approximate this spectrum. We present a new multifractal formalism for non concave and non increasing spectra based on wavelet leaders. In this talk, an implementation of this formalism is given and several numerical examples are presented. [less ▲] Detailed reference viewed: 49 (16 ULg)A multifractal formalism based on the Sν spaces: From theory to practice Esser, Céline ; Kleyntssens, Thomas ; Nicolay, Samuel E-print/Working paper (2014) We present an implementation of a multifractal formalism based on the Sν spaces and show that it effectively gives the right Hölder spectrum in numerous cases. In particular, it allows to recover non ... [more ▼] We present an implementation of a multifractal formalism based on the Sν spaces and show that it effectively gives the right Hölder spectrum in numerous cases. In particular, it allows to recover non-concave spectra, where other multifractal formalisms only lead to the concave hull of the spectra. [less ▲] Detailed reference viewed: 5 (0 ULg)Snu Spaces, from Theory to Practice Kleyntssens, Thomas ; Nicolay, Samuel Poster (2013, October 29) Computing the spectrum of singularities of a real-life signal by using the definition is impossible. One rather uses an indirect way to compute it: the multifractal formalism. The first multifractal ... [more ▼] Computing the spectrum of singularities of a real-life signal by using the definition is impossible. One rather uses an indirect way to compute it: the multifractal formalism. The first multifractal formalism was introduced by Frisch and Parisi in the context of fully developped turbulence (1985). Its main default is that it always leads to a concave spectrum. For this reason, Stéphane Jaffard has introduced the Snu spaces (2004). They lead to a new multifractal formalism which can detect non concave spectra. In practice, one has to avoid the concept of limit and to deal with finite size effects. I present a method to determine the spectrum based on the Snu spaces and I illustrate it numerically on theoretical functions. [less ▲] Detailed reference viewed: 35 (15 ULg)Mise en oeuvre du formalisme multifractal sur les espaces Snu Kleyntssens, Thomas ; Nicolay, Samuel Conference (2013, September 24) When considering very irregular functions, it does not make sense to try to characterize the pointwise irregularity because it can change from one point to another. It is more interesting to compute the ... [more ▼] When considering very irregular functions, it does not make sense to try to characterize the pointwise irregularity because it can change from one point to another. It is more interesting to compute the spectrum of singularities, ie "the size'' of the set of points which share the same pointwise irregularity; by size, one means the Hausdorff dimension. To compute the spectrum of singularities in practice, we use a multifractal formalism. In 1885, Frisch and Parisi have proposed a first formalism. Its main default is that it always leads to a concave spectrum. In 2004, Stéphane Jaffard has introduced the Snu spaces. They lead to a new multifractal formalism which can detect non concave spectra. In practice, one has to avoid the concept of limit and to deal with finite size effects (for example, one can only calculate a finite number of wavelet coefficients). I present a method to determine the spectrum based on the Snu spaces and I illustrate it numerically on theoretical functions. [less ▲] Detailed reference viewed: 31 (11 ULg)Implementation of the Multifractal Formalism on Snu Spaces Kleyntssens, Thomas ; Nicolay, Samuel Conference (2013, September 09) A multifractal formalism is a formula, numerically computable, which approximate the spectrum of singularities of a function. The first multifractal formalism (Frisch and Parisi, 1985) has the main ... [more ▼] A multifractal formalism is a formula, numerically computable, which approximate the spectrum of singularities of a function. The first multifractal formalism (Frisch and Parisi, 1985) has the main default is that it always leads to a concave spectrum. In 2004, Stéphane Jaffard has introduced a new multifractal formalism, based on the Snu spaces, which can detect non concave spectra. In practice, one has to avoid the concept of limit and to deal with finite size effects. In this talk, I present the first results of an implementation of the multifractal formalism on Snu spaces on several theoretical functions. [less ▲] Detailed reference viewed: 32 (6 ULg)Automatic Cargo Load Planning: Special shipments Kleyntssens, Thomas ; Limbourg, Sabine ; Schyns, Michael in ILS 2012 Proceedings (2012, August 28) The aircraft loading problem is a real-world combinatorial optimisation problem highly constrained. Indeed, loading the aircraft so the gross weight is less than the maximum allowable is not enough. This ... [more ▼] The aircraft loading problem is a real-world combinatorial optimisation problem highly constrained. Indeed, loading the aircraft so the gross weight is less than the maximum allowable is not enough. This weight must be distributed to keep the centre of gravity within specified limits. Moreover, an aircraft has usually several cargo compartments with specific contours and structural limitations such as floor loading, combined load limits and cumulative load limitations. Finally, some shipments are particularly restrictive to transport, like dangerous goods, live animals and perishable goods. This paper is concerned with the incorporation of these latter constraints in a mixed integer linear program for the problem of loading a set of Unit Loading Devices and bulk into an aircraft. Experimental results show that our method achieves optimal solutions within only few seconds. [less ▲] Detailed reference viewed: 87 (9 ULg) |
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