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A forecasting method using a wavelet-based mode decomposition and application to the ENSO index
http://hdl.handle.net/2268/183452
Title: A forecasting method using a wavelet-based mode decomposition and application to the ENSO index
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<br/>Author, co-author: Deliège, Adrien; Nicolay, Samuel; Fettweis, Xavier
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<br/>Abstract: 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.Distributive contact lattices with nontangential part-of relations
http://hdl.handle.net/2268/183277
Title: Distributive contact lattices with nontangential part-of relations
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<br/>Author, co-author: Raskin, Julien
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<br/>Abstract: In a Boolean algebra, the "contact" and "nontangential part-of" relations are linked and interdefinable thanks to the complement. These relations yield on the Stone dual of the algebra two closed relations that appear to be the same. In a distributive lattice, the contact and nontangential part-of relations also yield dual closed relations on the Priestley dual. However, as the link between them is lost, these relations are not equal anymore. We explore the conditions linking the contact and nontangential part-of relations one should add in order to recover these relations knowing the intersection of their dual.Robustifier nos analyses de données: un must!
http://hdl.handle.net/2268/183235
Title: Robustifier nos analyses de données: un must!
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<br/>Author, co-author: Aerts, Stéphanie
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<br/>Abstract: Financiers, biologistes, psychologues, économistes ou chimistes,… tous utilisent des outils statistiques pour prédire, expliquer, modéliser, comprendre des processus spécifiques à leur domaine. Néanmoins, les résultats de toute analyse statistique, de la plus simple à la plus complexe, peuvent être fortement influencés, voire renversés, par la présence d’une seule valeur atypique. S’il est assez aisé de surmonter ce problème pour de simples ensembles de données, lorsque plusieurs caractéristiques sont mesurées simultanément, disposer de méthodes robustes nécessite l’emploi d’algorithmes alliant à la puissance actuelle de nos ordinateurs, des théorèmes mathématiques parfois vieux de plus de 50 ans.
A l’heure où on dispose de bases de données de plus en plus grandes mais de qualité inégale, il est indispensable que les praticiens se familiarisent avec les statistiques robustes afin de fiabiliser leurs conclusions.
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<br/>Commentary: Invitation reçueA propos des fonctions continues nulle part dérivables
http://hdl.handle.net/2268/183188
Title: A propos des fonctions continues nulle part dérivables
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<br/>Author, co-author: Esser, Céline
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<br/>Abstract: En 1872, Karl Weierstrass choqua la communauté mathématique en présentant une famille de fonctions continues nulle part dérivables. Après la publication de ce résultat, beaucoup d'autres mathématiciens apportèrent leur propre contribution en construisant d'autres fonctions continues et nulle part dérivables.
Dans cet exposé, nous présenterons les fonctions de Weierstrass. De plus, nous montrerons que le th eorème de Baire permet d'affirmer que l'ensemble des fonctions nulle part dérivables est dense dans l'ensemble des fonctions continues. Nous étudierons également la régularité ponctuelle de telles fonctions en utilisant la notion d'exposant de Hölder.Abelian bordered factors and periodicity
http://hdl.handle.net/2268/182919
Title: Abelian bordered factors and periodicity
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<br/>Author, co-author: Charlier, Emilie; Harju, Tero; Puzynina, Svetlana; Zamboni, Luca
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<br/>Abstract: A finite word u is said to be bordered if u has a proper prefix which is also a suffix of u, and unbordered otherwise. Ehrenfeucht and Silberger proved that an infinite word is purely periodic if and only if it contains only finitely many unbordered factors. We are interested in abelian and weak abelian analogues of this result; namely, we investigate the following question(s): Let w be an infinite word such that all sufficiently long factors are (weakly) abelian bordered; is w (weakly) abelian periodic? In the process we answer a question of Avgustinovich et al. concerning the abelian critical factorization theorem.A generalization of the Snu spaces: getting rid of dyadic scales
http://hdl.handle.net/2268/182899
Title: A generalization of the Snu spaces: getting rid of dyadic scales
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<br/>Author, co-author: Kleyntssens, Thomas; Nicolay, Samuel
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<br/>Abstract: 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.Denjoy-Carleman classes and lineability
http://hdl.handle.net/2268/182889
Title: Denjoy-Carleman classes and lineability
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<br/>Author, co-author: Esser, Céline
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<br/>Abstract: The Denjoy-Carleman classes are spaces of smooth functions which satisfy growth conditions on their derivatives defined through weight sequences. In this talk, given a Denjoy-Carleman class E of Beurling type that strictly contains another non-quasianalytic class F of Roumieu type, we handle the question of knowing how large the set of functions in E that are nowhere in the class F is. In particular, we prove the dense-lineability of the set of functions of E which are nowhere in F. Consequences for the Gevrey classes are also given. We extend then these results to the case of classes of ultradifferentiable functions defined using weight functions.Proceedings of the 12th Workshop on Models and Algorithms for Planning and Scheduling Problems
