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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: 27 (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: 28 (7 ULg)[Beamer] A New Wavelet-Based Mode Decomposition for Oscillating Signals and Comparison with the Empirical Mode Decomposition Deliège, Adrien ; Nicolay, Samuel Conference (2016, April) We introduce a new method based on wavelets (EWMD) for decomposing a signal into quasi-periodic oscillating components with smooth time-varying amplitudes. This method is inspired by both the “classic” ... [more ▼] We introduce a new method based on wavelets (EWMD) 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 reconstruction skills and the period detection ability of the method with the well-established EMD on toys examples and the ENSO climate index. It appears that the EWMD accurately decomposes and reconstructs a given signal (with the same efficiency as the EMD), it is better at detecting prescribed periods and is less sensitive to noise. This work provides the first version of the EWMD. Even though there is still room for improvement, it turns out that preliminary results are highly promising. [less ▲] Detailed reference viewed: 20 (5 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)A New Wavelet-Based Mode Decomposition for Oscillating Signals and Comparison with the Empirical Mode Decomposition Deliège, Adrien ; Nicolay, Samuel in Information Technology: New Generations (2016, April) We introduce a new method based on wavelets (EWMD) for decomposing a signal into quasi-periodic oscillating components with smooth time-varying amplitudes. This method is inspired by both the “classic” ... [more ▼] We introduce a new method based on wavelets (EWMD) 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 reconstruction skills and the period detection ability of the method with the well-established EMD on toys examples and the ENSO climate index. It appears that the EWMD accurately decomposes and reconstructs a given signal (with the same efficiency as the EMD), it is better at detecting prescribed periods and is less sensitive to noise. This work provides the first version of the EWMD. Even though there is still room for improvement, it turns out that preliminary results are highly promising. [less ▲] Detailed reference viewed: 16 (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: 25 (15 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: 16 (6 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)Some characterizations about Generalized Hölder-Zygmund Spaces Λ_{σ,N}^{α}(R^d) Kreit, Damien ; Nicolay, Samuel E-print/Working paper (2016) Generalized Hölder-Zygmund spaces $\Lambda_{\sigma, N}^{\alpha}(\R^{d})$ were recently introduced and are based on a generalization of Besov spaces. Under some conditions, generalized Hölder-Zygmund and ... [more ▼] Generalized Hölder-Zygmund spaces $\Lambda_{\sigma, N}^{\alpha}(\R^{d})$ were recently introduced and are based on a generalization of Besov spaces. Under some conditions, generalized Hölder-Zygmund and Besov spaces are equal. It has been proved that most properties of classical Hölder-Zygmund spaces are held for spaces $\Lambda^{\sigma,\alpha}(\R^{d})$, which constitute a particular case of spaces $\Lambda_{\sigma, N}^{\alpha}(\R^{d})$ with $N_{j}=2^{j}$. The goal of the present document is to prove that most of these properties are kept for $\Lambda_{\sigma, N}^{\alpha}(\R^{d})$ spaces. [less ▲] Detailed reference viewed: 9 (1 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)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: 18 (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: 66 (12 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: 30 (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: 50 (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) |
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