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A graphical model is simply a representation of the results of an analysis of relationships between sets of variables. It can include the study of the dependence of one variable, or a set of variables on another variable or sets of variables, and can be extended to include variables which could be considered as intermediate to the others. This leads to the concept of representing these chains of relationships by means of a graph; where variables are represented by vertices, and relationships between the variables are represented by edges. These edges can be either directed or undirected, depending upon the type of relationship being represented. The thesis investigates a number of outstanding problems in the area of statistical modelling, with particular emphasis on representing the results in terms of a graph. The thesis will study models for multivariate discrete data and in the case of binary responses, some theoretical results are given on the relationship between two common models. In the more general setting of multivariate discrete responses, a general class of models is studied and an approximation to the maximum likelihood estimates in these models is proposed. This thesis also addresses the problem of measurement errors. An investigation into the effect that measurement error has on sample size calculations is given with respect to a general measurement error specification in both linear and binary regression models. Finally, the thesis presents, in terms of a graphical model, a re-analysis of a set of childhood growth data, collected in South Wales during the 1970s. Within this analysis, a new technique is proposed that allows the calculation of derived variables under the assumption that the joint relationships between the variables are constant at each of the time points.

Cette thèse s'inscrit dans le cadre de l'étude de réseaux de capteurs. L'objectif est de pouvoir comparer des réseaux en utilisant leurs structures d'indépendance conditionnelle. Cette structure représente les relations entre deux capteurs sachant l'information enregistrée par les autres capteurs du réseau. Nous travaillons sous l'hypothèse que les réseaux étudiés sont assimilables à des processus gaussiens multivariés. Sous cette hypothèse, estimer la structure d'indépendance conditionnelle d'un processus multivarié gaussien est équivalent à estimer son modèle graphique gaussien.Dans un premier temps, nous proposons une nouvelle méthode d'estimation de modèle graphique gaussien : elle utilise un score proportionnel à la probabilité d'un graphe de représenter la structure d'indépendance conditionnelle du processus étudié et est initialisée par Graphical lasso. Pour situer notre méthode par rapport aux méthodes existantes, nous avons développé une procédure d'évaluation des performances d'une méthode d'estimation de modèles graphiques gaussiens incluant notamment un algorithme permettant de générer des processus multivariés gaussiens dont la structure d'indépendance conditionnelle est connue.Dans un deuxième temps, nous classifions des processus à partir des estimées des structures d'indépendance conditionnelle de ces processus. Pour ce faire, nous introduisons comme métrique la divergence de Kullback-Leibler symétrisée entre les profils croisés normalisés des processus étudiés. Nous utilisons cette approche pour identifier des ensemble de régions cérébrales pertinentes pour l'étude de patients dans le coma à partir de données d'IRM fonctionnelle. / This thesis is motivated by the study of sensors networks. The goal is to compare networks using their conditional independence structures. This structure illustrates the relations between two sensors according to the information recorded by the others sensors in the network. We made the hypothesis that the studied networks are multivariate Gaussian processes. Under this assumption, estimating the conditional independence structure of a process is equivalent to estimate its Gaussian graphical model.First, we propose a new method for Gaussian graphical model estimation : it uses a score proportional to the probability of a graph to represent the conditional independence structure of the studied process and it is initialized by Graphical lasso. To compare our method to existing ones, we developed a procedure to evaluate the performances of Gaussian graphical models estimation methods. One part of this procedure is an algorithm to simulated multivariate Gaussian processes with known conditional independence structure.Then, we conduct a classification over processes thanks to their conditional independence structure estimates. To do so, we introduce a new metric : the symmetrized Kullback-Leibler divergence over normalized cross-profiles of studied processes. We use this approach to find sets of brain regions that are relevant to study comatose patients from functional MRI data.

This thesis is mainly concerned with the estimation of conditional probabilities in two-way contingency tables, that is probabilities of type P(R=i,S=j|X=x), for (i,j) in {1, . . . , r}×{1, . . . , s}, where R and S are the two categorical variables forming the contingency table, with r and s levels respectively, and X is a vector of explanatory variables possibly associated with R, S, or both. Analyzing such a conditional distribution is often of interest, as this allows to go further than the usual unconditional study of the behavior of the variables R and S. First, one can check an eventual effect of these covariates on the distribution of the individuals through the cells of the table, and second, one can carry out usual analyses of contingency tables, such as independence tests, taking into account, and removing in some sense, this effect. This helps for instance to identify the external factors which could be responsible for an eventual association between R and S. This also gives the possibility to adapt for a possible heterogeneity in the population of interest, when analyzing the table.

