Submitted by Etelvina Domingos (etelvina.domingos@ufpe.br) on 2015-03-12T18:21:25Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) TESE Ana Carla Percontini da Paixão.pdf: 2309750 bytes, checksum: 1f4caced5454dee673c1e41705168ad0 (MD5) / Made available in DSpace on 2015-03-12T18:21:26Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) TESE Ana Carla Percontini da Paixão.pdf: 2309750 bytes, checksum: 1f4caced5454dee673c1e41705168ad0 (MD5) Previous issue date: 2014 / Este trabalho está dividido em quatro capítulos independentes. Nos Capítulos 2 e 3 propomos extensões para a distribuição Weibull. A primeira delas, com cinco parâmetros, é uma composição das distribuições beta e Weibull Poisson. Essa nova distribuição tem como submodelos algumas importantes distribuições descritas na literatura e outras ainda não discutidas tais como: bata exponencial Poisson, Weibull Poisson exponencializada, Rayleigh Poisson exponencializada, beta Weibull, Weibull, exponencial, entre outras. Obtemos algumas propriedades matemáticas tais como momentos ordinários e incompletos, estatísticas de ordem e seus momentos e entropia de Rényi. Usamos o método da máxima verossimilhança para obter estimativas dos parâmetros. A potencialidade desse novo modelo é mostrada por meio de um conjunto de dados reais. A segunda extensão, com quatro parâmetros, é uma composição das distribuições Poisson generalizada e Weibull, tendo a Poisson generalizada exponencial, a Rayleigh Poisson, Weibull Poisson e Weibull como alguns de seus sub-modelos. Várias propriedades matemáticas foram investigadas, incluíndo expressões explícitas para os momentos ordinários e incompletos, desvios médios, função quantílica, curvas de Bonferroni e Lorentz, con abilidade e as entropias de Rényi e Shannon. Estatísticas de ordem e seus momentos são investigados. A estimativa de parâmetros é feita pelo método da máxima verossimilhança e é obtida a matriz de informação obsevada. Uma aplicação a um conjunto de dados reais mostra a utilidade do novo modelo. Nos dois últimos capítulos propomos duas novas classes de distribuições. No Capítulo 4 apresentamos a família G- Binomial Negativa com dois parâmetros extras. Essa nova família inclui como caso especial um modelo bastante popular, a Weibull binomial negativa, discutida por Rodrigues et al.(Advances and Applications in Statistics 22 (2011), 25-55.) Algumas propriedades matemáticas da nova classe são estudadas, incluindo momentos e função geradora. O método de máxima verossimilhança é utilizado para obter estimativas dos parâmetros. A utilidade da nova classe é mostrada através de um exemplo com conjuntos de dados reais. No Capítulo 5 apresentamos a classe Zeta-G com um parâmetro extra e algumas nova distribuições desta classe. Obtemos expressões explícitas para a função quantílica, momentos ordinários e incompletos, dois tipos de entropia, con abilidade e momentos das estatísticas de ordem. Usamos o método da máxima verossimilhança para estimar os parâmetros e a utilidade da nova classe é exempli cada com um conjunto de dados reais.

