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We introduce a multidimensional, neural-network approach to reveal and measure urban segregation phenomena, based on the Self-Organizing Map algorithm (SOM). The multidimensionality of SOM allows one to apprehend a large number of variables simultaneously, defined on census or other types of statistical blocks, and to perform clustering along them. Levels of segregation are then measured through correlations between distances on the neural network and distances on the actual geographical map. Further, the stochasticity of SOM enables one to quantify levels of heterogeneity across census blocks. We illustrate this new method on data available for the city of Paris., NCAA S.I. WSOM+ 2017

The purpose of this work is to establish a quantitative and constructive stability result for a class of subcritical Gagliardo-Nirenberg-Sobolev inequalities which interpolates between the logarithmic Sobolev inequality and the standard Sobolev inequality (in dimension larger than three), or Onofri's inequality in dimension two. We develop a new strategy, in which the flow of the fast diffusion equation is used as a tool: a stability result in the inequality is equivalent to an improved rate of convergence to equilibrium for the flow. The regularity properties of the parabolic flow allow us to connect an improved entropy - entropy production inequality during an initial time layer to spectral properties of a suitable linearized problem which is relevant for the asymptotic time layer. Altogether, the stability in the inequalities is measured by a deficit which controls in strong norms (a Fisher information which can be interpreted as a generalized Heisenberg uncertainty principle) the distance to the manifold of optimal functions. The method is constructive and, for the first time, quantitative estimates of the stability constant are obtained, including in the critical case of Sobolev's inequality. To build the estimates, we establish a quantitative global Harnack principle and perform a detailed analysis of large time asymptotics by entropy methods., Comment: This manuscript merges 2007.03674: Stability in Gagliardo-Nirenberg inequalities and 2007.03419: Stability in Gagliardo-Nirenberg inequalities. Supplementary material with several additions including: Chapter 1 (variational methods) and Chapter 6 (Sobolev's inequality)

International audience; Background:Percutaneous left atrial appendage closure (LAAC) exposes to the risk of device thrombosis in patients with atrial fibrillation who frequently have a contraindication to full anticoagulation. Thereby, dual antiplatelet therapy (DAPT) is usually preferred. No randomized study has evaluated nonvitamin K antagonist oral anticoagulant after LAAC, and we decided to evaluate the efficacy and safety of reduced doses of rivaroxaban after LAAC.Methods:ADRIFT (Assessment of Dual Antiplatelet Therapy Versus Rivaroxaban in Atrial Fibrillation Patients Treated With Left Atrial Appendage Closure) is a multicenter, phase IIb study, which randomized 105 patients after successful LAAC to either rivaroxaban 10 mg (R10, n=37), rivaroxaban 15 mg (R15, n=35), or DAPT with aspirin 75 mg and clopidogrel 75 mg (n=33). The primary end point was thrombin generation (prothrombin fragments 1+2) measured 2 to 4 hours after drug intake, 10 days after treatment initiation. Thrombin-antithrombin complex, D-dimers, rivaroxaban concentrations were also measured at 10 days and 3 months. Clinical end points were evaluated at 3-month follow-up.Results:The primary end point was reduced with R10 (179 pmol/L [interquartile range (IQR), 129–273], P

Sparse versions of principal component analysis (PCA) have imposed themselves as simple, yet powerful ways of selecting relevant features of high-dimensional data in an unsupervised manner. However, when several sparse principal components are computed, the interpretation of the selected variables is difficult since each axis has its own sparsity pattern and has to be interpreted separately. To overcome this drawback, we propose a Bayesian procedure called globally sparse probabilistic PCA (GSPPCA) that allows to obtain several sparse components with the same sparsity pattern. This allows the practitioner to identify the original variables which are relevant to describe the data. To this end, using Roweis' probabilistic interpretation of PCA and a Gaussian prior on the loading matrix, we provide the first exact computation of the marginal likelihood of a Bayesian PCA model. To avoid the drawbacks of discrete model selection, a simple relaxation of this framework is presented. It allows to find a path of models using a variational expectation-maximization algorithm. The exact marginal likelihood is then maximized over this path. This approach is illustrated on real and synthetic data sets. In particular, using unlabeled microarray data, GSPPCA infers much more relevant gene subsets than traditional sparse PCA algorithms., Comment: An earlier version of this paper appeared in the Proceedings of the 19th International Conference on Artificial Intelligence and Statistics (AISTATS 2016)

Over the last two decades, network structure analysis has experienced rapid growth with its construction and its intervention in many fields, such as: communication networks, financial transaction networks, gene regulatory networks, disease transmission networks, mobile telephone networks. Social networks are now commonly used to represent the interactions between groups of people; for instance, ourselves, our professional colleagues, our friends and family, are often part of online networks, such as Facebook, Twitter, email. In a network, many factors can exert influence or make analyses easier to understand. Among these, we find two important ones: the time factor, and the network context. The former involves the evolution of connections between nodes over time. The network context can then be characterized by different types of information such as text messages (email, tweets, Facebook, posts, etc.) exchanged between nodes, categorical information on the nodes (age, gender, hobbies, status, etc.), interaction frequencies (e.g., number of emails sent or comments posted), and so on. Taking into consideration these factors can lead to the capture of increasingly complex and hidden information from the data. The aim of this thesis is to define new models for graphs which take into consideration the two factors mentioned above, in order to develop the analysis of network structure and allow extraction of the hidden information from the data. These models aim at clustering the vertices of a network depending on their connection profiles and network structures, which are either static or dynamically evolving. The starting point of this work is the stochastic block model, or SBM. This is a mixture model for graphs which was originally developed in social sciences. It assumes that the vertices of a network are spread over different classes, so that the probability of an edge between two vertices only depends on the classes they belong to., Depuis les travaux précurseurs de Moreno (1934), l’analyse des réseaux est devenue une discipline forte, qui ne se limite plus à la sociologie et qui est à présent appliquée à des domaines très variés tels que la biologie, la géographie ou l’histoire. L’intérêt croissant pour l’analyse des réseaux s’explique d’une part par la forte présence de ce type de données dans le monde numérique d’aujourd’hui et, d’autre part, par les progrès récents dans la modélisation et le traitement de ces données. En effet, informaticiens et statisticiens ont porté leurs efforts depuis plus d’une dizaine d’années sur ces données de type réseau en proposant des nombreuses techniques permettant leur analyse. Parmi ces techniques on note les méthodes de clustering qui permettent en particulier de découvrir une structure en groupes cachés dans le réseau. De nombreux facteurs peuvent exercer une influence sur la structure d’un réseau ou rendre les analyses plus faciles à comprendre. Parmi ceux-ci, on trouve deux facteurs importants: le facteur du temps, et le contexte du réseau. Le premier implique l’évolution des connexions entre les nœuds au cours du temps. Le contexte du réseau peut alors être caractérisé par différents types d’informations, par exemple des messages texte (courrier électronique, tweets, Facebook, messages, etc.) échangés entre des nœuds, des informations catégoriques sur les nœuds (âge, sexe, passe-temps, Les fréquences d’interaction (par exemple, le nombre de courriels envoyés ou les commentaires affichés), et ainsi de suite. La prise en considération de ces facteurs nous permet de capturer de plus en plus d’informations complexes et cachées à partir des données. L’objectif de ma thèse été de définir des nouveaux modèles de graphes aléatoires qui prennent en compte les deux facteurs mentionnés ci-dessus, afin de développer l’analyse de la structure du réseau et permettre l’extraction de l’information cachée à partir des données. Ces modèles visent à regrouper les sommets d’un réseau en fonction de leurs profils de connexion et structures de réseau, qui sont statiques ou évoluant dynamiquement au cours du temps. Le point de départ de ces travaux est le modèle de bloc stochastique (SBM). Il s’agit d’un modèle de mélange pour les graphiques qui ont été initialement développés en sciences sociales. Il suppose que les sommets d’un réseau sont répartis sur différentes classes, de sorte que la probabilité d’une arête entre deux sommets ne dépend que des classes auxquelles ils appartiennent.

