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Regularity of functions: Genericity and multifractal analysis
http://hdl.handle.net/2268/174112
Title: Regularity of functions: Genericity and multifractal analysis
<br/>
<br/>Author, co-author: Esser, Céline
<br/>
<br/>Abstract: As surprising as it may seem, there exist functions of C∞(R) which are nowhere analytic. When such an unexpected object is found, a natural question is to ask whether many similar ones may exist. A classical technique is to use the Baire category theorem and the notion of residuality. This notion is purely topological and does not give any information about the measure of the set of objects satisfying such a property. In this
purpose, the notion of prevalence has been introduced. Moreover, one could also wonder whether large algebraic
structures of such objects can be constructed. This question is formalized by the notion of lineability.
The first objective of the thesis is to go further into the study of nowhere analytic functions. It is known that the set of nowhere analytic functions is residual and lineable in C∞([0, 1]). We prove that the set of nowhere analytic functions is also prevalent in C∞([0, 1]). Those results of genericity are then generalized using Gevrey classes, which can be seen as intermediate between the space of analytic functions and the space of infinitely differentiable functions. We also study how far such results of genericity could be extended to spaces of ultradifferentiable functions, defined using weight sequences or using weight functions.
The second main objective is to study the pointwise regularity of functions via their multifractal spectrum. Computing the multifractal spectrum of a function using directly its definition is an unattainable goal in most of the practical cases, but there exist heuristic methods, called multifractal formalisms, which allow to estimate this spectrum and which give satisfactory results in many situations. The Frisch-Parisi conjecture, classically used and based on Besov spaces, presents two disadvantages: it can only hold for spectra that are concave and
it can only yield the increasing part of spectra. Concerning the first problem, the use of Snu spaces allows to deal with non-concave increasing spectra. Concerning the second problem, a generalization of the Frisch-Parisi conjecture obtained by replacing the role played by wavelet coefficients by wavelet leaders allows to recover the decreasing part of concave spectra.
Our purpose in this thesis is to combine both approaches and define a new formalism derived from large deviations based on statistics of wavelet leaders. As expected, we show that this method yields non-concave spectra and is not limited to their increasing part. From the theoretical point of view, we prove that this formalism is more efficient than the previous wavelet-based multifractal formalisms. We present the underlying function space and endow it with a topology.Distribution and robustness of a distance-based multivariate coefficient of variation
http://hdl.handle.net/2268/173750
Title: Distribution and robustness of a distance-based multivariate coefficient of variation
<br/>
<br/>Author, co-author: Aerts, Stéphanie
<br/>
<br/>Abstract: When one wants to compare the homogeneity of a characteristic in several popula-
tions that have di erent means, the advocated statistic is the univariate coe cient of variation.
However, in the multivariate setting, comparing marginal coe cients may be inconclusive.
Therefore, several extensions that summarize multivariate relative dispersion in one single in-
dex have been proposed in the literature (see Albert & Zhang, 2010, for a review).
In this poster, focus is on a particular extension, due to Voinov & Nikulin (1996), based on the
Mahalanobis distance between the mean and the origin of the design space. Some arguments
are outlined for justifying this choice. Then, properties of its sample version under elliptical
symmetry are discussed. Under normality, this estimator is shown to be biased at nite samples.
In order to overcome this drawback, two bias corrections are proposed.
Moreover, the empirical estimator also su ers from a lack of robustness, which is illustrated
by means of in uence functions. A robust counterpart based on the Minimum Covariance
Determinant estimator is advocated.Corner asymptotics of the magnetic potential in the eddy-current model
http://hdl.handle.net/2268/173509
Title: Corner asymptotics of the magnetic potential in the eddy-current model
<br/>
<br/>Author, co-author: Dauge, Monique; Dular, Patrick; Krähenbühl, Laurent; Péron, Victor; Perrussel, Ronan; Poignard, Clair
<br/>
<br/>Abstract: In this paper, we describe the scalar magnetic potential in the vicinity of a corner of a conducting body embedded in a dielectric medium in a bidimensional setting. We make explicit the corner asymptotic expansion for this potential as the distance to the corner goes to zero. This expansion involves singular functions and singular coefficients. We introduce a method for the calculation of the singular functions near the corner and we provide two methods to compute the singular coefficients: the method of moments and the method of quasi-dual singular functions. Estimates for the convergence of both approximate methods are proven. We eventually illustrate the theoretical results with finite element computations. The specific non-standard feature of this problem lies in the structure of its singular functions: They have the form of series whose first terms are harmonic polynomials and further terms are genuine non-smooth functions generated by the piecewise constant zeroth order term of the operator.Abelian bordered factors and periodicity
http://hdl.handle.net/2268/173420
Title: Abelian bordered factors and periodicity