http://hdl.handle.net/2268/182862
Title: Proceedings of the 12th Workshop on Models and Algorithms for Planning and Scheduling Problems
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<br/>Editor: Marchetti-Spaccamela, Alberto; Crama, Yves; Goossens, Dries; Leus, Roel; Schyns, Michael; Spieksma, Frits
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<br/>Abstract: This volume contains abstracts of talks presented at the 12th Workshop on Models and Algorithms for Planning and Scheduling Problems (MAPSP 2015), held from June 8 to June 12, 2015, in La Roche-en-Ardenne, Belgium. MAPSP is a biennial workshop dedicated to all theoretical and practical aspects of scheduling, planning, and timetabling. The abstracts in this volume include 5 invited talks by Onno Boxma, Michel Goemans, Willem-Jan van Hoeve, Rolf Niedermeier, and Stephan Westphal, plus 88 contributed talks.Extreme phase sensitivity in systems with fractal isochrons
http://hdl.handle.net/2268/182791
Title: Extreme phase sensitivity in systems with fractal isochrons
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<br/>Author, co-author: Mauroy, Alexandre; Mezic, Igor
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<br/>Abstract: Sensitivity to initial conditions is usually associated with chaotic dynamics and strange attractors. However, even systems with (quasi)periodic dynamics can exhibit it. In this context we report on the fractal properties of the isochrons of some continuous-time asymptotically periodic systems. We define a global measure of phase sensitivity that we call the phase sensitivity coefficient and show that it is an invariant of the system related to the capacity dimension of the isochrons. Similar results are also obtained with discrete-time systems. As an illustration of the framework, we compute the phase sensitivity coefficient for popular models of bursting neurons, suggesting that some elliptic bursting neurons are characterized by isochrons of high fractal dimensions and exhibit a very sensitive (unreliable) phase response.About the Regularity of Cantor's Bijection
http://hdl.handle.net/2268/182199
Title: About the Regularity of Cantor's Bijection
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<br/>Author, co-author: Simons, Laurent; Nicolay, Samuel
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<br/>Abstract: 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.A Bayesian statistical model for the Ebola epidemic in Sierra Leone
http://hdl.handle.net/2268/182078
Title: A Bayesian statistical model for the Ebola epidemic in Sierra Leone
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<br/>Author, co-author: Frasso, Gianluca; Lambert, Philippe; Bonou, Wilfried
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<br/>Abstract: The 2014 Ebola outbreak in Sierra Leone is analyzed using an extension of the SEIR compartmental model. The unknown parameters of the system of differential equations are estimated by combining data on the number of new (laboratory confirmed) Ebola cases reported by the Ministry of Health and prior distributions for the transition rates elicited using information collected by the WHO Response Team (2014) during the follow-up of specific Ebola cases. The evolution over time of the disease transmission rate is modeled nonparametrically using penalized B-splines. Our framework represents a valuable and robust stochastic tool for the study of an epidemic dynamic from irregular and possibly aggregated case data. Simulations and the analysis of the 2014 Sierra Leone Ebola data highlight the merits of the proposed methodology.Dense-lineability in classes of ultradifferentiable functions
http://hdl.handle.net/2268/182070
Title: Dense-lineability in classes of ultradifferentiable functions
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<br/>Author, co-author: Esser, Céline
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<br/>Abstract: The Denjoy-Carleman classes are spaces of smooth functions which satisfy growth conditions on their derivatives defined through weight sequences. In this talk, given a Denjoy-Carleman class E of Beurling type that strictly contains another non-quasianalytic class F of Roumieu type, we handle the question of knowing how large the set of functions in E that are nowhere in the class $F$ is. In particular, we prove the dense-lineability of the set of functions of $E$ which are nowhere in F. Consequences for the Gevrey classes are also given. We extend then these results to the case of classes of ultradifferentiable functions defined imposing conditions on their Fourier Laplace transform using weight functions.Is Büchi's theorem useful for you?
http://hdl.handle.net/2268/181997
Title: Is Büchi's theorem useful for you?
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<br/>Author, co-author: Rigo, Michel
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<br/>Abstract: Almost a century ago, Presburger showed that the first order theory of the natural numbers with addition is decidable. Following the work of Büchi in 1960, this result still holds when adding a function $V_k$ to the structure, where $V_k(n)$ is the largest power of $k\ge 2$ diving $n$. In particular, this leads to a logical characterization of the $k$-automatic sequences. During the last few years, many applications of this result have been considered in combinatorics on words, mostly by J. Shallit and his coauthors.