In this thesis, we have attempted to contribute to the optimization of dynamical problems with delay in state space. We are specifically interested in the viewpoint of Pontryagin who outlined in his book published in 1962 the necessary conditions required for solving such problems. In his work published in 1972, Warga catalogued the possible solutions. Li and al. analyzed the case of periodic control. We will treat an optimal control problem governed by a Delay Functional Differential Equation. Our method is close to the one of P. Michel on dynamical system governed by Ordinary Differential Equations. The main problem ariving out in this approach is the use of the resolvent of the Delay Functional Differential Equation. We also consider with Euler-Lagrange condition in the framework of variational problems with delay., L'objectif de cette thèse est de contribuer à l'optimisation de problèmes dynamiques en présence de retard. Le point de vue qui nous intéressera est celui de Pontryagin qui dans son ouvrage publié en 1962 a donné les conditions nécessaires d'existence de solutions pour ce type de problème. Warga dans son ouvrage publié en 1972 a fait un catalogue des solutions possible, Li et al. ont étudié le cas de contrôle périodique. Notre méthode de démonstration est directement inspirée de la démonstration de P. Michel du cas des systèmes gouvernés par des équations différentielles ordinaires. La principale difficulté pour cette approche est l'utilisation de la résolvante de l'équation différentielle fonctionnelle linéarisée de l'équation différentielle fonctionnelle d'évolution qui gouverne le système. Nous traitons aussi de condition d'Euler-Lagrange dans le cadre d'un problème de calcul variationnel avec retard.

International audience; In recent years, many clustering methods have been proposed to extract information from networks. The principle is to look for groups of vertices with homogenous connection profiles. Most of these techniques are suitable for static networks, that is to say, not taking into account the temporal dimension. This work is motivated by the need of analyzing evolving networks where a decomposition of the networks into subgraphs is given. Therefore, in this paper, we consider the random subgraph model (RSM) which was proposed recently to model networks through latent clusters built within known partitions. Using a state space model to characterize the cluster proportions, RSM is then extended in order to deal with dynamic networks. We call the latter the dynamic random subgraph model (dRSM). A variational expectation maximization (VEM) algorithm is proposed to perform inference. We show that the variational approximations lead to an update step which involves a new state space model from which the parameters along with the hidden states can be estimated using the standard Kalman filter and Rauch-Tung-Striebel (RTS) smoother. Simulated data sets are considered to assess the proposed methodology. Finally, dRSM along with the corresponding VEM algorithm are applied to an original maritime network built from printed Lloyd's voyage records.

Due to the significant increase of communications between individuals via social media (Facebook, Twitter, Linkedin) or electronic formats (email, web, e-publication) in the past two decades, network analysis has become a unavoidable discipline. Many random graph models have been proposed to extract information from networks based on person-to-person links only, without taking into account information on the contents. This paper introduces the stochastic topic block model (STBM), a probabilistic model for networks with textual edges. We address here the problem of discovering meaningful clusters of vertices that are coherent from both the network interactions and the text contents. A classification variational expectation-maximization (C-VEM) algorithm is proposed to perform inference. Simulated data sets are considered in order to assess the proposed approach and to highlight its main features. Finally, we demonstrate the effectiveness of our methodology on two real-word data sets: a directed communication network and a undirected co-authorship network., Comment: in Statistics and Computing, 2016

International audience; The Cuban contact-tracing detection system set up in 1986 allowed the reconstruction and analysis of the sexual network underlying the epidemic (5,389 vertices and 4,073 edges, giant component of 2,386 nodes and 3,168 edges), shedding light onto the spread of HIV and the role of contact-tracing. Clustering based on modularity optimization provides a better visualization and understanding of the network, in combination with the study of covariates. The graph has a globally low but heterogeneous density, with clusters of high intraconnectivity but low interconnectivity. Though descriptive, our results pave the way for incorporating structure when studying stochastic SIR epidemics spreading on social networks.

International audience; The MLGL R-package, standing for Multi-Layer Group-Lasso, implements a new procedure of variable selection in the context of redundancy between explanatory variables, which holds true with high dimensional data. A sparsity assumption is made that is, only a few variables are assumed to be relevant for predicting the response variable. In this context, the performance of classical Lasso-based approaches strongly deteriorates as the redundancy strengthens. The proposed approach combines variables aggregation and selection in order to improve interpretability and performance. First, a hierarchical clustering procedure provides at each level a partition of the variables into groups. Then, the set of groups of variables from the different levels of the hierarchy is given as input to group-Lasso, with weights adapted to the structure of the hierarchy. At this step, group-Lasso outputs sets of candidate groups of variables for each value of regularization parameter. The versatility offered by MLGL to choose groups at different levels of the hierarchy a priori induces a high computational complexity. MLGL however exploits the structure of the hierarchy and the weights used in group-Lasso to greatly reduce the final time cost. The final choice of the regularization parameter – and therefore the final choice of groups – is made by a multiple hierarchical testing procedure.

— In recent years, many random graph models have been proposed to extract information from networks. The principle is to look for com-munities or groups of vertices with homogenous connection profiles. Most of these models are suitable for static networks, that is to say, not taking into account the temporal dimension, but can handle different types of edges, whether binary or discrete. This work is motivated by the need of analysing an evolving network describing email communications between employees of the Enron compagny where social positions play an important role. Therefore, in this paper, we consider the random subgraph model (RSM) which was pro-posed recently to model networks through latent clusters built within known partitions. Using a state space model to characterize the cluster proportions, RSM is then extended in order to deal with dynamic networks. We call the latter the dynamic random subgraph model (dRSM). A variational expectation maximisation (VEM) algorithm is proposed to perform inference. We show that the variational approximations lead to a new state space model from which the parameters along with hidden states can be estimated using the standard Kalman filter and Rauch-Tung-Striebel (RTS) smoother. The me-thodology is finally applied to the Enron email dataset and allows to discover a early reaction of the partners and directors compared to the other employees regarding the coming scandal., Résumé. — Ces dernières années, de nombreux modèles de graphes aléatoires ont été proposés pour extraire des informations à partir de réseaux dans des domaines variés. Le principe de ces modèles consiste à chercher des groupes de nœuds ayant des profils de connexion homogènes. La plupart de ces modèles sont adaptés pour des réseaux statiques ayant des arêtes binaires ou discrètes mais sans prendre en compte la dimension temporelle. Ce travail est motivé par la nécessité d'analyser un réseau dynamique décrivant les communications électroniques (e-mail) entre les employés de l'entreprise Enron où les positions sociales jouent un rôle important. Nous proposons dans cet article une extension au cadre dynamique du modèle de graphe aléatoire RSM qui a été récemment proposé pour modéliser à l'aide de groupes latents des réseaux statiques pour lesquels une partition en sous-graphes est connue. Notre approche est basée sur l'utilisation d'un state-space model pour modéliser l'évolution au cours du temps des proportions des groupes latents. Le modèle ainsi obtenu est appelé modèle de sous-graphes aléatoires dynamiques (dRSM) et un algorithme de type EM variationnel (VEM) est proposé pour en effectuer l'inférence. Nous montrons que les approximations variationnelles conduisent à un nouveau state-space model à partir duquel les paramètres ainsi que les états cachés peuvent être estimés en utilisant le filtre de Kalman et le Rauch-Tung-Striebel (RTS) smoother. La méthodologie est finalement appliquée au jeu des données d'e-mails de l'entreprise Enron et permet de mettre en évidence une réaction anticipée des cadres par rapport aux autres employés concernant le scandale à venir.

