References of "Nicolay, Samuel"
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See detailA multifractal formalism for non-concave and non-increasing spectra: the leaders profile method
Esser, Céline ULg; Kleyntssens, Thomas ULg; Nicolay, Samuel ULg

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 ▲]

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See detailOn generalized Hölder spaces
Kreit, Damien; Nicolay, Samuel ULg

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.

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See detailUse of the wavelet theory as a tool to investigate the l-abelian complexity of a sequence
Kleyntssens, Thomas ULg; Nicolay, Samuel ULg; Vandomme, Elise ULg 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 ▲]

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See detailA wavelet-based mode decomposition compared to the EMD
Deliège, Adrien ULg; Nicolay, Samuel ULg

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 ▲]

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See detailThe leaders profile method: detection of distinct processes in a signal
Kleyntssens, Thomas ULg; Nicolay, Samuel ULg

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 ▲]

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See detailLes nombres
Nicolay, Samuel ULg

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.

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See detailLes algorithmes : entre quotidien et créativité
Nicolay, Samuel ULg; Kleyntssens, Thomas ULg; Mainz, Isabelle ULg

Conference given outside the academic context (2015)

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See detailA generalization of the Snu spaces: getting rid of dyadic scales
Kleyntssens, Thomas ULg; Nicolay, Samuel ULg

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 ▲]

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See detailA forecasting method using a wavelet-based mode decomposition and application to the ENSO index
Deliège, Adrien ULg; Nicolay, Samuel ULg; Fettweis, Xavier ULg

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 ▲]

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See detailAbout the Regularity of Cantor's Bijection
Simons, Laurent ULg; Nicolay, Samuel ULg

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 ▲]

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See detailEGU2015 - ENSO forecast using a wavelet-based mode decomposition
Deliège, Adrien ULg; Nicolay, Samuel ULg; Fettweis, Xavier ULg

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 ▲]

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See detailPar A plus B
Deliège, Adrien ULg; Nicolay, Samuel ULg

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.

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See detailAbout the Uniform Hölder Continuity of Generalized Riemann Function
Bastin, Françoise ULg; Nicolay, Samuel ULg; Simons, Laurent ULg

in Mediterranean Journal of Mathematics (2014)

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 ▲]

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See detailENSO forecast using a wavelet-based mode decomposition
Deliège, Adrien ULg; Nicolay, Samuel ULg; Fettweis, Xavier ULg

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 ▲]

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See detailDe l’importance des échelles dyadiques dans les espaces Snu
Kleyntssens, Thomas ULg; Nicolay, Samuel ULg

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 ▲]

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See detailFonction de Riemann généralisée
Simons, Laurent ULg; Bastin, Françoise ULg; Nicolay, Samuel ULg

Conference (2014, September 22)

Dans cet exposé, nous étudions la régularité de la fonction de Riemann généralisée~$R_{\alpha,\beta}$ (avec $\alpha>1$ et $\beta>0$) définie par \[ R_{\alpha,\beta}(x)=\sum_{n=1}^{+\infty}\frac{\sin(\pi n ... [more ▼]

Dans cet exposé, nous étudions la régularité de la fonction de Riemann généralisée~$R_{\alpha,\beta}$ (avec $\alpha>1$ et $\beta>0$) définie par \[ R_{\alpha,\beta}(x)=\sum_{n=1}^{+\infty}\frac{\sin(\pi n^\beta x)}{n^\alpha},\quad x\in\R. \] En particulier, nous déterminons son exposant de Hölder uniforme. Pour terminer, nous analysons le comportement de~$R_{\alpha,\beta}$ lorsque le paramètre $\alpha$ ou $\beta$ tend vers l'infini. Cet exposé est basé sur un travail en collaboration avec F. Bastin et S. Nicolay. [less ▲]

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See detailA multifractal analysis of air temperature signals based on the wavelet leaders method
Deliège, Adrien ULg; Nicolay, Samuel ULg

Conference (2014, August 20)

We present the wavelet leaders method (introduced by S. Jaffard) as a tool to study the Hölder regularity of signals, which is closely related to some functional spaces. We use the associated multifractal ... [more ▼]

We present the wavelet leaders method (introduced by S. Jaffard) as a tool to study the Hölder regularity of signals, which is closely related to some functional spaces. We use the associated multifractal formalism to show that surface air temperature signals are monofractal, i.e. these are regularly irregular. Then we use this result to establish a climate classification of weather stations in Europe which matches the Köppen-Geiger climate classification. This result could give rise to new criteria to determine the effectiveness of current climatic models. [less ▲]

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See detailA wavelet leaders-based climate classification of European surface air temperature signals
Deliège, Adrien ULg; Nicolay, Samuel ULg

in Ruiz Garcia, Gonzalo; Rojas Ruiz, Ignacio (Eds.) Proceedings of the International work-conference on Time Series: Volume 1, Granada 25-27 June 2014 (2014, June 25)

We explain the wavelet leaders method, a tool to study the pointwise regularity of signals, which is closely related to some functional spaces. We use the associated multifractal formalism to show that ... [more ▼]

We explain the wavelet leaders method, a tool to study the pointwise regularity of signals, which is closely related to some functional spaces. We use the associated multifractal formalism to show that surface air temperature signals are monofractal, i.e. these climate time series are regularly irregular. Then we use this result to establish a climate classification of weather stations in Europe which matches the Köppen-Geiger climate classification. This result could give rise to new criteria to determine the efficiency of current climatic models. [less ▲]

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See detailA wavelet leaders-based climate classification of European surface air temperature signals
Deliège, Adrien ULg; Nicolay, Samuel ULg

Conference (2014, June 25)

We present the wavelet leaders method as a tool to study the pointwise regularity of signals, which is closely related to some functional spaces. We use the associated multifractal formalism to show that ... [more ▼]

We present the wavelet leaders method as a tool to study the pointwise regularity of signals, which is closely related to some functional spaces. We use the associated multifractal formalism to show that the surface air temperature signals are monofractal, i.e. these are regularly irregular. Then we use this result to establish a climate classification of weather stations in Europe which matches the Köppen-Geiger climate classification. This result could give rise to new criteria to determine the effectiveness of current climatic models. [less ▲]

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See detailA wavelet-based analysis of surface air temperature regularity
Deliège, Adrien ULg; Nicolay, Samuel ULg

Conference (2014, June 19)

The aim of the talk is to present the "wavelet transform microscope" and the wavelet leaders method to show that surface air temperature signals of weather stations selected in Europe are monofractal, i.e ... [more ▼]

The aim of the talk is to present the "wavelet transform microscope" and the wavelet leaders method to show that surface air temperature signals of weather stations selected in Europe are monofractal, i.e. all the points have the same Hölder (regularity) exponent. This study reveals that the information obtained in this way are richer than previous works studying long range correlations in meteorological stations. The approach presented here allows to bind the Hölder exponents with the pressure anomalies, and such a link does not exist with methods previously carried out. Moreover, this regularity is a signature of the type of climate the stations are associated to: indeed, it is possible to establish a climate classification of weather stations in Europe which matches the Köppen-Geiger climate classification. A blind test is performed in order to confirm the results, which can be partly explained by the influence of the North Atlantic Oscillation. This result could give rise to new criteria to determine the efficiency of current climatic models. [less ▲]

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