Orientador: Hildete Prisco Pinheiro / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-08T17:00:10Z (GMT). No. of bitstreams: 1 Mendes_ClariceCamargo_M.pdf: 1127292 bytes, checksum: f015352011300a41bb50c17c81a49bb1 (MD5) Previous issue date: 2007 / Resumo: Ao lidarmos com dados de contagem, uma abordagem possível é estimar um Modelo Linear Generalizado com distribuição de Poisson. Freqüentemente nestes modelos costuma surgir o problema da superdispersão, um fenômeno que aparece quando estamos diante de uma variabilidade dos dados maior do que a média. Temos basicamente três soluções para este problema: abordagem bayesiana, assumindo que o parâmetro do modelo possui uma distribuição de probabilidade; estimação por Quase-verossimilhança, incluindo um fator de dispersão diferente da unidade ou uma função de variância diversa e, finalmente, o emprego de modelos mistos, com a separação de efeitos fixos e aleatórios. Outra ocorrência comum para dados de contagem é encontrarmos amostras que apresentem um número excessivo de zeros. Detectamos a presença da superdispersão, mas agora ela é devida à ocorrência de mais valores zero na amostra do que seria esperado para dados que seguissem a distribuição de Poisson. Para este caso Lambert (1982) apresenta a chamada regressão de Poisson inflacionada de zeros (ZIP - Zero lnflated Poisson). Através de uma aplicação a dados reais, em estudo referente à alimentação de rãs da espécie Adenomera, identificamos os melhores modelos para explicar a quantidade de comida ingerida em função dos efeitos de sexo e da estação do ano. Utilizamos técnicas de diagnóstico para avaliar o impacto que uma determinada observação exerce na estimativa dos parâmetros. / Abstract: When one deals with count data, a possible approach is to fit a generalized linear model with Poisson distribution. Usually it may occur the problem of superdispersion, when the variability of the data is greater then the mean. There are three basic solutions to this problem: the Bayesian approach, when we assume that the parameter of the mo deI has a distribution of probability; the Quasilikelihood estimation, including a non-unitary dispersion parameter or a different variance function and, finally, the mixed models. Another possible occurrence to count data is the presence of samples with an excess of zeros. We detect the presence of the superdispersion, but now it is due to more zero counts than expected from the Poisson distribution. For this case, Lambert (1982) presents the Zero Infiated Poisson (ZIP) mode. As an application to real data, in the study of frogs' nourishment from the species Adenomera, we identify the best models to explain the quantity of swallowed food related to sex and season effects. We employ techniques of diagnosis to verify the impact of a specific observation in the parameter estimations / Mestrado / Bioestatistica / Mestre em Estatística

As florestas na Amazônia tem sido exploradas para a provisão de madeira. A exploração era feita sem planejamento das atividades (Exploração Convencional ou EC), resultando em grandes áreas de florestas degradadas. A partir da década de 90, foram estabelecidas técnicas para minimizar os danos da exploração à vegetação remanescente, visando ciclos sequenciais de corte, sem prejuízo à produção, através da Exploração de Impacto Reduzido (EIR). Embora muito tenha sido discutido sobre o quão mais eficiente a EIR seja em relação a EC, ainda existem lacunas, principalmente acerca da organização e manutenção da diversidade dessas áreas. A exploração é, de maneira muito resumida, o corte de algumas árvores de interesse comercial e a morte acidental de outras árvores que não as de interesse, pela queda e arraste das toras, construção das estradas e dos pátios de estocagem. As clareiras formadas, em geral, são maiores que as que ocorrem naturalmente na floresta. Essas clareiras são ocupadas por espécies de rápido crescimento, potencialmente mudando a organização da comunidade. Assim, são necessárias abordagens para detectar e então caracterizar quais os impactos que a exploração tem na diversidade de comunidades arbóreas. É importante ressaltar que muitas decisões sobre a sustentabilidade do manejo são tomadas considerando como essas comunidades respondem a distúrbios. Diante do apresentado, esta tese teve como objetivo analisar a dinâmica da diversidade de florestas submetidas à exploração florestal. A distribuição de abundância entre espécies (DAE) é uma abordagem paramétrica para caracterização de comunidades, baseada em um dos poucos padrões que se mantem em todas as comunidades: muitas espécies raras e poucas espécies abundantes, sendo também o reflexo de como as espécies partilham recursos. Distúrbios mudam a identidade das espécies dominantes, podendo ser refletido em alterações nos parâmetros e/ou curvas da DAE. Nós analisamos a DAE de áreas submetidas ao manejo florestal (EIR e EC) e uma área Controle, antes do manejo e ao longo de 16 anos em Paragominas, Pará, Brasil. Antes da exploração, em 1993, foram estabelecidas três parcelas permanentes, onde os indivíduos arbóreos com DAP &ge; 25 cm foram cadastrados. Indivíduos menores (DAP &ge; 10 cm) foram mensurados