We study average-case complexity of branch-and-bound for maximum independent set in random graphs under the $\mathcal{G}(n,p)$ distribution. In this model every pair $(u,v)$ of vertices belongs to $E$ with probability $p$ independently on the existence of any other edge. We make a precise case analysis, providing phase transitions between subexponential and exponential complexities depending on the probability $p$ of the random model.

International audience; We address the problem of Bayesian variable selection for high-dimensional lin-ear regression. We consider a generative model that uses a spike-and-slab-like prior distribution obtained by multiplying a deterministic binary vector, which traduces the sparsity of the problem, with a random Gaussian parameter vector. The origi-nality of the work is to consider inference through relaxing the model and using a type-II log-likelihood maximization based on an EM algorithm. Model selection is performed afterwards relying on Occam's razor and on a path of models found by the EM algorithm. Numerical comparisons between our method, called spinyReg, and state-of-the-art high-dimensional variable selection algorithms (such as lasso, adap-tive lasso, stability selection or spike-and-slab procedures) are reported. Competitive variable selection results and predictive performances are achieved on both simulated and real benchmark data sets. An original regression data set involving the predic-tion of the number of visitors of the Orsay museum in Paris using bike-sharing system data is also introduced, illustrating the efficiency of the proposed approach. An R package implementing the spinyReg method is currently under development and is available at https://r-forge.r-project.org/projects/spinyreg.

Proceedings of the 23-th European Symposium on Artificial Neural Networks, Computational Intelligence and Machine Learning (ESANN 2015); International audience; In recent years, many random graph models have been proposed to extract information from networks. The principle is to look for groups of vertices with homogenous connection profiles. Most of these models are suitable for static networks and can handle different types of edges. This work is motivated by the need of analyzing an evolving network describing email communications between employees of the Enron compagny where social positions play an important role. Therefore, in this paper, we consider the random subgraph model (RSM) which was proposed recently to model networks through latent clusters built within known partitions. Using a state space model to characterize the cluster proportions, RSM is then extended in order to deal with dynamic networks. We call the latter the dynamic random subgraph model (dRSM).

Objective To analyze the effect of a fertilin-derived peptide (cFEE) on in vitro maturation of human oocytes. Design Randomized study. Setting Fertility center in an academic hospital. Patients Not applicable. Intervention Human immature germinal vesicle-stage oocytes (n=1629), donated for research according to French bioethics laws were randomly allocated to groups treated with 1 μM, 100 μM cFEE or to a control group. They were incubated at 37° C in 6% CO2 and 5% O2 and their maturation was assessed using time-lapse microscopy over 24 hours. In vitro-maturated metaphase II oocytes were analyzed for chromosomal content using microarray-comparative genomic hybridization (CGH-array) and their transcriptomes were analyzed using Affymetrix Clariom D microarrays. Main outcome measures The percentage of oocytes undergoing maturation in vitro was observed. Aneuploidy and euploidy were assessed for all chromosomes, and differential gene expression was analyzed in oocytes treated with cFEE compared to the control to obtain insights into its mechanism of action. Results cFEE significantly increased the percentage of oocytes that matured in vitro and improved euploidy in meiosis II oocytes by upregulation of FMN1 and FLNA genes, both of which encode proteins involved in spindle structure. Conclusion cFEE improves human oocyte maturation in vitro and reduces aneuploidy. It might prove useful for treating oocytes prior to fertilization in Assisted Reproduction Technology and for in vitro maturation in fertility preservation programs to improve oocyte quality and the chances for infertile couples to conceive.

We introduce the information geometry module of the Python package Geomstats. The module first implements Fisher-Rao Riemannian manifolds of widely used parametric families of probability distributions, such as normal, gamma, beta, Dirichlet distributions, and more. The module further gives the Fisher-Rao Riemannian geometry of any parametric family of distributions of interest, given a parameterized probability density function as input. The implemented Riemannian geometry tools allow users to compare, average, interpolate between distributions inside a given family. Importantly, such capabilities open the door to statistics and machine learning on probability distributions. We present the object-oriented implementation of the module along with illustrative examples and show how it can be used to perform learning on manifolds of parametric probability distributions.

International audience; In this paper we address an important economic question. Is there, as mainstream economic theory asserts it, an homogeneous labor market with mechanisms which govern supply and demand for work, producing an equilibrium with its remarkable properties? Using the Panel Study of Income Dynamics (PSID) collected on the period 1984-2003, we study the situations of American workers with respect to employment. The data include all heads of household (men or women) as well as the partners who are on the labor market, working or not. They are extracted from the complete survey and we compute a few relevant features which characterize the worker's situations. To perform this analysis, we suggest using a Self-Organizing Map (SOM, Kohonen algorithm) with specific structure based on planar graphs, with disconnected components (called D-SOM), especially interesting for clustering. We compare the results to those obtained with a classical SOM grid and a star-shaped map (called SOS). Each component of D-SOM takes the form of a string and corresponds to an organized cluster. From this clustering, we study the trajectories of the individuals among the classes by using the transition probability matrices for each period and the corresponding stationary distributions. As a matter of fact, we find clear evidence of heterogeneous parts, each one with high homo-geneity, representing situations well identified in terms of activity and wage levels and in degree of stability in the workplace. These results and their interpretation in economic terms contribute to the debate about flexibility which is commonly seen as a way to obtain a better level of equilibrium on the labor market.

