Using drones for cellular coverage enhancement is a recent technology that has shown a great potential in various practical scenarios. However, one of the main challenges that limits the performance of drone-enabled wireless networks is the limited flight time. In particular, due to the limited on-board battery size, the drone needs to frequently interrupt its operation and fly back to a charging station to recharge/replace its battery. In addition, the charging station might be responsible to recharge multiple drones. Given that the charging station has limited capacity, it can only serve a finite number of drones simultaneously. Hence, in order to accurately capture the influence of the battery limitation on the performance, it is required to analyze the dynamics of the time spent by the drones at the charging stations. In this thesis, we first use tools from queuing theory and stochastic geometry to study the influence of each of the charging stations limited capacity and spatial density on the performance of a drone-enabled wireless network. We then extend our work to rural areas where users are greatly impacted by low income, high cost of backhaul connectivity, and limited resources. Considering the limitation of the electricity supply scarcity in some rural regions, we investigate the possibility and performance enhancement of the deployment of renewable energy (RE) charging stations. We outline three practical scenarios, and use simulation results to demonstrate that RE charging stations can be a possible solution to address the limited on-board battery of UAVs in rural areas, specially when they can harvest and store enough energy.

<div>Model-based clustering, with its flexibility and solid statistical foundations, is an important tool for unsupervised learning, and has numerous applications in a variety of fields. This dissertation focuses on nonparametric Bayesian approaches to model-based clustering under structural restrictions. These are additional constraints on the model that embody prior knowledge, either to regularize the model structure to encourage interpretability and parsimony or to encourage statistical sharing through underlying tree or network structure.</div><div><br></div><div>The first part in the dissertation focuses on the most commonly used model-based clustering models, mixture models. Current approaches typically model the parameters of the mixture components as independent variables, which can lead to overfitting that produces poorly separated clusters, and can also be sensitive to model misspecification. To address this problem, we propose a novel Bayesian mixture model with the structural restriction being that the clusters repel each other.The repulsion is induced by the generalized Matérn type-III repulsive point process. We derive an efficient Markov chain Monte Carlo (MCMC) algorithm for posterior inference, and demonstrate its utility on a number of synthetic and real-world problems. <br></div><div><br></div><div>The second part of the dissertation focuses on clustering populations with a hierarchical dependency structure that can be described by a tree. A classic example of such problems, which is also the focus of our work, is the phylogenetic tree with nodes often representing biological species. The structure of this problem refers to the hierarchical structure of the populations. Clustering of the populations in this problem is equivalent to identify branches in the tree where the populations at the parent and child node have significantly different distributions. We construct a nonparametric Bayesian model based on hierarchical Pitman-Yor and Poisson processes to exploit this, and develop an efficient particle MCMC algorithm to address this problem. We illustrate the efficacy of our proposed approach on both synthetic and real-world problems.</div>

We consider random simplicial complexes constructed on a Poisson point process within a convex set in a Euclidean space, especially the Vietoris-Rips complex and the Cech complex both of whose 1-skeleton is the Gilbert graph. We investigate at first the Vietoris-Rips complex by considering the volume-power functionals defined by summing powers of the volume of all k-dimensional faces in the complex. The asymptotic behaviour of these functionals is investigated as the intensity of the underlying Poisson point process tends to infinity and the distance parameter goes to zero. This behaviour is observed in different regimes. Univariate and multivariate central limit theorems are proven, and analogous results for the Cech complex are then given. Finally we provide a Poisson limit theorem for the components of the f-vector in the sparse regime.

