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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: 25 (10 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: 24 (6 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: 16 (7 ULg)A Refined Method for Estimating the Global Hölder Exponent Nicolay, Samuel ; in Latifi, Shahram (Ed.) Information Technology: New Generations (2016, April) In this paper, we recall basic results we have obtained about generalized Hölder spaces and present a wavelet characterization that holds under more general hypothesis than previously stated. This ... [more ▼] In this paper, we recall basic results we have obtained about generalized Hölder spaces and present a wavelet characterization that holds under more general hypothesis than previously stated. This theoretical tool gives rise to a method for estimating the global Hölder exponent which seems to be more precise than other wavelet-based approaches. This work should prove helpful for estimating long range correlations. [less ▲] Detailed reference viewed: 11 (3 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: 16 (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: 19 (6 ULg)About the Uniform Hölder Continuity of Generalized Riemann Function Bastin, Françoise ; Nicolay, Samuel ; Simons, Laurent in Mediterranean Journal of Mathematics (2016), 13(1), 101-117 In this paper, we study the uniform H\"{o}lder continuity of the generalized Riemann function~$R_{\alpha,\beta}$ (with $\alpha>1$ and $\beta>0$) defined by \[ R_{\alpha,\beta}(x)=\sum_{n=1}^{+\infty}\frac ... [more ▼] In this paper, we study the uniform H\"{o}lder continuity of the generalized Riemann function~$R_{\alpha,\beta}$ (with $\alpha>1$ and $\beta>0$) defined by \[ R_{\alpha,\beta}(x)=\sum_{n=1}^{+\infty}\frac{\sin(\pi n^\beta x)}{n^\alpha},\quad x\in\mathbb{R}, \] using its continuous wavelet transform. In particular, we show that the exponent we find is optimal. We also analyse the behaviour of~$R_{\alpha,\beta}$ as $\beta$ tends to infinity. [less ▲] Detailed reference viewed: 47 (15 ULg)On generalized Hölder spaces ; Nicolay, Samuel Conference (2015, September 24) We introduce generalized pointwise Hölder spaces as the point wise version of generalized uniform Hölder spaces. These last ones can be seen as a special case of generalized Besov spaces. Detailed reference viewed: 19 (3 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: 17 (5 ULg)A wavelet-based mode decomposition compared to the EMD Deliège, Adrien ; Nicolay, Samuel Poster (2015, September 08) We introduce a new method based on wavelets for decomposing a signal into quasi-periodic oscillating components with smooth time-varying amplitudes. This method is inspired by both the "classic" wavelet ... [more ▼] We introduce a new method based on wavelets for decomposing a signal into quasi-periodic oscillating components with smooth time-varying amplitudes. This method is inspired by both the "classic" wavelet-based decomposition and the empirical mode decomposition (EMD). We compare the efficiency of the method with the well-established EMD on toys examples and the ENSO climate index. [less ▲] Detailed reference viewed: 61 (11 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: 29 (9 ULg)Les nombres Nicolay, Samuel Book published by Hermann (2015) Cet ouvrage présente une construction axiomatique des nombres basée sur la théorie des ensembles. Les nombres naturels, entiers, rationnels, réels et hyperréels sont introduits. Detailed reference viewed: 70 (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: 29 (8 ULg)A forecasting method using a wavelet-based mode decomposition and application to the ENSO index Deliège, Adrien ; Nicolay, Samuel ; Fettweis, Xavier Conference (2015, June) This work consists of a presentation and applications of a forecasting methodology based on a mode decomposition performed through a continuous wavelet transform. The idea is comparable to the Fourier ... [more ▼] This work consists of a presentation and applications of a forecasting methodology based on a mode decomposition performed through a continuous wavelet transform. The idea is comparable to the Fourier series decomposition but where the amplitudes of the components are not constant anymore: the signal is written as a sum of periodic components with smooth time-varying amplitudes. This leads to a drastic decrease in the number of terms needed to decompose and rebuild the original signal without loss of precision. Once the decomposition is performed, the components are separately extrapolated, which leads to an extrapolation of the reconstructed signal that stands for a forecast of the original one. The quality of the forecast is assessed through a hindcast procedure (running retroactive probing forecasts) and Pearson correlations and root mean square errors are computed as functions of the lead time. This technique is first illustrated in details with a toy example, then with the El Niño Southern Oscillation (ENSO) time series. This signal consists of monthly-sampled sea surface temperature (SST) anomalies in the Eastern Pacific Ocean and is well-known to be one of the most influential climate patterns on the planet, inducing many consequences worldwide (hurricanes, droughts, flooding,…) and affecting human