National audience; Ces dernières années, de nombreux modèles de graphes aléatoires ont été proposé pour extraire des informations à partir de réseaux. Le principe consiste à chercher des groupes de sommets ayant des profils de connexions homogènes. La plupart de ces modèles sont adaptés pour les réseaux statiques, c'est-à-dire ne prenant pas en compte la dimension temporelle, mais pouvant gérer différents types d'arêtes, qu'elles soient binaires ou discrètes. Les modèles existant doivent donc être adaptés pour être capables de gérer l'aspect temporel. Nous nous intéressons ici au modèle de sous-graphes aléatoires proposé par [1]. Ce modèle permet de modéliser les sous-graphes connus d'un réseau à l'aide de structures latentes cachées. Nous proposons d'intégrer un processus à espace d'états à ce modèle afin d'autoriser les classes, ainsi que la topologie du réseau, à évaluer au cours du temps. L'inférence est réalisée par l'intermédiaire d'un algorithme Bayes variationnel. Un critère de sélection de modèles est également dérivé. Nous appliquons cette méthodologie à un réseau décrivant les relations entre évêques en Gaule Mérovingienne. Mots-clés. Réseau, sous-graphe, classification, algorithme VBEM. 1 Contexte de notre étude Ce travail a été motivé par la nécessité d'analyser un réseau historique évoluant dans le temps, pour lequel une partition des sommets est donnée et dont les arêtes sont de type catégorielle. L'analyse porte sur les relations interindividuelles tissées dans le cadre des conciles réunis en Gaule au VIe siècle. Le concile est une assemblée d'ecclésiastiques réunie sous l'autorité d'un évêque pour débattre des affaires ecclésiastiques. Pour le VIe siècle, 46 conciles furent réunis en Gaule, qui est alors, au gré des partages successoraux, divisée entre plusieurs royaumes : Neustrie, Austrasie et Bourgogne, la Bourgogne et l'Aquitaine changeant plusieurs fois de mains au cours du VIe siècle. La base de données regroupe 1331 personnages ayant occupé une ou plusieurs fonctions en Gaule entre 480 et 614 et les individus pour lesquels nous savons qu'ils ont eu des liens relationnels avec d'autres que ce soient des liens de parenté ou de connaissance. D'un point de vue historique, l'analyse et la comparaison de la structure organisationnelle de chacun des royaumes ou provinces pourrait fournir des éléments clés de compréhension de cette période.

In the last two decades many random graph models have been proposed to extract knowledge from networks. Most of them look for communities or, more generally, clusters of vertices with homogeneous connection profiles. While the first models focused on networks with binary edges only, extensions now allow to deal with valued networks. Recently, new models were also introduced in order to characterize connection patterns in networks through mixed memberships. This work was motivated by the need of analyzing a historical network where a partition of the vertices is given and where edges are typed. A known partition is seen as a decomposition of a network into subgraphs that we propose to model using a stochastic model with unknown latent clusters. Each subgraph has its own mixing vector and sees its vertices associated to the clusters. The vertices then connect with a probability depending on the subgraphs only, while the types of edges are assumed to be sampled from the latent clusters. A variational Bayes expectation-maximization algorithm is proposed for inference as well as a model selection criterion for the estimation of the cluster number. Experiments are carried out on simulated data to assess the approach. The proposed methodology is then applied to an ecclesiastical network in Merovingian Gaul. An R code, called Rambo, implementing the inference algorithm is available from the authors upon request., Comment: Published in at http://dx.doi.org/10.1214/13-AOAS691 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

L’analyse fractale des réseaux hydrographiques a donné lieu à de nombreux travaux (Tarboton et al., 1988 ; Rodriguez-Iturbe et Rinaldo, 1997 ; Hauchard et al., 1999 ; Forriez et al., 2010), alors même que les valeurs du principal indicateur fractal – la dimension fractale – n’ont que rarement alimenté des comparaisons selon leur mode d’obtention ou des discussions sur l’incertitude qui les caractérise. Les réseaux hydrographiques sont un cas particulier, relativement simple, de la grande famille des réseaux de transport qui de Hagget et Chorley (1969) à Strano et al. (2012) ont suscité réflexion théorique et applications en géographie prospective et aménagement (Dupuy, 1991 ; Frankhauser et Genre-Grandpierre, 1998 ; Porta et al., 2006). Nous nous intéressons ici aux réseaux vectoriels, typiquement des réseaux construits comme assemblage déterministe ou aléatoire de segments unitaires (fig. 2) ou des réseaux extraits à partir de modèles numériques de terrain de type raster avec une taille de cellule fixée (fig. 4, fig. 7), au-dessous de laquelle il n’existe plus de données informatives. Nous nous focalisons sur les méthodes d’estimation de dimensions fractales, qui soient, autant que faire se peut, dans le droit fil des méthodes mathématiques de calcul d’une dimension asymptotique et privilégions une analyse de type monofractal.Partant, d’une part, de la distinction entre fractale mathématique infinie et fractale de la nature, d’autre part, du rappel des différentes dimensions théoriques (asymptotiques) auxquelles se rattachent les dimensions empiriques que nous utilisons, nous mettons ici l’accent sur trois résultats méthodologiques. Le premier, l’apport des réseaux simulés – dont le mode de construction est connu –, permet d’apprécier la pertinence des divers estimateurs selon les caractéristiques des réseaux (leur ramification, leur degré de hiérarchisation, par exemple) et, au total, autorise une mise en garde vis-à-vis de certaines procédures très répandues – lorsqu’elles sont appliquées à de petits réseaux notamment. Le second concerne la détermination nécessaire à nos yeux d’un domaine de fractalité enserré dans des limites obtenues selon une procédure non subjective, et entre lesquelles est calculée la valeur de l’estimateur fractal (fig. 8 et fig. 11) ; cette procédure exigeante peut conduire à une réduction conséquente de l’intervalle de validité mais est utile pour des comparaisons. Enfin, l’importance de la notion d’instabilité des résultats selon le niveau hiérarchique du réseau (fig. 13, fig. 14, tab. 1), distincte de l’incertitude qui, pour les réseaux aléatoires, a pu être évaluée pour chaque niveau hiérarchique en répétant les mesures un grand nombre de fois (50 et 100 réalisations – réseau de Scheidegger et réseau binaire bruité, respectivement ; tab. 1) est mise en valeur.Les trois estimateurs de dimension fractale que nous comparons sont, l’un, un estimateur traditionnel en hydrographie (DHS) car fondé sur la topologie du réseau, plus particulièrement sur les rapports de Horton-Strahler (Horton, 1945 ; Strahler, 1957) ; les autres, la dimension obtenue par comptage de boîtes (DB) et la dimension de corrélation (DC). Le cas d’un réseau de talwegs extrait à partir de MNT semble montrer que les résultats d’estimation les plus stables sont obtenus à partir de cette dernière méthode (tab. 3). Streams networks are part of transport networks and more generally of “spatial networks” that gave rise to fundamental researches (Barthelemy, 2011) and applied geography (Strano et al., 2012). Numerous studies have focused on fractal analysis of stream networks. However, only few papers compare or discuss the estimation methods and the uncertainty of the main fractal indicator, the fractal dimension. This work focuses on the fractal properties of vector networks both virtual and actual. We first mention the essential distinction between infinite mathematical fractal and nature fractal. Then, we present different theoretical and empirical dimensions that we use. In particular, we compare three fractal dimension estimators: the most classical estimator for stream networks, based on a topological approach with the Horton-Strahler ratios, and two other estimators based on a geometric approach, the box-counting dimension and the correlation dimension. Three main methodological results can be highlighted: 1- the study of virtual network contributes to the assessment of the various estimators’ relevance, according to the characteristics of networks; 2- an empirical fractal domain must be determined with an objective method to estimate fractal dimensions that can be compared; 3- the observation of uncertainty and stability of the fractal dimension is necessary for any valid comparison.