Les neurones interagissent à travers leur potentiel de membrane qui a en général une évolution temporelle complexe due aux nombreuses entrées synaptiques irrégulières reçues. Cette évolution est mieux décrite en termes probabilistes, en raison de ces entrées irrégulières ou «bruit synaptique». L'évolution temporelle du potentiel de membrane est stochastique mais aussi déterministe: stochastique, car conduite par des entrées synaptiques qui arrivent de façon aléatoire dans le temps, et déterministe, car un neurone biologique a une évolution temporelle très similaire quand soumis à une même séquence d'entrées synaptiques. Nous étudions les propriétés statistiques d'un modèle simplifié de neurone soumis à des entrées à taux variable d'où en résulte l'évolution non-stationnaire du potentiel de membrane. Nous considérons un modèle passif de membrane neuronale, sans mécanisme de décharge neuronale, soumis à des entrées à courant ou à conductance sous la forme d'un processus de «shot noise». Les fluctuations du potentiel de membrane sont aussi modélisées par un processus stochastique similaire, de «shot noise» filtré. Nous avons analysé les propriétés statistiques de ces processus dans le cadre des transformations de processus ponctuels de Poisson. Des propriétés de ces transformations sont dérivées les statistiques non-stationnaires du processus. Nous obtenons ainsi des expressions analytiques exactes pour les moments et cumulants du processus filtré dans le cas général des taux d'entrée variables. Ce travail ouvre de nombreuses perspectives pour l'analyse de neurones dans les conditions in vivo, en présence d'entrées synaptiques intenses et bruitées. / Neurons interact through their membrane potential that generally has a complex time evolution due to numerous irregular synaptic inputs received. This complex time evolution is best described in probabilistic terms due to this irregular or "noisy" activity. The time evolution of the membrane potential is therefore both stochastic and deterministic: it is stochastic since it is driven by random input arrival times, but also deterministic, since subjecting a biological neuron to the same sequence of input arrival times often results in very similar membrane potential traces. In this thesis, we investigated key statistical properties of a simplified neuron model under nonstationary input from other neurons that results in nonstationary evolution of membrane potential statistics. We considered a passive neuron model without spiking mechanism that is driven by input currents or conductances in the form of shot noise processes. Under such input, membrane potential fluctuations can be modeled as filtered shot noise currents or conductances. We analyzed the statistical properties of these filtered processes in the framework of Poisson Point Processes transformations. The key idea is to express filtered shot noise as a transformation of random input arrival times and to apply the properties of these transformations to derive its nonstationary statistics. Using this formalism we derive exact analytical expressions, and useful approximations, for the mean and joint cumulants of the filtered process in the general case of variable input rate. This work opens many perspectives for analyzing neurons under in vivo conditions, in the presence of intense and noisy synaptic inputs.

Extreme Value Theory nds application in problems concerning low probability but high consequence events. In hydrology the study of heavy rainfall is important in regional ood risk assessment. In particular, the N-year return level is a key output of an extreme value analysis, hence care needs to be taken to ensure that the model is accurate and that the level of imprecision in the parameter estimates is made explicit. Rainfall is a process that evolves over time and space. Therefore, it is anticipated that at extreme levels the process would continue to show temporal and spatial correlation. In this study interest is in whether any trends in heavy rainfall can be detected for the Western Cape. The focus is on obtaining the 50-year daily winter rainfall return level and investigating whether this quantity is homogenous over the study area. The study is carried out in two stages. In the rst stage, the point process approach to extreme value theory is applied to arrive at the return level estimates at each of the fteen sites. Stationarity is assumed for the series at each station, thus an issue to deal with is that of short-range temporal correlation of threshold exceedances. The proportion of exceedances is found to be smaller (approximately 0.01) for stations towards the east such as Jonkersberg, Plettenbergbay and Tygerhoek. This can be attributed to rainfall values being mostly low, with few instances where large amounts of rainfall were observed. Looking at the parameters of the point process extreme value model, the location parameter estimate appears stable over the region in contrast to the scale parameter estimate which shows an increase towards in a south easterly direction. While the model is shown to t exceedances at each station adequately, the degree of uncertainty is large for stations such as Tygerhoek, where the maximum observed rainfall value is approximately twice as large as the high rainfall values. This situation was also observed at other stations and in such cases removal of these high rainfall values was avoided to minimize the risk of obtaining inaccurate return level estimates. The key result is an N-year rainfall return level estimate at each site. Interest is in mapping an estimate of the 50-year daily winter rainfall return level, however to evaluate the adequacy of the model at each site the 25-year return level is considered since a 25 year return period is well within the range of the observed data. The 25-year daily winter rainfall return level estimate for Ladismith is the smallest at 22:42 mm. This can be attributed to the station's generally low observed winter rainfall values. In contrast, the return level estimate for Tygerhoek is high, almost six times larger than that of Ladismith at 119:16 mm. Visually design values show di erences between sites, therefore it is of interest to investigate whether these di erences can be modelled. The second stage is the geostatistical analysis of the 50-year 24-hour rainfall return level The aim here is to quantify the degree of spatial variation in the 50-year 24-hour rainfall return level estimates and to use that association to predict values at unobserved sites within the study region. A tool for quantifying spatial variation is the variogram model. Estimation of the parameters of this model require a su ciently large sample, which is a challenge in this study since there is only fteen stations and therefore only fteen observations for the geostatistical analysis. To address this challenge, observations are expanded in space and time and then standardized and to create a larger pool of data from which the variogram is estimated. The obtained estimates are used in ordinary and universal kriging to derive the 50-year 24-hour winter rainfall return level maps. It is shown that 50-year daily winter design rainfall over most of the Western Cape lies between 40 mm and 80 mm, but rises sharply as one moves towards the east coast of the region. This is largely due to the in uence of large design values obtained for Tygerhoek. In ordinary kriging prediction uncertainty is lowest around observed values and is large if the distance from these points increases. Overall, prediction uncertainty maps show that ordinary kriging performs better than universal kriging where a linear regional trend in design values is included.