em sub-parcelas. Após essa primeira medição, foi conduzida exploração em duas parcelas (EIR e EC) e a terceira parcela foi mantida como Controle. As parcelas foram medidas novamente em 1994, 1998, 2000, 2006 e 2009. Para testar se a DAE muda devido ao manejo, usamos a distribuição Poisson Lognormal (PLN) para descrever cada ano e cada tratamento. Para detectar mudanças na similaridade, em termos de composição de espécies e abundância relativa de cada espécie, nós usamos o parâmetro de correlação da PLN bivariada para comparar a comunidade antes da exploração com os outros anos. Ao contrário do esperado, não foram encontradas mudanças estatisticamente relevantes nos parâmetros da DAE ao longo do tempo em nenhum dos três tratamentos. Entretanto o parâmetro de correlação diminui para a comunidade com o menor critério de inclusão de DAP, nas últimas medições das áreas exploradas. Nós também analisamos a DAE dessas mesmas áreas considerando como medida de abundância a área basal de cada espécie. Não foram encontradas mudanças relvantes na DAE, apenas diminuição do parâmetro de correlação. Embora o manejo tenha resultado na diminuição de até 22% dos indivíduos e 27% da área basal, ele não é imediatamente detectado na DAE pois as mortes acidentais são a maioria no manejo. Para cada árvore explorada, em média 19 árvores com DAP &ge; 10 cm são mortas acidentalmente. Essa morte acidental é de certa forma independente da espécie, assim todas as espécies tem suas abundâncias diminuídas. A distribuição PLN considera os efeitos da amostragem através do processo Poisson, então mesmo que a porcentagem de indivíduos mortos fosse maior, dificilmente seria detectável na DAE. No entanto, a diminuição do parâmetro de correlação ao longo do tempo nas áreas exploradas é devido ao aumento da abundância e da área basal de espécies de rápido crescimento como as do gênero Cecropia. A DAE da Ilha de Barro Colorado (BCI), Panamá, também foi analisada (28 anos de medições, DAP &ge; 1 cm). Nós dividimos o conjunto de dados em quatro critérios de inclusão de DAP (1, 10, 25 e 45 cm). Visto que a área não é submetida a grandes distúrbios e, como era esperado, não foram encontradas mudanças relevantes nem na DAE nem na identidade das espécies dominantes. Muitos indivíduos morreram durante esse período, devido a elevada mortalidade, comum para árvores pequenas (DAP < 10 cm), e outra grande parcela da comunidade não atingiu os critérios de DAP que geralmente são amostrados (10, 25 e 45 cm). Mesmo os indivíduos considerados ingressantes, de acordo com esses critérios, já estavam presentes na parcela na primeira medição, apenas eram menores que esses critérios. Diante do apresentado, um distúrbio pode demorar décadas para aparecer nas classes de DAP que geralmente são amostradas, por exemplo DAP &ge; 10 cm. Assim, reiteramos que efeito ou falta de efeito de distúrbios na diversidade de comunidades arbóreas deve ser interpretado com muita cautela. / Amazon tropical forests in Brazil have been exploited for the provisioning of timber, mainly using conventional logging (CL) practices. Little operational planning has been considered and, as a result, large forest areas in the Amazon have become highly degraded over time. Since the 1990\'s, reduced impact logging (RIL) has been adopted as a means of reducing the damage caused by timber exploitation and of establishing more sustainable practices, trying to make sequential logging cycles possible. Much has been discussed about the higher efficiency of RIL or CL, but there are still important knowledge gaps, mainly regarding tree diversity and forest functioning in logged areas. The logging of commercial species and the accidental death of non-targeted trees may alter environmental conditions, leading to higher abundance of some species and to the reduction of others, thus modifying forest structure and diversity. These changes promoted by exploitation have a signature in the structure of the remaining community and tracking them at the community level is still a great challenge, but important decisions are mainly taken considering tree diversity. The development of tools is crucial to quantify the effects of disturbance and to predict its consequence over communities. The main goal of this thesis was to analyze the temporal dynamics of tree diversity in the context of forest management. Species abundance distribution models (SAD) are a parametric approach, based on the most strong pattern in community ecology: few species have high abundance and rarity is the rule. It is also independent of sampling size and related to how species share resources. Species dominance change as a result of disturbances, and these changes may be detectable by changes on the parameters and/or curves of the SAD. We used inventory data from three permanent plots established in 1993 in Paragominas, Pará, Brazil. All the trees with Diameter at Breast Height (DBH) &ge; 25 cm were determined and measured and smaller individuals were measured within subplots (DBH &ge; 10 cm). After this first survey, we harvested two plots using different techniques (RIL and CL) and a third plot was kept as Control. The plots were surveyed again in 1994, 