Networks are a commonly used mathematical model to describe the rich set of interactions between objects of interest. Many clustering methods have been developed in order to partition such structures, among which several rely on underlying probabilistic models, typically mixture models. The relevant hidden structure may however show overlapping groups in several applications. The Overlapping Stochastic Block Model (2011) has been developed to take this phenomenon into account. Nevertheless, the problem of the choice of the number of classes in the inference step is still open. To tackle this issue, we consider the proposed model in a Bayesian framework and develop a new criterion based on a non asymptotic approximation of the marginal log-likelihood. We describe how the criterion can be computed through a variational Bayes EM algorithm, and demonstrate its efficiency by running it on both simulated and real data., Comment: article

Cette étude visait à quantifier le pouvoir prédictif du métabolome sur des phénotypes de production dans la filière porcine. Des données métabolomiques ont été obtenues par Résonance Magnétique Nucléaire à partir d’une prise de sang réalisée pendant la période de croissance de l’animal. Selon le type de phénotype considéré, un spectre métabolomique peut être envisagé comme une source de biomarqueurs intéressants. Ainsi, le taux de muscle des pièces de la carcasse a pu être correctement prédit à partir des mesures prises sur des prélèvements réalisés à environ 60 kg., This study was aimed at quantifying the predictive power of metabolomic data on production phenotypes in the pig. Metabolomic data were obtained by Nuclear Magnetic Resonance from one blood sample during the growing period of the animals. Depending on the phenotype, a metabolomic spectrum can be envisioned as a source of suitable biomarkers. In particular, the lean meat percentage was correctly predicted from samples taken at around 60 kg

We emphasize that it is possible to improve the principle of unbiased risk estimation for model selection by addressing excess risk deviations in the design of penalization procedures. Indeed, we propose a modification of Akaike's Information Criterion that avoids overfitting, even when the sample size is small. We call this correction an over-penalization procedure. As proof of concept, we show the nonasymptotic optimality of our histogram selection procedure in density estimation by establishing sharp oracle inequalities for the Kullback-Leibler divergence. One of the main features of our theoretical results is that they include the estimation of unbounded logdensities. To do so, we prove several analytical and probabilistic lemmas that are of independent interest. In an experimental study, we also demonstrate state-of-the-art performance of our over-penalization criterion for bin size selection, in particular outperforming AICc procedure., This is a further version of the preprint entitled "Model Selection as a Multiple Testing Procedure: Improving Akaike's Information Criterion"

We study the behaviour of a Schelling-class system in which a fraction $f$ of spatially-fixed switching agents is introduced. This new model allows for multiple interpretations, including: (i) random, non-preferential allocation (\textit{e.g.} by housing associations) of given, fixed sites in an open residential system, and (ii) superimposition of social and spatial mobility in a closed residential system.\\ We find that the presence of switching agents in a segregative Schelling-type dynamics can lead to the emergence of intermediate patterns (\textit{e.g.} mixture of patches, fuzzy interfaces) as the ones described in Ref. 1. We also investigate different transitions between segregated and mixed phases both at $f=0$ and along lines of increasing $f$, where the nature of the transition changes., Comment: 7 pages, 6 figures, to appear in EPJB

Weight control diets favorably affect parameters of the metabolic syndrome and delay the onset of diabetic complications. The adaptations occurring in adipose tissue (AT) are likely to have a profound impact on the whole body response as AT is a key target of dietary intervention. Identification of environmental and individual factors controlling AT adaptation is therefore essential. Here, expression of 271 transcripts, selected for regulation according to obesity and weight changes, was determined in 515 individuals before, after 8-week low-calorie diet-induced weight loss, and after 26-week ad libitum weight maintenance diets. For 175 genes, opposite regulation was observed during calorie restriction and weight maintenance phases, independently of variations in body weight. Metabolism and immunity genes showed inverse profiles. During the dietary intervention, network-based analyses revealed strong interconnection between expression of genes involved in de novo lipogenesis and components of the metabolic syndrome. Sex had a marked influence on AT expression of 88 transcripts, which persisted during the entire dietary intervention and after control for fat mass. In women, the influence of body mass index on expression of a subset of genes persisted during the dietary intervention. Twenty-two genes revealed a metabolic syndrome signature common to men and women. Genetic control of AT gene expression by cis signals was observed for 46 genes. Dietary intervention, sex, and cis genetic variants independently controlled AT gene expression. These analyses help understanding the relative importance of environmental and individual factors that control the expression of human AT genes and therefore may foster strategies aimed at improving AT function in metabolic diseases., Author Summary In obesity, an excess of adipose tissue is associated with dyslipidemia and diabetic complications. Gene expression is under the control of various genetic and environmental factors. As a central organ for the control of metabolic disturbances in conditions of both weight gain and loss, a comprehensive understanding of the control of adipose tissue gene expression is of paramount interest. We analyzed adipose tissue gene expression in obese individuals from the DiOGenes protocol, one of the largest dietary interventions worldwide. We found evidence for composite control of adipose tissue gene expression by nutrition, metabolic syndrome, body mass index, sex, and genotype with two main novel features. First, we observed a preeminent effect of sex on adipose tissue gene expression, which was independent of nutritional status, fat mass, and sex chromosomes. Second, the control of gene expression by cis genetic factors was unaffected by sex and nutritional status. Altogether, the effects of the investigated factors were most often independent of each other. Comprehension of the relative importance of environmental and individual factors that control the expression of human adipose tissue genes may help deciphering strategies aimed at controlling adipose tissue function during metabolic disorders.

The issue of detecting abnormal vibrations from spectra is addressed in this article, when little is known both on the mechanical behavior of the system, and on the characteristic patterns of potential faults. With vibration measured from a bearing test rig and from an aircraft engine, we show that when only a small learning set is available, probabilistic approaches have several advantages, including modelling healthy vibrations, and thus ensuring fault detection. To do so, we compare two original algorithms: the first one relies on the statistics of the maximum of log-periodograms. The second one computes the probability density function (pdf) of the wavelet transform of log-periodograms, and a likelihood index when new periodograms are presented. A by-product of it is the ability to generate random log-periodograms according with respect to the learning dataset. Receiver Operator Characteristic (ROC) curves are built in several experimental settings, and show the superiority of one of our algorithms over state-of-the-art machine-learning-oriented fault detection methods; lastly we generate random samples of aircraft engine log-periodograms.

Malaria remains endemic in tropical areas, especially in Africa. For the evaluation of new tools and to further ourunderstanding of host-parasite interactions, knowing the environmental risk of transmission-even at a very local scale-isessential. The aim of this study was to assess how malaria transmission is influenced and can be predicted by local climaticand environmental factors. As the entomological part of a cohort study of 650 newborn babies in nine villages in the ToriBossito district of Southern Benin between June 2007 and February 2010, human landing catches were performed to assessthe density of malaria vectors and transmission intensity. Climatic factors as well as household characteristics were recordedthroughout the study. Statistical correlations between Anopheles density and environmental and climatic factors weretested using a three-level Poisson mixed regression model. The results showed both temporal variations in vector density(related to season and rainfall), and spatial variations at the level of both village and house. These spatial variations could belargely explained by factors associated with the house's immediate surroundings, namely soil type, vegetation index andthe proximity of a watercourse. Based on these results, a predictive regression model was developed using a leave-one-outmethod, to predict the spatiotemporal variability of malaria transmission in the nine villages. This study points up theimportance of local environmental factors in malaria transmission and describes a model to predict the transmission risk ofindividual children, based on environmental and behavioral characteristics., Comment: PLoS ONE, Public Library of Science, 2012

International audience; Using the Panel Study of Income Dynamics (PSID) collected on the period 1984-2003, we study the situations of American workers with respect to employment. The data include all heads of household (men or women) as well as the partners who are on the labor market, working or not. They are extracted from the complete survey by computing a few relevant features which characterize the worker's situations. To perform this analysis, we suggest to use a Self-Organizing Map (Kohonen algorithm) with specific topologies. Self Organizing Maps (SOM) have been successfully applied in a lot of real world hard problems since their apparition. In this paper we present a new topology for SOM based on a planar graph with disconnected components (called D-SOM) which is especially interesting for clustering. Each component takes the form of a string and corresponds to an organized cluster. This algorithm operates on elementary observations that are pairs (year, individual). It allows us to group the data into some clusters. Each cluster can be described by its mean features (good situation, precarious, no job at all, etc.) as well as the personal descriptors of the individuals it contains. From this clustering, we study the dynamics at the individual level, that is the trajectories of the individuals among the classes during the observed period. Then we estimate the transition probability matrices for each year and the corresponding stationary distributions. Finally, we try to give an answer to the question: is there a significant change in 1992 (new economic policies after the "Reaganomics").