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<br/>Author, co-author: Charlier, Emilie; Harju, Tero; Puzynina, Svetlana; Zamboni, Luca
<br/>
<br/>Abstract: A finite word is bordered if it has a non-empty proper prefix which is equal to its suffix, 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 an abelian modification of this fact. Namely, we have the following question: Let w be an infinite
word such that all sufficiently long factors are abelian bordered. Is w (abelian) periodic? We also consider a weakly abelian modification of this question, when only the frequencies of letters are taken into account. Besides that, we answer a question of Avgustinovich, Karhumaki and Puzynina concerning abelian central factorization theorem.The freeness problem for products of matrices defined on bounded languages
http://hdl.handle.net/2268/173417
Title: The freeness problem for products of matrices defined on bounded languages
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<br/>Author, co-author: Charlier, Emilie
<br/>
<br/>Abstract: In this talk, I presented a joint work with Juha Honkala. We study the freeness problem for matrix semigroups. We show that the freeness problem is decidable for upper-triangular 2x2 matrices with rational entries when the products are restricted to certain bounded languages. We also show that this problem becomes undecidable for large enough matrices.Infinite self-shuffling words
http://hdl.handle.net/2268/173414
Title: Infinite self-shuffling words
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<br/>Author, co-author: Charlier, Emilie
<br/>
<br/>Abstract: In this paper we introduce and study a new property of infinite words: An infinite word x, with values in a finite set A, is said to be k-self-shuffling (k≥2) if there exists a shuffle of k copies of x which produces x. We are particularly interested in the case k=2, in which case we say x is self-shuffling. This property of infinite words is shown to be independent of the complexity of the word as measured by the number of distinct factors of each length. Examples exist from bounded to full complexity. It is also an intrinsic property of the word and not of its language (set of factors). For instance, every aperiodic word contains a non-self-shuffling word in its shift orbit closure. While the property of being self-shuffling is a relatively strong condition, many important words arising in the area of symbolic dynamics are verified to be self-shuffling. They include for instance the Thue–Morse word fixed by the morphism 0↦01, 1↦10. As another example we show that all Sturmian words of intercept 0<ρ<1 are self-shuffling (while those of intercept ρ=0 are not). Our characterization of self-shuffling Sturmian words can be interpreted arithmetically in terms of a dynamical embedding and defines an arithmetic process we call the stepping stone model. One important feature of self-shuffling words stems from their morphic invariance: The morphic image of a self-shuffling word is self-shuffling. This provides a useful tool for showing that one word is not the morphic image of another. In addition to its morphic invariance, this new notion has other unexpected applications particularly in the area of substitutive dynamical systems. For example, as a consequence of our characterization of self-shuffling Sturmian words, we recover a number theoretic result, originally due to Yasutomi, on a classification of pure morphic Sturmian words in the orbit of the characteristic.Detection of non-concave and non-increasing spectra: Snu spaces revisited with wavelet leaders
http://hdl.handle.net/2268/172473
Title: Detection of non-concave and non-increasing spectra: Snu spaces revisited with wavelet leaders
<br/>
<br/>Author, co-author: Esser, Céline
<br/>
<br/>Abstract: Our objective is to study the pointwise regularity of functions via their multifractal spectrum. Computing the multifractal spectrum of a function using directly its definition is an unattainable goal in most of the practical cases, but there exist heuristic methods, called multifractal formalisms, which allow to estimate this spectrum and which give satisfactory results in many situations. The Frisch-Parisi conjecture, classically used and based on Besov spaces, presents two disadvantages: it can only hold for spectra that are concave and it can only yield the increasing part of spectra. Concerning the first problem, the use of S spaces allows to deal with non-concave
increasing spectra. Concerning the second problem, a generalization of the Frisch-Parisi conjecture obtained by replacing the role played by wavelet coefficients by wavelet leaders allows to recover the decreasing part of concave spectra. We present a combination of both approaches to define a new formalism derived from large deviations based on statistics of wavelet leaders. We also present the associated function space.Genericity and classes of ultradifferentiable functions
http://hdl.handle.net/2268/172472
Title: Genericity and classes of ultradifferentiable functions
<br/>
<br/>Author, co-author: Esser, Céline
<br/>
<br/>Abstract: As surprising as it may seem, there exist infinitely differentiable functions which are nowhere analytic. When such an unexpected object is found, a natural question is to ask whether many similar ones may exist. A classical technique is to use the Baire category theorem and the notion of residuality. This notion is purely topological and does not give any information about the measure of the set of objects satisfying such a property. In this purpose, the notion of prevalence has been introduced. Moreover, one could also wonder whether large algebraic structures of such objects can be constructed. This question is formalized by the notion of lineability.
