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

Conference (2014, June 06)

We study the Hölder regularity of surface air temperature signals using the wavelet leaders method (WLM). This method has been successfully applied in several domains such as DNA analysis, fully ... [more ▼]

We study the Hölder regularity of surface air temperature signals using the wavelet leaders method (WLM). This method has been successfully applied in several domains such as DNA analysis, fully developped turbulence analysis, internet data traffic analysis,... to name just a few, and we now use it in climatology. We first define the notions of Hölder exponent, monofractal functions and spectrum of singularities before explaining the WLM. Then we use it to study surface air temperature signals from weather stations spread across Western and Eastern Europe and show that they are monofractal, i.e. their irregularity (in the sense of variability) is regular. After, we show that the stations can be classified according to their Hölder exponent and that this classification matches with the worldwide used 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. Our results can be helpful to test the accuracy of current climatic models. [less ▲]

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See detailMultifractal analysis of air temperature signals using the wavelet leaders method
Deliège, Adrien ULg; Nicolay, Samuel ULg

Conference (2014, May 22)

We use the discrete "wavelet transform microscope" to show that the surface air temperature signals of weather stations selected in Europe are monofractal. This study reveals that the information obtained ... [more ▼]

We use the discrete "wavelet transform microscope" to show that the surface air temperature signals of weather stations selected in Europe are monofractal. 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 climate variability. We also establish that such a link does not exist with methods previously carried out, such as the detrended fluctuation analysis. [less ▲]

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See detailDetection of non concave and non increasing multifractal spectra using wavelet leaders (Part I)
Esser, Céline ULg; Kleyntssens, Thomas ULg; Bastin, Françoise ULg 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 ▲]

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See detailDetection of non concave and non increasing multifractal spectra using wavelet leaders (Part II)
Kleyntssens, Thomas ULg; Esser, Céline ULg; Nicolay, Samuel ULg

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 ▲]

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See detailA multifractal-based climate analysis
Deliège, Adrien ULg; Nicolay, Samuel ULg

Conference (2014, March 25)

This work consists of a multifractal analysis of surface air temperature in Western and Eastern Europe. The wavelet leaders method enables us to exhibit the monofractal nature of these signals. Then, we ... [more ▼]

This work consists of a multifractal analysis of surface air temperature in Western and Eastern Europe. The wavelet leaders method enables us to exhibit the monofractal nature of these signals. Then, we define a climate classification relying on the Zygmund-Hölder spaces and show that it matches the Köppen classification. By doing so, we evidence the existing relation between the regularity of the temperature data (obtained through functional spaces) and the type of climate. Finally, we give a climatic interpretation of these results. [less ▲]

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See detailA Weak Local Irregularity Property in $S^\nu$ spaces
Clausel, Marianne; Nicolay, Samuel ULg

Conference (2014, March 25)

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See detailA new multifractal formalism based on wavelet leaders : detection of non concave and non increasing spectra (Part I)
Esser, Céline ULg; Kleyntssens, Thomas ULg; Nicolay, Samuel ULg 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 ▲]

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See detailA new multifractal formalism based on wavelet leaders: detection of non concave and non increasing spectra (Part II)
Kleyntssens, Thomas ULg; Esser, Céline ULg; Nicolay, Samuel ULg

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 ▲]

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

Conference (2014, March 25)

In this talk, we present the Cantor's bijection between the irrational numbers of the unit interval [0,1] and the irrational numbers of the unit square [0,1]². We explore the regularity and the fractal ... [more ▼]

In this talk, we present the Cantor's bijection between the irrational numbers of the unit interval [0,1] and the irrational numbers of the unit square [0,1]². We explore the regularity and the fractal nature of this map. This talk is based on a joint work with S. Nicolay. [less ▲]

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See detailAnalysis of PSII antenna size heterogeneity of Chlamydomonas reinhardtii during state transitions
de Marchin, Thomas ULg; Ghysels, Bart ULg; Nicolay, Samuel ULg et al

in Biochimica et Biophysica Acta-Bioenergetics (2014), 1837(1), 121-130

PSII antenna size heterogeneity has been intensively studied in the past. Based on DCMU fluorescence rise kinetics, multiple types of photosystems with different properties were described. However, due to ... [more ▼]