Face à la croissance exponentielle des réseaux mobiles très haut débit, les opérateurs de téléphonie mobile se sont lancé dans le déploiement des réseaux dits hiérarchiques (HetNet), composés par des sous-réseaux avec des caractéristiques divergentes en termes de type des cellules déployées et des technologies d’accès radio utilisées. Avec ce caractère hétérogène des réseaux cellulaire, l’exploitation de ces derniers devienne de plus en plus compliquée et coûteuse impliquant le déploiement, la configuration et la reconfiguration de stations de base et d’équipements de différentes caractéristiques. Ainsi, l’intégration dans les réseaux HetNet de fonctionnalités d’auto-configuration automatisant et simplifiant l’exploitation des réseaux deviennent une demande forte des opérateurs. Cette thèse a pour objectif l’étude et le développement de solutions de gestion dynamique de l’équilibrage de charges entre les différentes couches composant un même HetNet, pour une expérience utilisateur (QoE) améliorée. Dans ce contexte, une classe des algorithmes d’équilibrage de charges dite ‘équilibrage de charges par adaptation dynamique des paramètres de la procédure de handover’ est étudiée. Pour commencer, nous développons un modèle théorique basé sur des solutions et des outils de la géométrie stochastique et incorporant le caractère hétérogène des réseaux cellulaires. Ensuite nous exploitons ce modèle pour introduire des algorithmes d’adaptation des paramètres de handover basés sur la maximisation de la puissance reçue et du rapport signal/brouillage plus bruit (SINR). Nous exploitons ces résultats pour implémenter et étudier, par simulation à évènements discrets, des algorithmes d’équilibrage de charges dans le contexte des réseaux LTE HetNet auto-organisés basés sur les spécifications 3GPP. Ces travaux soulignent l’importance de l’équilibrage de charges afin de booster les performances des réseaux cellulaires en termes de débit global transmis, perte de paquets de données et utilisation optimisée des ressources radio. / High demands on mobile networks provide a fresh opportunity to migrate towardsmulti-tier deployments, denoted as heterogeneous network (HetNet), involving a mix of cell types and radio access technologies working together seamlessly. In this context, network optimisation functionalities such as load balancing have to be properly engineered so that HetNet benefit are fully exploited. This dissertation aims to develop tractable frameworks to model and analyze load balancing dynamics while incorporating the heterogeneous nature of cellular networks. In this context we investigate and analyze a class of load balancingstrategies, namely adaptive handover based load balancing strategies. These latter were firstly studied under the general heading of stochastic networks using independent and homogeneous Poisson point processes based network model. We propose a baseline model to characterize rate coverage and handover signalling in K-tier HetNet with a general maximum power based cell association and adaptive handover strategies. Tiers differ in terms of deployment density and cells characteristics (i.e. transmit power, bandwidth, and path loss exponent). One of the main outcomes is demonstrating the impact of offloading traffic from macro- to small-tier. This impact was studied in terms of rate coverage and HO signalling. Results show that enhancement in rate coverage is penalized by HO signalling overhead. Then appropriate algorithms of LB based adaptive HO are designed and their performance is evaluated by means of extensive system level simulations. These latter are conducted in 3GPP defined scenarios, including representation of mobility procedures in both connectedstate. Simulation results show that the proposed LB algorithms ensure performance enhancement in terms of network throughput, packet loss ratio, fairness and HO signalling.