Automatic Root Cause Analysis (RCA) systems aim to streamline the process of identifying the underlying cause of software failures in complex cloud-native environments. These systems employ graph-like structures to represent causal relationships between different components of a software application. These relationships are typically learned through performance and resource utilization metrics of the microservices in the system. To accomplish this objective, numerous RCA systems utilize statistical algorithms, specifically those falling under the category of causal discovery. These algorithms have demonstrated their utility not only in RCA systems but also in a wide range of other domains and applications. Nonetheless, there exists a research gap in the exploration of the feasibility and efficacy of multivariate time series causal discovery algorithms for deriving causal graphs within a microservice framework. By harnessing metric time series data from Prometheus and applying these algorithms, we aim to shed light on their performance in a cloudnative environment. Furthermore, we have introduced an adaptation in the form of an ensemble causal discovery algorithm. Our experimentation with this ensemble approach, conducted on datasets with known causal relationships, unequivocally demonstrates its potential in enhancing the precision of detected causal connections. Notably, our ultimate objective was to ascertain reliable causal relationships within Ericsson’s cloud-native system ’X,’ where the ground truth is unavailable. The ensemble causal discovery approach triumphs over the limitations of employing individual causal discovery algorithms, significantly augmenting confidence in the unveiled causal relationships. As a practical illustration of the utility of the ensemble causal discovery techniques, we have delved into the domain of anomaly detection. By leveraging causal graphs within our study, we have successfully applied this technique to anomaly detection within the Ericsson system. / System för automatisk rotorsaksanalys (RCA) syftar till att effektivisera process för att identifiera den underliggande orsaken till programvarufel i komplexa molnbaserade miljöer. Dessa system använder grafliknande strukturer att representera orsakssamband mellan olika komponenter i en mjukvaruapplikation. Dessa relationer lär man sig vanligtvis genom prestanda och resursutnyttjande mätvärden för mikrotjänsterna i systemet. För att uppnå detta mål använder många RCAsystem statistiska algoritmer, särskilt de som faller under kategorin orsaksupptäckt. Dessa algoritmer har visat att de inte är användbara endast i RCA-system men även inom en lång rad andra domäner och applikationer. Icke desto mindre finns det en forskningslucka i utforskningen av genomförbarhet och effektivitet av orsaksupptäckt av multivariat tidsserie algoritmer för att härleda kausala grafer inom ett mikrotjänstramverk. Genom att utnyttja metriska tidsseriedata från Prometheus och tillämpa Dessa algoritmer strävar vi efter att belysa deras prestanda i ett moln- inhemsk miljö. Dessutom har vi infört en anpassning i formen av en ensemble kausal upptäcktsalgoritm. Vårt experiment med denna ensemblemetod, utförd på datauppsättningar med kända orsakssamband relationer, visar otvetydigt sin potential för att förbättra precisionen hos upptäckta orsakssamband. Särskilt vår ultimata Målet var att fastställa tillförlitliga orsakssamband inom Ericssons molnbaserade systemet ’X’, där grundsanningen inte är tillgänglig. De ensemble kausal discovery approach segrar över begränsningarna av att använda individuella kausala upptäcktsalgoritmer, avsevärt öka förtroendet för de avslöjade orsakssambanden. Som en praktisk illustration av nyttan av ensemblens kausal upptäcktstekniker har vi fördjupat oss i anomalidomänen upptäckt. Genom att utnyttja kausala grafer inom vår studie har vi framgångsrikt tillämpat denna teknik för att detektera anomali inom Ericsson system

This work focuses on statistical methods to understand how frequently rare events occur and what the magnitude of extreme values such as large losses is. It lies in a field called extreme value analysis whose scope is to provide support for scientific decision making when extreme observations are of particular importance such as in environmental applications, insurance and finance. In the univariate case, I propose new techniques to model tails of discrete distributions and illustrate them in an application on word frequency and multiple birth data. Suitably rescaled, the limiting tails of some discrete distributions are shown to converge to a discrete generalized Pareto distribution and generalized Zipf distribution respectively. In the multivariate high-dimensional case, I suggest modeling tail dependence between random variables by a graph such that its nodes correspond to the variables and shocks propagate through the edges. Relying on the ideas of graphical models, I prove that if the variables satisfy a new notion called asymptotic conditional independence, then the density of the joint distribution can be simplified and expressed in terms of lower dimensional functions. This generalizes the Hammersley- Clifford theorem and enables us to infer tail distributions from observations in reduced dimension. As an illustration, extreme river flows are modeled by a tree graphical model whose structure appears to recover almost exactly the actual river network. A fundamental concept when studying limiting tail distributions is regular variation. I propose a new notion in the multivariate case called one-component regular variation, of which Karamata's and the representation theorem, two important results in the univariate case, are generalizations. Eventually, I turn my attention to website visit data and fit a censored copula Gaussian graphical model allowing the visualization of users' behavior by a graph.