The first objective of this talk is to go further into the study of nowhere analytic functions. It is known that the set of nowhere analytic functions is residual and lineable in C^infty([0, 1]). We prove that the set of nowhere analytic functions is also prevalent in this space. Those results of genericity are then generalized using Gevrey classes, which can be seen as intermediate between the space of analytic functions and the space of infinitely differentiable functions. We also study how far such results of genericity could be extended to spaces of ultradifferentiable functions, defined using weight sequences.De l’importance des échelles dyadiques dans les espaces Snu
http://hdl.handle.net/2268/172239
Title: De l’importance des échelles dyadiques dans les espaces Snu
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<br/>Author, co-author: Kleyntssens, Thomas; Nicolay, Samuel
<br/>
<br/>Abstract: 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.Fonction de Riemann généralisée
http://hdl.handle.net/2268/172189
Title: Fonction de Riemann généralisée
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<br/>Author, co-author: Simons, Laurent; Bastin, Françoise; Nicolay, Samuel
<br/>
<br/>Abstract: 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.Large-scale optimization for component analysis of fMRI resting brain data
http://hdl.handle.net/2268/172172
Title: Large-scale optimization for component analysis of fMRI resting brain data
<br/>
<br/>Author, co-author: Liegeois, RaphaëlNote on how cerebral functional connectivity encodes structural constraints of the human brain
http://hdl.handle.net/2268/172171
Title: Note on how cerebral functional connectivity encodes structural constraints of the human brain
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<br/>Author, co-author: Liegeois, RaphaëlAnalyse de la régularité hölderienne : De la théorie à l'application à des séries temporelles de températures
http://hdl.handle.net/2268/172055
Title: Analyse de la régularité hölderienne : De la théorie à l'application à des séries temporelles de températures
<br/>
<br/>Author, co-author: Deliège, Adrien
<br/>
<br/>Abstract: Part 1 is divided in 3 chapters. Chapter 1 is devoted to the definition of the notion of Hölder exponent and its properties. Chapter 2 introduces the Hausdorff measure and dimension and their properties. Chapter 3 is about multiresolution analysis and the wavelet leaders method, the multifractal formalism used in Part 2. Part 2 consists of applications of the wavelet leaders method to analyze the Hölder regularity of some well-known functions (Chapter 4) and of surface air temperature time series (Chapter 5).
<br/>
<br/>Commentary: Mémoire de fin d'études.A wavelet leaders-based climate classification of European surface air temperature signals
http://hdl.handle.net/2268/171961
Title: A wavelet leaders-based climate classification of European surface air temperature signals
<br/>
<br/>Author, co-author: Deliège, Adrien; Nicolay, Samuel
<br/>
<br/>Abstract: 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.A multifractal analysis of air temperature signals based on the wavelet leaders method
http://hdl.handle.net/2268/171956
Title: A multifractal analysis of air temperature signals based on the wavelet leaders method
<br/>
<br/>Author, co-author: Deliège, Adrien; Nicolay, Samuel
<br/>
<br/>Abstract: 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.A wavelet leaders-based climate classification of European surface air temperature signals
http://hdl.handle.net/2268/171949
Title: A wavelet leaders-based climate classification of European surface air temperature signals
<br/>
<br/>Author, co-author: Deliège, Adrien; Nicolay, Samuel
<br/>
<br/>Abstract: 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.A wavelet-based analysis of surface air temperature regularity
http://hdl.handle.net/2268/171945
Title: A wavelet-based analysis of surface air temperature regularity
<br/>
<br/>Author, co-author: Deliège, Adrien; Nicolay, Samuel
<br/>
<br/>Abstract: 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.A wavelet-based analysis of surface air temperature variability
http://hdl.handle.net/2268/171943
Title: A wavelet-based analysis of surface air temperature variability
<br/>
<br/>Author, co-author: Deliège, Adrien; Nicolay, Samuel
<br/>
<br/>Abstract: 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.Mathématiques convoquées par le registre graphique au sein du cours de physique
http://hdl.handle.net/2268/171813
Title: Mathématiques convoquées par le registre graphique au sein du cours de physique
<br/>
<br/>Author, co-author: Renkens, Céline; Henry, ValérieComparison of robust detection techniques for local outliers in multivariate spatial data
http://hdl.handle.net/2268/171721
Title: Comparison of robust detection techniques for local outliers in multivariate spatial data
<br/>
<br/>Author, co-author: Ernst, Marie; Haesbroeck, Gentiane
<br/>
<br/>Abstract: Spatial data are characterized by statistical units, with known geographical positions, on which non spatial attributes are measured. Spatial data may contain two types of atypical observations: global and/or local outliers. The attribute values of a global outlier are outlying with respect to the values taken by the majority of the data points while the attribute values of a local outlier are extreme when compared to those of its neighbors.
Usual outlier detection techniques may be used to find global outliers as the geographical positions of the data is not taken into account in this specific search. The detection of local outliers is more complex, especially when there are more than one non spatial attributes. This talk focuses on local detection with two main objectives.
First, we will shortly review some of the local detection techniques that seem to perform well in practice. Among these, one can find robust ``Mahalanobis-type'' detection techniques and a wheighted PCA approach. We suggest an adaptation to one of these to further develop its local characteristic.
Then, examples and simulations, based on linear model of co-regionalisation with Matern models, are reported and discussed in order to compare in an objective way the different detection techniques.