La thèse est consacrée à l'étude de modèles bivariés en Fabilité, qui tiennent compte de différents types de dépendance entre composants. Dans un premier temps, nous nous intéressons au cas d'un système formé de deux composants, dont la dégradation est modélisée par un processus de Lévy croissant bivarié (subordinateur bivarié). Sous cette hypothèse, eux études sont faites : l'une sous l'hypothèse de surveillance continue et de réparation parfaite du système, l'autre sous une hypothèse d'inspections périodiques et de réparation imparfaite. Dans un deuxième temps, la thèse est consacrée à un autre modèle de survie bivarié, sous influence d'un environnement stochastique stressant ponctuel. La dépendance entre composants est ici induite par un environnement stressant commun, qui induit des détériorations différentes sur chacun des composants (augmentation du taux de panne pour l'un, du niveau de détérioration pour l'autre). Pour chacun des modèles étudiés, nos résultats montrent l'importance de la prise en compte de la dépendance entre les composants d'un système. / The thesis is devoted to the study of bivariate models in reliability, which take into account several types of dependence between components. As a first step, we are interested in a two-component system with accumulating deterioration modeled by a bivariate increasing Lévy process (bivariate subordinator). Under this hypothesis, two different studies are made : one under the assumption of continuous monitoring and perfect repair, the other one under the assumption of periodic inspections and imperfect repair. In a second step, the thesis is devoted to the study of another bivariate survivalmodel, under the influence of a stochastic and stressful environment. The dependence between components is here induced by the common stressful environment, with different incidence on the two components (increment of failure rate for one, of deterioration level for the other). For each of the studied models, our results show the importance of taking into account the dependence between the components of a system.

No description available.

Dans la modélisation et l'évaluation des performances de la communication cellulaire sans fil, la géométrie stochastique est largement appliquée afin de fournir des solutions plus efficaces et plus précises. Le processus ponctuel de Poisson homogène (H-PPP), est le processus ponctuel le plus largement utilisé pour modéliser les emplacements spatiaux des stations de base (BS) en raison de sa facilité de traitement mathématique et de sa simplicité. Pour les fortes corrélations spatiales entre les emplacements des stations de base, seuls les processus ponctuels (PP) avec inhibitions et attractions spatiales peuvent être utiles. Cependant, le temps de simulation long et la faible aptitude mathématique rendent les PP non-Poisson non adaptés à l'évaluation des performances au niveau du système. Par conséquent, pour surmonter les problèmes mentionnés, nous avons les contributions suivantes dans cette thèse: Premièrement, nous introduisons une nouvelle méthodologie de modélisation et d’analyse de réseaux cellulaires de liaison descendante, dans laquelle les stations de base constituent un processus ponctuel invariant par le mouvement qui présente un certain degré d’interaction entre les points. L'approche proposée est basée sur la théorie des PP inhomogènes de Poisson (I-PPP) et est appelée approche à double amincissement non homogène (IDT). L’approche proposée consiste à approximer le PP initial invariant par le mouvement avec un PP équivalent constitué de la superposition de deux I-PPP conditionnellement indépendants. Les inhomogénéités des deux PP sont créées du point de vue de l'utilisateur type ``centré sur l'utilisateur''. Des conditions suffisantes sur les paramètres des fonctions d'amincissement qui garantissent une couverture meilleure ou pire par rapport au modèle de PPP homogène de base sont identifiées. La précision de l'approche IDT est justifiée à l'aide de données empiriques sur la distribution spatiale des stations de base. Ensuite, sur la base de l’approche IDT, une nouvelle expression analytique traitable du rapport de brouillage moyen sur signal (MISR) des réseaux cellulaires où les stations de base présentent des corrélations spatiales est introduite. Pour les PP non-Poisson, nous appliquons l'approche IDT proposée pour estimer les performances des PP non-Poisson. En prenant comme exemple le processus de points β-Ginibre ( β -GPP), nous proposons de nouvelles fonctions d’approximation pour les paramètres clés dans l’approche IDT afin de modéliser différents degrés d’inhibition spatiale et de prouver que MISR est constant en densification de réseau. Nous prouvons que la performance MISR dans le cas β-GPP ne dépend que du degré de répulsion spatiale, c'est-à-dire β , quelles que soient les densités de BS. Les nouvelles fonctions d'approximation et les tendances sont validées par des simulations