1998, 2000, 2006 and 2009. To test if it is possible to detect changes in the SAD due to logging we used the Poisson Lognormal distribution (PLN) to describe the data from each year. To detect changes in species similarity, we compared species composition and abundance over time, using the correlation parameter of the bivariate PLN. We compared the same plots in 1993 (before exploitation) and in the years that followed. We observed no relevant changes in SAD, but only small changes in the correlation parameter in the last surveys. We also analyzed species\' basal area distribution, and found no statistically relevant changes apart from small changes in the correlation parameter in harvested areas. After exploitation, we observed a reduction of as much as 22 % of the number of trees and 27% of the basal area, and mortality was mainly attributed to the accidental death of non-targeted trees. For each tree harvested, an average 19 trees died accidentally. Because of the dominant role of these random accidental deaths, the signature of management operations could not be detected immediately after logging, using SAD or the correlation parameter. Since SAD models consider sampling effects (Poisson process), any disturbance which causes the random death of trees may not be detected in the short run. However, the start of small changes could be noticed within a couple of decades. The relative abundance of some species has been altered in exploited forests, which was detected by changes in the correlation parameter. Pioneer species of the genus Cecropia have become the most abundant trees in the last survey, considering DBH &ge; 10 cm. We also evaluated species abundance distribution in the Barro Colorado Island, Panama (28 years, DBH &ge; 1 cm). We established four different inclusion criteria (DBH 1, 10, 25 and 45). The area was not subject to major disturbances and, as expected, we observed no relevant changes in the SAD during this period. A large number of individuals died during this period, due to the high mortality common to small trees (DBH &ge; 10 cm), and many others have not reached the used inclusion criteria (10, 25 and 45 cm). Using these inclusion criteria, we showed that the community recruited during this period was already present in the forest in the first survey, but those trees just did not reach the size to be measured. Therefore, a recent disturbance may take decades to be detected within the DBH classes usually sampled. We highlight the role of time scale in the interpretation of tree diversity dynamics.

[pt] O presente trabalho tem como objetivo principal investigar por meio de simulação Monte Carlo algumas propriedades estatísticas dos modelos GARMA (Generalized Autoregressive Moving Average) para séries temporais de dados de contagem. Os modelos GARMA são uma extensão dos Modelos Lineares Generalizados de McCullagh e Nelder para situações de dados dependentes, caracterizando-se pela adição de um termo extra ao preditor linear, o qual passa a incorporar termos autoregressivos (AR) e de médias móveis (MA). As propriedades estatísticas investigadas foram às condições de estacionariedade dos modelos GARMA e os critérios de identificação da ordem (p,q) dos polinômios AR e MA que definem o modelo. Os resultados encontrados indicam que os critérios AIC BIC e Hannan-Quin utilizados foram razoavelmente eficazes na identificação da ordem dos modelos e que as condições de estacionariedade estabelecidas empiricamente em termo de restrições no espaço paramétrico são bastante complexas exigindo um estudo mais detalhado. Como objetivo secundário testamos os modelo GARMA em séries reais, ajustando os modelos GARMA- Poissson e GARMA-Binomial Negativa ao número de caso de poliomielite nos EUA e ao número de infartos do miocárdio no município do Rio de Janeiro. Os resultados indicam que os modelos foram capazes de explicar, de forma econômica, a variação destas séries. / [en] The main objective of this dissertation is to investigate, using Monte Carlo simulations, some statistical properties of GARMA (Generalized Autoregressive Moving Average ) models for time series of count data. GARMA models are extensions of the Generalized Linear Models to dependent data, in which autoregressive (AR) and/or moving average (MA) terms are incorporated into the linear predictor. The statistical properties targeted in our investigation were the model stationarity conditions and the identification criteria for selection of model orders, the lag structure (p,q) associated with the AR and MA terms. Our results suggest that AIC, BIC and Hann-Quinn criteria worked relatively well in identifying the model order, and that the conditions for stationarity established empirically in terms of parameter space restrictions were not totally conclusive, requiring further investigation. As a secondary objective we tested the model against real data, by fitting both a GARMA-Poisson and a GARMA-Negative Binomial to the series of number of cases of poliomyelitis on the US and the number of heart-attacks in Rio de Janeiro city. The results we found indicate that these models were able to explain, in a parsimonious way, the variation of both series.