In this talk, we will present this theorem of Büchi where decidability relies on finite automata.Then we will review some results about automatic sequences or morphic words that can be proved automatically (i.e., the proof is carried on by an algorithm). Finally, we will sketch the limitation of this technique. With a single line formula, one can prove automatically that the Thue-Morse word has no overlap but, hopefully, not all the combinatorial properties of morphic words can be derived in this way.Hölder Continuity and Wavelets
http://hdl.handle.net/2268/181715
Title: Hölder Continuity and Wavelets
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<br/>Author, co-author: Simons, Laurent
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<br/>Abstract: There exist a lot of continuous nowhere differentiable functions, but these functions do not have the same irregularity. Hölder continuity, and more precisely Hölder exponent, allow to quantify this irregularity. If the Hölder exponent of a function takes several values, the function is said multifractal. In the first part of this thesis, we study in details the regularity and the multifractality of some functions: the Darboux function, the Cantor bijection and a generalization of the Riemann function.
The theory of wavelets notably provides a tool to investigate the Hölder continuity of a function. Wavelets also take part in other contexts. In the second part of this thesis, we consider a nonstationary version of the classical theory of wavelets. More precisely, we study the nonstationary orthonormal bases of wavelets and their construction from a nonstationary multiresolution analysis. We also present the nonstationary continuous wavelet transform.
For some irregular functions, it is difficult to determine its Hölder exponent at each point. In order to get some information about this one, new function spaces based on wavelet leaders have been introduced. In the third and last part of this thesis, we present these new spaces and their first properties. We also define a natural topology on them and we study some properties.Syntactic polygraphs - A formalism extending both constituency and dependency
http://hdl.handle.net/2268/181587
Title: Syntactic polygraphs - A formalism extending both constituency and dependency
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<br/>Author, co-author: Kahane, Sylvain; Mazziotta, Nicolas
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<br/>Abstract: Syntactic analyses describe grouping operations that explain how words are combined to form utterances. The nature of these operations depends on the approach. In a constituency-based approach, grouping operations are ordered, or stratified, part-whole relations. In a dependency-based approach, grouping operations identify a governor (or head), i.e. they are directed hierarchical relations between words. It is possible to convert a constituency tree into a dependency tree by dereifying the nodes, by identifying the governor and by removing the stratification of the part-whole relations. Polygraphs combine the two types of information into a single structure and are therefore a more powerful formalism. By relaxing constraints, polygraphs also allow to underspecify both kinds of information.
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<br/>Commentary: Full paper acceptedExtensions du concept de fonction
http://hdl.handle.net/2268/181230
Title: Extensions du concept de fonction
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<br/>Author, co-author: Bair, Jacques; Henry, Valérie
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<br/>Abstract: Extensions modernes du concept de fonction
Comme la plupart des concepts mathématiques, celui de fonction s’est transformé progressivement pour devenir ce qui nous en est enseigné aujourd’hui. Il évolue encore de nos jours : il est toujours plus général, mais également plus abstrait et sophistiqué. Nous donnons ici un petit aperçu intuitif de quelques-unes de ses généralisations modernes.Petit voyage à travers les âges: de l'expression à la fonction
http://hdl.handle.net/2268/181228
Title: Petit voyage à travers les âges: de l'expression à la fonction
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<br/>Author, co-author: Bair, Jacques; Henry, Valérie
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<br/>Abstract: Le concept de fonction est assurément fondamental en mathématiques. Il a été construit lentement, par de multiples améliorations successives. Dans cet article, nous souhaitons évoquer succinctement quelques étapes marquantes de son élaboration.Des statistiques pour détecter les équipements de laboratoire peu fiables
http://hdl.handle.net/2268/181141
Title: Des statistiques pour détecter les équipements de laboratoire peu fiables
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<br/>Author, co-author: Aerts, StéphaniePar A plus B
http://hdl.handle.net/2268/180796
Title: Par A plus B
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<br/>Author, co-author: Deliège, Adrien; Nicolay, Samuel
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<br/>Abstract: Exposé de vulgarisation sur l'état de l'art concernant les ondelettes et ses applications, en particulier l'étude du phénomène El Niño.EGU2015 - ENSO forecast using a wavelet-based mode decomposition
http://hdl.handle.net/2268/180789
Title: EGU2015 - ENSO forecast using a wavelet-based mode decomposition
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<br/>Author, co-author: Deliège, Adrien; Nicolay, Samuel; Fettweis, Xavier
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<br/>Abstract: 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.