activities. Therefore, short-term predictions are of first importance in order to plan actions before the occurrence of these phenomena. As far as the ENSO time series is concerned, the wavelet-based mode decomposition leads to four components corresponding to periods of about 20, 31, 43 and 61 months respectively and the reconstruction recovers 97% of the El Niño/La Niña events (anomalous warming/cooling of the SST) of the last 65 years. Also, it turns out that more than 78% of these extreme events can be retrieved up to three years in advance. Finally, a forecast of the ENSO index is issued: the next La Niña event should start early in 2018 and should be followed soon after by a strong El Niño event in the second semester of 2019. [less ▲] Detailed reference viewed: 49 (9 ULg)About the Regularity of Cantor's Bijection Simons, Laurent ; Nicolay, Samuel Conference (2015, May 25) In 1878, Cantor proved that there exists a one-to-one correspondence between the points of the unit line segment [0,1] and the points of the unit square [0,1]². Since this application is defined via ... [more ▼] In 1878, Cantor proved that there exists a one-to-one correspondence between the points of the unit line segment [0,1] and the points of the unit square [0,1]². Since this application is defined via continued fractions, it is very hard to have any intuition about its smoothness. In this talk, we explore the regularity and the fractal nature of Cantor's bijection, using some notions concerning the metric theory and the ergodic theory of continued fractions. This talk is based on a joint work with S. Nicolay. [less ▲] Detailed reference viewed: 19 (0 ULg)EGU2015 - ENSO forecast using a wavelet-based mode decomposition Deliège, Adrien ; Nicolay, Samuel ; Fettweis, Xavier Conference (2015, April 13) The aim of this work is to introduce a new method for forecasting major El Niño/ La Niña events with the use of a wavelet-based mode decomposition. These major events are related to sea surface ... [more ▼] The aim of this work is to introduce a new method for forecasting major El Niño/ La Niña events with the use of a wavelet-based mode decomposition. These major events are related to sea surface temperature anomalies in the tropical Pacific Ocean: anomalous warmings are known as El Niño events, while excessive coolings are referred as La Niña episodes. These climatological phenomena are of primary importance since they are involved in many teleconnections ; predicting them long before they occur is therefore a crucial concern. First, we perform a wavelet transform (WT) of the monthly sampled El Niño Southern Oscillation 3.4 index (from 1950 to present) and compute the associated scale spectrum, which can be seen as the energy carried in the WT as a function of the scale. It can be observed that the spectrum reaches five peaks, corresponding to time scales of about 7, 20, 31, 43 and 61 months respectively. Therefore, the Niño 3.4 signal can be decomposed into five dominant oscillating components with time-varying amplitudes, these latter being given by the modulus of the WT at the associated pseudo-periods. The reconstruction of the index based on these five components is accurate since more than 93% of the El Niño/ La Niña events of the last 60 years are recovered and no major event is erroneously predicted. Then, the components are smoothly extrapolated using polynomials and added together, giving so several years forecasts of the Niño 3.4 index. In order to increase the reliability of the forecasts, we perform several months hindcasts (i.e. retroactive probing forecasts) which can be validated with the existing data. It turns out that most of the major events can be accurately predicted up to three years in advance, which makes our methodology competitive for such forecasts. Finally, we discuss the El Niño conditions currently undergone and give indications about the next La Niña event. [less ▲] Detailed reference viewed: 58 (10 ULg)Par A plus B Deliège, Adrien ; Nicolay, Samuel Conference given outside the academic context (2015) Exposé de vulgarisation sur l'état de l'art concernant les ondelettes et ses applications, en particulier l'étude du phénomène El Niño. Detailed reference viewed: 28 (4 ULg)ENSO forecast using a wavelet-based mode decomposition Deliège, Adrien ; Nicolay, Samuel ; Fettweis, Xavier Poster (2014, December) We introduce a new method for forecasting major El Niño/ La Niña events based on a wavelet mode decomposition. This methodology allows us to approximate the ENSO time series with a superposition of three ... [more ▼] We introduce a new method for forecasting major El Niño/ La Niña events based on a wavelet mode decomposition. This methodology allows us to approximate the ENSO time series with a superposition of three periodic signals corresponding to periods of about 31, 43 and 61 months respectively with time-varying amplitudes. This pseudo-periodic approximation is then extrapolated to give forecasts. While this last one only resolves the large variations in the ENSO time series, three years hindcast as retroactive prediction allows to recover most of the El Niño/ La Niña events of the last 60 years. [less ▲] Detailed reference viewed: 45 (9 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) |
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