Le but de cette thèse est de développer la théorie de Freidlin-Wentzell pour des modèles des épidémies, afin de prédire le temps mis par les perturbations aléatoires pour éteindre une situation endémique "stable". Tout d'abord nous proposons une nouvelle démonstration plus courte par rapport à celle établit récemment (sous une hypothèse un peu différente, mais satisfaite dans tous les exemples de modèles de maladie infectieuses que nous avons à l'esprit) par Kratz et Pardoux (2017) sur le principe de grandes déviations pour les modèles des épidémies. Ensuite nous établissons un principe de grandes déviations pour des EDS poissoniennes réfléchies au bord d'un ouvert suffisamment régulier. Nous établissons aussi un résultat concernant la zone du bord la plus probable par laquelle le processus solution de l'EDS de Poisson va sortir du domaine d'attraction d'un équilibre stable de sa loi des grands nombres limite. Nous terminons cette thèse par la présentation des méthodes "non standard aux différences finis", appropriés pour approcher numériquement les solutions de nos EDOs ainsi que par la résolution d'un problème de contrôle optimal qui permet d'avoir une bonne approximation du temps d'extinction d'un processus d'infection. / In this thesis, we develop the Freidlin-Wentzell theory for the "natural'' Poissonian random perturbations of the above ODE in Epidemic Dynamics (and similarly for models in Ecology or Population Dynamics), in order to predict the time taken by random perturbations to extinguish a "stable" endemic situation. We start by a shorter proof of a recent result of Kratz and Pardoux (under a somewhat different hypothesis which is satisfied in all the cases we have examined so far), which establishes the large deviations principle for epidemic models. Next, we establish the large deviations principle for reflected Poisonian SDE at the boundary of a sufficiently regular open set. Then, we establish the result for the most likely boundary area by which the process will exit the domain of attraction of a stable equilibrium of an ODE. We conclude this thesis with the presentation of the "non - standard finite difference" methods, suitable to approach numerically the solutions of our ODEs as well as the resolution of an optimal control problem which allows to have a good approximation of the time of extinction of an endemic situation.

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.

[pt] O estudo de deformações e degenerações de estruturas de Poisson ocupa posição especial dentro do marco clássico de análise de degenerações de estruturas geométricas. Nesta tese como resultado principal construímos uma deformação não trivial na qual a estrutura quadrática canônica do espaço projetivo complexo n-dimensional é limite contínuo de estruturas Kahlerianas. Além disso, como resultado segundário de estudos de deformações mostramos que uma estrutura Poisson invariante numa variedade tórica com número finito de folhas não pode ser exata na cohomologia Poisson. Nosso estudo também inclui considerações sobre cohomologia Poisson da estrutura quadrática canônica do espaço vetorial complexo n-dimensional. / [en] The study of deformations and degenerations of Poisson structures occupies a special position within the classical framework of analysis of degenerations of geometric structures. In this thesis as the main result we build a non-triavial deformation in which the canonical quadratic structure in CP(n) is a continuous limit of Kahlerian structures. Furthermore, as a secondary result of deformation studies we have shown that an invariant Poisson structure in a toric variety with finite number of leaves cannot be exact in Poisson cohomology. Our study also includes considerations about Poisson cohomology of the canonical quadratic structure of C(n).

Evaluation of crash count data as a function of roadway characteristics allows Departments of Transportation to predict expected average crash risks in order to assist in identifying segments that could benefit from various treatments. Currently, the evaluation is performed using negative binomial regression, as a function of average annual daily traffic (AADT) and other variables. For this thesis, a crash study was carried out for the interstate, primary and secondary routes, in the Salem District of Virginia. The data used in the study included the following information obtained from Virginia Department of Transportation (VDOT) records: 2010 to 2012 crash data, 2010 to 2012 AADT, and horizontal radius of curvature (CV). Additionally, tire-pavement friction or skid resistance was measured using a continuous friction measurement, fixed-slip device called a Grip Tester. In keeping with the current practice, negative binomial regression was used to relate the crash data to the AADT, skid resistance and CV. To determine which of the variables to include in the final models, the Akaike Information Criterion (AIC) and Log-Likelihood Ratio Tests were performed. By mathematically combining the information acquired from the negative binomial regression models and the information contained in the crash counts, the parameters of each network's true average crash risks were empirically estimated using the Empirical Bayes (EB) approach. The new estimated average crash risks were then used to rank segments according to their empirically estimated crash risk and to prioritize segments according to their expected crash reduction if a friction treatment were applied. / Master of Science