PSII antenna size heterogeneity has been intensively studied in the past. Based on DCMU fluorescence rise kinetics, multiple types of photosystems with different properties were described. However, due to the complexity of fluorescence signal analysis, multiple questions remain unanswered. The number of different types of PSII is still debated as well as their degree of connectivity. In Chlamydomonas reinhardtii we found that PSIIα possesses a high degree of connectivity and an antenna 2-3 times larger than PSIIβ, as described previously. We also found some connectivity for PSIIβ in contrast with the majority of previous studies. This is in agreement with biochemical studies which describe PSII mega-, super- and core- complexes in Chlamydomonas. In these studies, the smallest unit of PSII in vivo would be a dimer of two core complexes hence allowing connectivity. We discuss the possible relationships between PSIIα and PSIIβ and the PSII mega-, super- and core- complexes. We also showed that strain and medium dependent variations in the half-time of the fluorescence rise can be explained by variations in the proportions of PSIIα and PSIIβ. When analyzing the state transition process in vivo, we found that this process induces an inter-conversion of PSIIα and PSIIβ. During a transition from state 2 to state 1, DCMU fluorescence rise kinetics are satisfactorily fitted by considering two PSII populations with constant kinetic parameters. We discuss our findings about PSII heterogeneity during state transitions in relation with recent results on the remodeling of the pigment-protein PSII architecture during this process. [less ▲]

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See detailGeneralized Pointwise Hölder Spaces
Nicolay, Samuel ULg; Kreit, Damien

Conference (2013, October 31)

In [8,7], the properties of generalized uniform Hölder spaces have been investigated. The idea underlying the definition is to replace the exponent α of the usual spaces Λ^α(R^d) (see e.g. [6]) with a ... [more ▼]

In [8,7], the properties of generalized uniform Hölder spaces have been investigated. The idea underlying the definition is to replace the exponent α of the usual spaces Λ^α(R^d) (see e.g. [6]) with a sequence σ satisfying some conditions. The so-obtained spaces Λ^σ(R^d) generalize the spaces Λ^α(R^d); the spaces Λ^σ(R^d) are actually the spaces B^{1/σ_{∞,∞}(R^d), but they present specific properties (induced by L^∞-norms) when compared to the more general spaces B^{1/σ}_{p,q}(R^d) studied in [2,4,1,5,9,10] for example. Indeed it is shown in [8,7] that most of the usual properties holding for the spaces Λ^α(R^d) can be transposed to the spaces Λ^σ(R^d). Here, we introduce the pointwise version of these spaces: the spaces Λ^{σ,M}(x_0), with x_0∈R^d. Let us recall that a function f∈L^∞_loc(R^d) belongs to the usual pointwise Hölder space Λ^α(x_0) (α>0) if and only if there exist C,J>0 and a polynomial P of degree at most α such that sup_{|h|≤2^{−j}} |f(x_0+h)−P(h)|≤C2^{−jα}. As in [8,7], the idea is again to replace the sequence (2^{−jα})_j appearing in this inequality with a positive sequence (σ_j)j such that σ_{j+1}/σ_j is bounded (for any j); the number M stands for the maximal degree of the polynomial (this degree can not be induced by a sequence σ). By doing so, one tries to get a better characterization of the regularity of the studied function f. Generalizations of the pointwise Hölder spaces have already been proposed (see e.g. [3]), but, to our knowledge, the definition we give here is the most general version and leads to the sharpest results. [1] Alexandre Almeida. Wavelet bases in generalized Besov spaces. J. Math. Anal. Appl., 304(1):198–211, 2005. [2] António M. Caetano and Susana D. Moura. Local growth envelopes of spaces of generalized smoothness: the critical case. Math. Inequal. Appl., 7(4):573–606, 2004. [3] Marianne Clausel. Quelques notions d'irrégularité uniforme et ponctuelle : le point de vue ondelettes. PhD thesis, University of Paris XII, 2008. [4] Walter Farkas. Function spaces of generalised smoothness and pseudo-differential operators associated to a continuous negative definite function. Habilitation Thesis, 2002. [5] Walter Farkas and Hans-Gerd Leopold. Characterisations of function spaces of generalised smoothness. Ann. Mat. Pura Appl., IV. Ser., 185(1):1–62, 2006. [6] Steven G. Krantz. Lipschitz spaces, smoothness of functions, and approximation theory. Exposition. Math., 1(3):193–260, 1983. [7] Damien Kreit and Samuel Nicolay. Characterizations of the elements of generalized Hölder-Zygmund spaces by means of their representation. J. Approx. Theory, to appear, 10.1016/j.jat.2013.04.003. [8] Damien Kreit and Samuel Nicolay. Some characterizations of generalized Hölder spaces. Math. Nachr., 285(17-18):2157–2172, 2012. [9] Thomas Kühn, Hans-Gerd Leopold, Winfried Sickel, and Leszek Skrzypczak. Entropy numbers of embeddings of weighted Besov spaces II. Proceedings of the Edinburgh Mathematical Society (Series 2), 49(02):331–359, 2006. [10] Susana D. Moura. On some characterizations of Besov spaces of generalized smoothness. Math. Nachr., 280(9-10):1190–1199, 2007. [less ▲]