International audience; The issue of detecting abnormal vibrations is addressed in this article, when little is known both on the mechanical behavior of the system, and on the characteristic patterns of potential faults. With data from a bearing test rig and from an aircraft engine, we show that when only a small learning set is available, Bayesian inference has several advantages in order to compute a model of healthy vibrations, and thus ensure fault detection. To do so, we compute the wavelet transform of many log-periodograms, and show that their probability density can be easily modelled. This allows us to compute a likelihood index when a new log-periodogram is presented, thanks to marginal likelihood approximation. A by-product of this computation is the ability to generate random log-periodograms according to the learning dataset probability density. Finally, we first detect the degradation of a bearing on a test rig; then we generate random samples of aircraft engine log-periodograms.

International audience; Aircraft engines are designed to be used during several tens of years. Ensuring a proper operation of engines over their lifetime is therefore an important and difficult task. The maintenance can be improved if efficient procedures for the understanding of data flows produced by sensors for monitoring purposes are implemented. This paper details such a procedure aiming at visualizing in a meaningful way successive data measured on aircraft engines and finding for every possible request sequence of data measurement similar behaviour already observed in the past which may help to anticipate failures. The core of the procedure is based on Self-Organizing Maps (SOM) which are used to visualize the evolution of the data measured on the engines. Rough measurements can not be directly used as inputs, because they are influenced by external conditions. A preprocessing procedure is set up to extract meaningful information and remove uninteresting variations due to change of environmental conditions. The proposed procedure contains four main modules to tackle these difficulties: environmental conditions normalization (ECN), change detection and adaptive signal modeling (CD), visualization with Self-Organizing Maps (SOM) and finally minimal Edit Distance search (SEARCH). The architecture of the procedure and of its modules is described in this paper and results on real data are also supplied.

International audience; When regression analysis is carried out in a prediction purpose, an evolution in the modeled phenomenon between the training and the prediction stages obliges the practitioner to perform a new and complete analysis. Similarly, when regression aims to explain the modeled phenomenon, a new regression model must be estimated whenever the phenomenon or its study conditions change. This paper shows how a previous regression analysis can be used for the estimation of the regression model in a new situation avoiding a new and expensive collect of data. Two case studies are considered in the paper. On the one hand, a regression model of the house price versus house and household features is adapted from a city of the US South-East (Birmingham, AL) to a city of the US West coast (San Jose, CA). On the other hand, the link between CO$ emissions and gross national product in 1999 is analyzed based on a previous analysis dating from 1980.

We study the convergence of $N-$particle systems described by SDEs driven by Brownian motion and Poisson random measure, where the coefficients depend on the empirical measure of the system. Every particle jumps with a jump rate depending on its position and on the empirical measure of the system. Jumps are simultaneous, that is, at each jump time, all particles of the system are affected by this jump and receive a random jump height that is centred and scaled in $N^{-1/2}.$ This particular scaling implies that the limit of the empirical measures of the system is random, describing the conditional distribution of one particle in the limit system. We call such limits {\it conditional McKean-Vlasov limits}. The conditioning in the limit measure reflects the dependencies between coexisting particles in the limit system such that we are dealing with a {\it conditional propagation of chaos property}. As a consequence of the scaling in $N^{-1/2}$ and of the fact that the limit of the empirical measures is not deterministic the limit system turns out to be solution of a non-linear SDE, where not independent martingale measures and white noises appear having an intensity that depends on the conditional law of the process., 35 pages, 1 figure

International audience; Dimensionality reduction can be efficiently achieved by generative latent variable models such as probabilistic principal component analysis (PPCA) or independent component analysis (ICA), aiming to extract a reduced set of variables (latent variables) from the original ones. In most cases, the learning of these methods is achieved within the unsupervised framework where only unlabeled samples are used. In this paper we investigate the possibility of estimating independent factor analysis model (IFA) and thus projecting original data onto a lower dimensional space, when prior knowledge on the cluster membership of some training samples is incorporated. In the basic IFA model, latent variables are only recovered from their linear observed mixtures (original features). Both the mapping matrix (assumed to be linear) and the latent variable densities (that are assumed to be mutually independent and generated according to mixtures of Gaussians) are learned from observed data. We propose to learn this model within semisupervised framework where the likelihood of both labeled and unlabeled samples is maximized by a generalized expectation-maximization (GEM) algorithm. Experimental results on real data sets are provided to demonstrate the ability of our approach to find law dimensional manifold with good explanatory power.

International audience; Nous étudions les situations de chefs de famille américains vis-à-vis de l'emploi. Nous proposons pour cela une méthode de classification et de visualisation de données de grande dimension basée sur l'algorithme de Kohonen permettant l'apprentissage des cartes auto-organisatrices. Cette méthode que nous nommerons cartes auto-organisée classifiante nous permet d'étudier la segmentation du marché du travail américain en classes aux caractéristiques différentes et d'étudier la dynamique des trajectoires professionnelles des foyers appartenant au panel parmi ces classes.

Fractal dimension is an index which can be used to characterize urban areas. The use of the curve of scaling behaviour is less common. However, its shape gives local information about the morphology of the built-up area. This paper suggests a method based on a k-medoid for clustering these curves. It is applied to forty-nine wards of European cities, and shows that the curves add interesting intraward information to our knowledge of the spatial variation of the urban texture. Moreover, morphological similarities are observed between cities: living, architectural, and planning trends are not specific to individual cities.

We study the focusing stochastic nonlinear Schrodinger equation in one spatial dimension with multiplicative noise, driven by a Wiener process white in time and colored in space, in the $$L^2$$ -critical and supercritical cases. The mass ( $$L^2$$ -norm) is conserved due to the multiplicative noise defined via the Stratonovich integral, the energy (Hamiltonian) is not preserved. We first investigate both theoretically and numerically how the energy is affected by various spatially correlated random perturbations and its dependence on the discretization parameters and the schemes. We then perform numerical investigation of the noise influence on the global dynamics measuring the probability of blow-up versus scattering behavior depending on parameters of correlation kernels. Finally, we study numerically the effect of the spatially correlated noise on the blow-up behavior, and conclude that such random perturbations do not influence the blow-up dynamics, except for shifting of the blow-up center location. This is similar to what we observed for a space-time white driving noise in Millet et al. (Numerical study of solutions behavior to the 1d stochastic $$L^2$$ -critical and supercritical nonlinear Schrodinger equation, 2020. arXiv:2006.10695 ).