The mathematical modeling and performance analysis of cellular networks have seen a major paradigm shift with the application of stochastic geometry. The main purpose of stochastic geometry is to endow probability distributions on the locations of the base stations (BSs) and users in a network, which, in turn, provides an analytical handle on the performance evaluation of cellular networks. To preserve the tractability of analysis, the common practice is to assume complete spatial randomness} of the network topology. In other words, the locations of users and BSs are modeled as independent homogeneous Poisson point processes (PPPs). Despite its usefulness, the PPP-based network models fail to capture any spatial coupling between the users and BSs which is dominant in a multi-tier cellular network (also known as the heterogeneous cellular networks (HetNets)) consisting of macro and small cells. For instance, the users tend to form hotspots or clusters at certain locations and the small cell BSs (SBSs) are deployed at higher densities at these locations of the hotspots in order to cater to the high data demand. Such user-centric deployments naturally couple the locations of the users and SBSs. On the other hand, these spatial couplings are at the heart of the spatial models used in industry for the system-level simulations and standardization purposes. This dissertation proposes fundamentally new spatial models based on stochastic geometry which closely emulate these spatial couplings and are conductive for a more realistic and fine-tuned performance analysis, optimization, and design of cellular networks. First, this dissertation proposes a new class of spatial models for HetNets where the locations of the BSs and users are assumed to be distributed as Poisson cluster process (PCP). From the modeling perspective, the proposed models can capture different spatial couplings in a network topology such as the user hotspots and user BS coupling occurring due to the user-centric deployment of the SBSs. The PCP-based model is a generalization of the state-of-the-art PPP-based HetNet model. This is because the model reduces to the PPP-based model once all spatial couplings in the network are ignored. From the stochastic geometry perspective, we have made contributions in deriving the fundamental distribution properties of PCP, such as the distance distributions and sum-product functionals, which are instrumental for the performance characterization of the HetNets, such as coverage and rate. The focus on more refined spatial models for small cells and users brings to the second direction of the dissertation, which is modeling and analysis of HetNets with millimeter wave (mm-wave) integrated access and backhaul (IAB), an emerging design concept of the fifth generation (5G) cellular networks. While the concepts of network densification with small cells have emerged in the fourth generation (4G) era, the small cells can be realistically deployed with IAB since it solves the problem of high capacity wired backhaul of SBSs by replacing the last-mile fibers with mm-wave links. We have proposed new stochastic geometry-based models for the performance analysis of IAB-enabled HetNets. Our analysis reveals some interesting system-design insights: (1) the IAB HetNets can support a maximum number of users beyond which the data rate drops below the rate of a single-tier macro-only network, and (2) there exists a saturation point of SBS density beyond which no rate gain is observed with the addition of more SBSs. The third and final direction of this dissertation is the combination of machine learning and stochastic geometry to construct a new class of data driven network models which can be used in the performance optimization and design of a network. As a concrete example, we investigate the classical problem of wireless link scheduling where the objective is to choose an optimal subset of simultaneously active transmitters (Tx-s) from a ground set of Tx-s which will maximize the network-wide sum-rate. Since the optimization problem is NP-hard, we replace the computationally expensive heuristic by inferring the point patterns of the active Tx-s in the optimal subset after training a determinantal point process (DPP). Our investigations demonstrate that the DPP is able to learn the spatial interactions of the Tx-s in the optimal subset and gives a reasonably accurate estimate of the optimal subset for any new ground set of Tx-s. / Doctor of Philosophy / The high speed global cellular communication network is one of the most important technologies, and it continues to evolve rapidly with every new generation. This evolution greatly depends on observing performance-trends of the emerging technologies on the network models through extensive system-level simulations. Since these simulation models are extremely time-consuming and error prone, the complementary analytical models of cellular networks have been an area of active research for a long time. These analytical models are intended to provide crisp insights on the network behavior such as the dependence of network performance metrics (such as coverage or rate) on key system-level parameters (such as transmission powers, base station (BS) density) which serve as the prior knowledge for more fine-tuned simulations. Over the last decade, the analytical modeling of the cellular networks has been driven by stochastic geometry. The main purpose of stochastic geometry is to endow the locations of the base stations (BSs) and users with probability distributions and then leverage the properties of these distributions to average out the spatial randomness. This process of spatial averaging allows us to derive the analytical expressions of the system-level performance metrics despite the presence of a large number of random variables (such as BS and user