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See detailSnu Spaces, from Theory to Practice
Kleyntssens, Thomas ULg; Nicolay, Samuel ULg

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 ▲]

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

Poster (2013, October 29)

The Hölder spaces provide a natural way for measuring the smoothness of a function. These spaces appear in different areas such as approximation theory and multifractal analysis. The purpose of this ... [more ▼]

The Hölder spaces provide a natural way for measuring the smoothness of a function. These spaces appear in different areas such as approximation theory and multifractal analysis. The purpose of this poster is to present a generalization of such spaces as well as some recent results about their characterizations. These spaces are a particular case of a generalization of Besov Spaces who have recently been extensively studied. [less ▲]

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See detailMise en oeuvre du formalisme multifractal sur les espaces Snu
Kleyntssens, Thomas ULg; Nicolay, Samuel ULg

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 ▲]

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See detailImplementation of the Multifractal Formalism on Snu Spaces
Kleyntssens, Thomas ULg; Nicolay, Samuel ULg

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 ▲]

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See detailAnalysis of PSII antenna size heterogeneity of Chlamydomonas reinhardtii during state transitions - Colloque annuel de la Société Française de Photosynthèse
de Marchin, Thomas ULg; Ghysels, Bart ULg; Nicolay, Samuel ULg et al

Conference (2013, June 18)

PSII antenna size heterogeneity has been extensively studied in the past. Based on in vivo DCMU fluorescence rise kinetics, at least two types of photosystems were described. They differ by their apparent ... [more ▼]

PSII antenna size heterogeneity has been extensively studied in the past. Based on in vivo DCMU fluorescence rise kinetics, at least two types of photosystems were described. They differ by their apparent antenna size and connectivity (this last term refers to the transfer of absorbed energy from a closed PSII unit to an open neighboring unit). In this study, we analysed PSII heterogeneity in Chlamydomonas reinhardtii using non-linear linear regression fitting on in vivo DCMU fluorescence rise kinetics, with a focus on changes in PSII heterogeneity associated with state transitions. We found that PSIIα possesses a high degree of connectivity and an antenna about 3 times larger than PSIIβ, as described previously. In contrast with most earlier studies, we found some connectivity for PSIIβ (although it was highly variable). This is in agreement with recent models based on biochemical and structural analysis of PSII after gel filtration separation which describe PSII mega-, super- and core- complexes in Chlamydomonas. According to these studies, the smallest unit of PSII in vivo would be a dimer of two core complexes hence still allowing connectivity. We also showed that strain and medium dependent variations in the half-time of the fluorescence rise, generally taken as an indicator of the average cross-section of PSII, can be explained by variations in the proportions of PSIIα and PSIIβ. When analyzing the state transition process, we showed for the first time in vivo that it induces an inter-conversion of PSIIα and PSIIβ. These findings are discussed with respect to the latest insights on the remodeling of the pigment-protein PSII architecture during this process. [less ▲]

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