Count data is becoming more and more ubiquitous in a wide range of applications, with datasets growing both in size and in dimension. In this context, an increasing amount of work is dedicated to the construction of statistical models directly accounting for the discrete nature of the data. Moreover, it has been shown that integrating dimension reduction to clustering can drastically improve performance and stability. In this paper, we rely on the mixture of multinomial PCA, a mixture model for the clustering of count data, also known as the probabilistic clustering-projection model in the literature. Related to the latent Dirichlet allocation model, it offers the flexibility of topic modeling while being able to assign each observation to a unique cluster. We introduce a greedy clustering algorithm, where inference and clustering are jointly done by mixing a classification variational expectation maximization algorithm, with a branch & bound like strategy on a variational lower bound. An integrated classification likelihood criterion is derived for model selection, and a thorough study with numerical experiments is proposed to assess both the performance and robustness of the method. Finally, we illustrate the qualitative interest of the latter in a real-world application, for the clustering of anatomopathological medical reports, in partnership with expert practitioners from the Institut Curie hospital., 34 pages, 11 figures, published in : Computational Statistics

International audience; This paper presents a general methodology for nonparametric estimation of a function s related to a nonnegative real random variable X, under a constraint of type s(0) = c. Three dierent examples are investigated: the direct observations model (X is observed), the multiplicative noise model (Y = XU is observed, with U following a uniform distribution) and the additive noise model (Y = X + V is observed where V is a nonnegative nuisance variable with known density). When a projection estimator of the target function is available, we explain how to modify it in order to obtain an estimator which satises the constraint. We extend risk bounds from the initial to the new estimator. Moreover if the previous estimator is adaptive in the sense that a model selection procedure is available to perform the squared bias/variance trade-o, we propose a new penalty also leading to an oracle type inequality for the new constrained estimator. The procedure is illustrated on simulated data, for density and survival function estimation.

International audience; We consider a sequence of systems of Hawkes processes having mean field interactions in a diffusive regime. The stochastic intensity of each process is a solution of a stochastic differential equation driven by N independent Poisson random measures. We show that, as the number of interacting components N tends to infinity, this intensity converges in distribution in Skorohod space to a CIR-type diffusion. Moreover, we prove the convergence in distribution of the Hawkes processes to the limit point process having the limit diffusion as intensity. To prove the convergence results, we use analytical technics based on the convergence of the associated infinitesimal generators and Markovian semigroups.; Nous considérons des systèmes de processus de Hawkes, avec des interactions de type champ moyen, dans un régime diffusif. L'intensité stochastique de chaque processus est solution d'une équation différentielle stochastique dirigée par N mesures aléatoires de Poisson indépendantes. Nous montrons que, quand le nombre N de composants qui interagissent tend vers l'infini, cette intensité converge en loi dans l'espace de Skorokhod vers une diffusion de type CIR. De plus, nous démontrons la convergence en loi du système des processus de Hawkes vers un processus ponctuel ayant comme intensité stochastique la diffusion limite. Pour prouver cette convergence, nous utilisons des techniques analytiques basées sur la convergence des générateurs infinitésimaux et des semi-groupes associés à des processus de Markov.

We consider the stochastic system of interacting neurons introduced in De Masi et al. (2015) and in Fournier and L\"ocherbach (2016) and then further studied in Erny, L\"ocherbach and Loukianova (2021) in a diffusive scaling. The system consists of $N$ neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the potential of the spiking neuron is reset to $0$ and all other neurons receive an additional amount of potential which is a centred random variable of order $ 1 / \sqrt{N}.$ In between successive spikes, each neuron's potential follows a deterministic flow. In our previous article Erny, L\"ocherbach and Loukianova (2021) we proved the convergence of the system, as $N \to \infty$, to a limit nonlinear jumping stochastic differential equation. In the present article we complete this study by establishing a strong convergence result, stated with respect to an appropriate distance, with an explicit rate of convergence. The main technical ingredient of our proof is the coupling introduced in Koml\'os, Major and Tusn\'ady (1976) of the point process representing the small jumps of the particle system with the limit Brownian motion.

This paper develops a simple and computationally efficient parametric approach to the estimation of general hidden Markov models (HMMs). For non-Gaussian HMMs, the computation of the maximum likelihood estimator (MLE) involves a high-dimensional integral that has no analytical solution and can be difficult to approach accurately. We develop a new alternative method based on the theory of estimating functions and a deconvolution strategy. Our procedure requires the same assumptions as the MLE and deconvolution estimators. We provide theoretical guarantees about the performance of the resulting estimator; its consistency and asymptotic normality are established. This leads to the construction of confidence intervals. Monte Carlo experiments are investigated and compared with the MLE. Finally, we illustrate our approach using real data for ex-ante interest rate forecasts.

This paper studies the model selection problem in a large class of causal time series models, which includes both the ARMA or AR($\infty$) processes, as well as the GARCH or ARCH($\infty$), APARCH, ARMA-GARCH and many others processes. We first study the asymptotic behavior of the ideal penalty that minimizes the risk induced by a quasi-likelihood estimation among a finite family of models containing the true model. Then, we provide general conditions on the penalty term for obtaining the consistency and efficiency properties. We notably prove that consistent model selection criteria outperform classical AIC criterion in terms of efficiency. Finally, we derive from a Bayesian approach the usual BIC criterion, and by keeping all the second order terms of the Laplace approximation, a data-driven criterion denoted KC'. Monte-Carlo experiments exhibit the obtained asymptotic results and show that KC' criterion does better than the AIC and BIC ones in terms of consistency and efficiency.

International audience; When a continuous-time diffusion is observed only at discrete times with measurement noise, in most cases the transition density is not known and the likelihood is in the form of a high-dimensional integral that does not have a closed-form solution and is difficult to compute accurately. Using Hermite expansions and deconvolution strategy, we provide a general explicit sequence of closed-form contrast for noisy and discretely observed diffusion processes. This work allows the estimation of many diffusion processes. We show that the approximation is very accurate and prove that minimizing the sequence results in a consistent and asymptotically normal estimator. Monte Carlo evidence for the Ornstein-Uhlenbeck process reveals that this method works well and outperforms the Euler expansion of the transition density in situations relevant for financial models.

We establish differentiability properties of the value function of problems of Static Optimization in an abstract infinite dimensional setting and we apply that to problems of Calculus of Variations. We lighten the assumptions of existing results, notably by using G{\^a}teaux and Hadamard differentials. Moreover we use recently established Multipliers Rules.