numériques.Troisièmement nous étudions plus avant la méta-distribution du SIR à l’aide de l’approche IDT. La méta-distribution est la distribution de la probabilité de réussite conditionnelle compte tenu du processus de points. Nous dérivons et comparons l'expression sous forme fermée pour le b-ème moment dans les cas PP H-PPP et non-Poisson. Le calcul direct de la fonction de distribution cumulative complémentaire (CCDF) pour la méta-distribution n'étant pas disponible, nous proposons une méthode numérique simple et précise basée sur l'inversion numérique des transformées de Laplace. L'approche proposée est plus efficace et stable que l'approche conventionnelle utilisant le théorème de Gil-Pelaez. La valeur asymptotique de la CCDF de la méta distribution est calculée dans la nouvelle définition de la probabilité de réussite. En outre, la méthode proposée est comparée à certaines autres approximations et limites, par exemple l’approximation bêta, les bornes de Markov et les liaisons de Paley-Zygmund. Cependant, les autres modèles et limites d'approximation sont comparés pour être moins précis que notre méthode proposée. / In the modeling and performance evaluation of wireless cellular communication, stochastic geometry is widely applied, in order to provide more efficient and accurate solutions. Homogeneous Poisson point process (H-PPP) with identically independently distributed variables, is the most widely used point process to model the spatial locations of base stations (BSs) due to its mathematical tractability and simplicity. For strong spatial correlations between locations of BSs, only point processes (PPs) with spatial inhibitions and attractions can help. However, the long simulation time and weak mathematical tractability make non-Poisson PPs not suitable for system level performance evaluation. Therefore, to overcome mentioned problems, we have the following contributions in this thesis: First, we introduce a new methodology for modeling and analyzing downlink cellular networks, where the base stations constitute a motion-invariant point process that exhibits some degree of interactions among the points. The proposed approach is based on the theory of inhomogeneous Poisson PPs (I-PPPs) and is referred to as inhomogeneous double thinning (IDT) approach. The proposed approach consists of approximating the original motion-invariant PP with an equivalent PP that is made of the superposition of two conditionally independent I-PPPs. The inhomogeneities of both PPs are created from the point of view of the typical user. The inhomogeneities are mathematically modeled through two distance-dependent thinning functions and a tractable expression of the coverage probability is obtained. Sufficient conditions on the parameters of the thinning functions that guarantee better or worse coverage compared with the baseline homogeneous PPP model are identified. The accuracy of the IDT approach is substantiated with the aid of empirical data for the spatial distribution of the BSs. Then, based on the IDT approach, a new tractable analytical expression of mean interference to signal ratio (MISR) of cellular networks where BSs exhibits spatial correlations is introduced.For non-Poisson PPs, we apply proposed IDT approach to approximate the performance of non-Poisson PPs. Taking β-Ginibre point process (β -GPP) as an example, we propose new approximation functions for key parameters in IDT approach to model different degree of spatial inhibition and we successfully prove that MISR for β -GPP is constant under network densification with our proposed approximation functions. We prove that of MISR performance under β-GPP case only depends on the degree of spatial repulsion, i.e., β , regardless of different BS densities. We also prove that with the increase of β or (given fixed γ or β respectively), the corresponding MISR for β-GPP decreases. The new approximation functions and the trends are validated by numerical simulations. Third, we further study meta distribution of the SIR with the help of the IDT approach. Meta distribution is the distribution of the conditional success probability given the point process. We derive and compare the closed-form expression for the b-th moment under H-PPP and non-Poisson PP case. Since the direct computation of the complementary cumulative distribution function (CCDF) for meta distribution is not available, we propose a simple and accurate numerical method based on numerical inversion of Laplace transforms. The proposed approach is more efficient and stable than the conventional approach using Gil-Pelaez theorem. The asymptotic value of CCDF of meta distribution is computed under new definition of success probability. Furthermore, the proposed method is compared with some other approximations and bounds, e.g., beta approximation, Markov bounds and Paley-Zygmund bound. However, the other approximation models and bounds are compared to be less accurate than our proposed method.