Time series analysis is a very active research topic in Statistics and Data Science. Theabundance of this type of data has created a huge need for efficient and accurate methodologies.Thus, several families of models have emerged. Given this multitude of models,how do you choose one to model a time series? The purpose of this thesis is to proposeand study model selection criteria for a large family of models containing autoregressivetime series such as ARMA and conditionally heteroscedastic time series such as GARCH.We start with a brief presentation of time series, in particular of the family of causalaffine models, while recalling some previous results useful for this thesis. We then describesome classical model selection criteria obtained for time series and end with a brief summaryof our main contributions.The rest of our work presents our four contributions. The first chapter gives sufficientconditions on the penalty depending on the regularity of the dependence of the process onits past in order to obtain a consistent criterion. We also propose a goodness-of-fit test ofthe selected model based on the autocorrelation of the square of the model residuals. Thenumerical simulations have shown satisfactory results.In Chapter 3, we propose a generalization of the Hannan and Quinn criterion to theclass of causal affine series. This generalization induces a certain constant known for classicalmodels (ARMA, GARCH or APARCH type) and can be data-driven estimated forcomplex models like ARMA-GARCH. Here again, some simulation studies have attestedto the quality of the criteria obtained.In the third contribution, we construct asymptotically efficient criteria. We propose ageneralization of Akaike’s AIC criterion based on the so-called ideal penalty. The asymptoticbehavior of this ideal penalty suggested a penalty term which is exactly $2\,D_m$ asin the AIC for simple models, and for complex models, we give a less explicit formula.Following Schwartz, we also derive the BIC criterion based on the maximization of the aposteriori probability of choosing the true model.In Chapter 5, we restricted ourselves to the non-asymptotic study of a particular processof the class of causal affines. A penalized least-squares estimator is built on a datadriven selected model among a collection of linear models. We showed that the final estimatorperforms almost as well as the best over the considered collection, i.e. it achieves,up to a constant, the bias-variance tradeoff. The penalty obtained generalizes Mallows’penalty and depends on a constant that is estimated with data-driven calibration algorithms.Finally, we give some research directions in the General Conclusion of the work., L’analyse des séries temporelles est un sujet de recherche très actif en Statistique eten Data Science. L’abondance de ce type de données a créé d’énormes besoins de méthodologiesefficaces et précises. C’est ainsi que plusieurs familles de modèles ont vu lejour. Étant donnée cette multitude de modèles, comment en choisir un pour modéliserune série temporelle ? L’objet de cette thèse est de proposer et d’étudier des critères desélection de modèles pour une grande famille de modèles contenant les séries temporellesautorégressives telles les ARMA ainsi que les séries temporelles conditionnellement hétéroscédastiquestelles les GARCH.Nous commençons par une brève présentation des séries temporelles, en particulier dela famille des modèles affines causaux tout en rappelant quelques précédents résultatsutiles pour cette thèse. Nous décrivons ensuite quelques critères classiques de sélection demodèles obtenus pour les séries temporelles et terminons par un succinct résumé de nosprincipales contributions.Dans la suite, nous allons présenter les quatre contributions originales de cette thèse.Le chapitre 2 donne des conditions suffisantes sur la pénalité en fonction de la régularitéde la dépendance du processus par rapport à son passé afin d’obtenir un critère consistanten probabilité. Nous proposons également un test d’adéquation du modèle sélectionnébasé sur l’autocorrélation du carré des résidus du modèle. Les simulations numériques ontmontré des résultats satisfaisants.Au chapitre 3, nous proposons une généralisation du critère de Hannan et Quinn à laclasse des séries affines causales. Cette généralisation induit une certaine constante connuepour les modèles classiques (type ARMA, GARCH ou APARCH) et pour les modèlescomplexes tels les ARMA-GARCH, la constante est inconnue mais peut être estimée demanière adaptative via l’heuristique de pente. Là également, quelques études de simulationont attesté de la qualité des critères obtenus.Dans la troisième contribution, nous construisons des critères asymptotiquement efficients.Nous proposons une généralisation du critère AIC d’Akaike se basant sur la pénalité diteidéale. Le comportement asymptotique de cette pénalité idéale nous a suggéré un termede pénalité qui vaut exactement $2\,D_m$ comme dans l’AIC pour des modèles assez simples,et pour des modèles complexes, nous avons donné une formule moins explicite. A la suitede Schwartz, nous dérivons également le critère BIC qui s’appuie sur la maximisation dela probabilité a posteriori de choisir le vrai modèle.Au chapitre 5, nous nous sommes restreints à l’étude non asymptotique d’un processusparticulier de la classe des modèles affines causaux. Un estimateur des moindres carréspénalisé est construit à partir d’un critère de sélection adaptatif et la sélection est opéréeparmi une collection de modèles linéaires. Nous avons montré que l’estimateur final estpresque aussi performant que le meilleur sur la collection considérée, i.e. qu’il réalise, àune constante près, le compromis biais-variance. La pénalité obtenue généralise celle deMallows et dépend d’une constante que l’on pourrait estimer avec des algorithmes de calibrationadaptative.Enfin, nous donnons quelques pistes de recherche dans la Conclusion générale du travail.

High-dimensional data clustering has become and remains a challenging task for modern statistics and machine learning, with a wide range of applications. We consider in this work the powerful discriminative latent mixture model, and we extend it to the Bayesian framework. Modeling data as a mixture of Gaussians in a low-dimensional discriminative subspace, a Gaussian prior distribution is introduced over the latent group means and a family of twelve submodels are derived considering different covariance structures. Model inference is done with a variational EM algorithm, while the discriminative subspace is estimated via a Fisher-step maximizing an unsupervised Fisher criterion. An empirical Bayes procedure is proposed for the estimation of the prior hyper-parameters, and an integrated classification likelihood criterion is derived for selecting both the number of clusters and the submodel. The performances of the resulting Bayesian Fisher-EM algorithm are investigated in two thorough simulated scenarios, regarding both dimensionality as well as noise and assessing its superiority with respect to state-of-the-art Gaussian subspace clustering models. In addition to standard real data benchmarks, an application to single image denoising is proposed, displaying relevant results. This work comes with a reference implementation for the R software in the FisherEM package accompanying the paper., The FisherEM package is available on CRAN, see https://github.com/nicolasJouvin/FisherEM for additional information

We prove that an asymmetric normed space is never a Baire space if the topology induced by the asymmetric norm is not equivalent to the topology of a norm. More precisely, we show that a biBanach asymmetric normed space is a Baire space if and only if it is isomorphic to its associated normed space.

We introduce the notion of Hypergraph Weighted Model (HWM), a computational model that generically associates a tensor network to a graph or a hypergraph and then computes a value by generalized tensor contractions directed by its hyperedges. A series r defined on a hypergraph family is said to be recognizable if there exists a HWM that computes it. This model generalizes the notion of recognizable series on strings and trees. We present some examples on non classical graphs families such as circular strings and pictures and we study properties of the model such as closure properties and recognizability of finite support series. We conclude by a section exploring the learnability of HWMs defined over the family of circular strings.

In numerous tasks, deep networks are state of the art. However, they are still not well understood from a statistical point of view. In this article, we try to contribute to filling this gap, and we consider regression models involving deep multilayer perceptrons (MLP) with rectified linear (ReLU) functions for activation units. It is a difficult task to study the statistical properties of such models. The main reason is that in practice these models may be heavily overparameterized. For the sake of simplicity, we focus here on the sum of square errors (SSE) cost function which is the standard cost function for regression purposes. In this framework, we study the asymptotic behavior of the difference between the SSE of estimated models and the SSE of the theoretical best model. This behavior gives us information on the overfitting properties of such models. We use in this paper new methodology introduced to deal with models with a loss of identifiability, i.e. in the case that the true parameter cannot be identified uniquely. Hence, we don’t have to assume that a unique parameter vector realizes the best regression function which seems to be a too strong assumption for heavily overparameterized models. Our results shed new light on the overfitting behavior of MLP models.