locations, channel gains) under some reasonable assumptions. The simplest stochastic geometry based model of cellular networks, which is also the most tractable, is the so-called Poisson point process (PPP) based network model. In this model, users and BSs are assumed to be distributed as independent homogeneous PPPs. This is equivalent to saying that the users and BSs independently and uniformly at random over a plane. The PPP-based model turned out to be a reasonably accurate representation of the yesteryear’s cellular networks which consisted of a single tier of macro BSs (MBSs) intended to provide a uniform coverage blanket over the region. However, as the data-hungry devices like smart-phones, tablets, and application like online gaming continue to flood the consumer market, the network configuration is rapidly deviating from this baseline setup with different spatial interactions between BSs and users (also termed spatial coupling) becoming dominant. For instance, the user locations are far from being homogeneous as they are concentrated in specific areas like residential and commercial zones (also known as hotspots). Further, the network, previously consisting of a single tier of macro BSs (MBSs), is becoming increasingly heterogeneous with the deployment of small cell BSs (SBSs) with small coverage footprints and targeted to serve the user hotspots. It is not difficult to see that the network topology with these spatial couplings is quite far from complete spatial randomness which is the basis of the PPP-based models. The key contribution of this dissertation is to enrich the stochastic geometry-based mathematical models so that they can capture the fine-grained spatial couplings between the BSs and users. More specifically, this dissertation contributes in the following three research directions. Direction-I: Modeling Spatial Clustering. We model the locations of users and SBSs forming hotspots as Poisson cluster processes (PCPs). A PCP is a collection of offspring points which are located around the parent points which belong to a PPP. The coupling between the locations of users and SBSs (due to their user-centric deployment) can be introduced by assuming that the user and SBS PCPs share the same parent PPP. The key contribution in this direction is the construction of a general HetNet model with a mixture of PPP and PCP-distributed BSs and user distributions. Note that the baseline PPP-based HetNet model appears as one of the many configurations supported by this general model. For this general model, we derive the analytical expressions of the performance metrics like coverage probability, BS load, and rate as functions of the coupling parameters (e.g. BS and user cluster size). Direction-II: Modeling Coupling in Wireless Backhaul Networks. While the deployment of SBSs clearly enhances the network performance in terms of coverage, one might wonder: how long network densification with tens of thousands of SBSs can meet the everincreasing data demand? It turns out that in the current network setting, where the backhaul links (i.e. the links between the BSs and core network) are still wired, it is not feasible to densify the network beyond some limit. This backhaul bottleneck can be overcome if the backhaul links also become wireless and the backhaul and access links (link between user and BS) are jointly managed by an integrated access and backhaul (IAB) network. In this direction, we develop the analytical models of IAB-enabled HetNets where the key challenge is to tackle new types of couplings which exist between the rates on the wireless access and backhaul links. Such couplings exist due to the spatial correlation of the signal qualities of the two links and the number of users served by different BSs. Two fundamental insights obtained from this work are as follows: (1) the IAB HetNets can support a maximum number of users beyond which the network performance drops below that of a single-tier macro-only network, and (2) there exists a saturation point of SBS density beyond which no performance gain is observed with the addition of more SBSs. Direction-III: Modeling Repulsion. In this direction, we focus on modeling another aspect of spatial coupling imposed by the intra-point repulsion. Consider a device-to-device (D2D) communication scenario, where some users are transmitting some on-demand content locally cached in their devices using a common channel. Any reasonable multiple access scheme will ensure that two nearly users are never simultaneously active as they will cause severe mutual interference and thereby reducing the network-wide sum rate. Thus the active users in the network will have some spatial repulsion. The locations of these users can be modeled as determinantal point processes (DPPs). The key property of DPP is that it forms a bridge between stochastic geometry and machine learning, two otherwise non-overlapping paradigms for wireless network modeling and design. The main focus in this direction is to explore the learning framework of DPP and bring together advantages of stochastic geometry and machine learning to construct a new class of data-driven analytical network models.

Including energy harvesting capability in a wireless network is attractive for multiple reasons. First and foremost, powering base stations with renewable resources could significantly reduce their reliance on the traditional energy sources, thus helping in curtailing the carbon footprint. Second, including this capability in wireless devices may help in increasing their lifetime, which is especially critical for devices for which it may not be easy to charge or replace batteries. This will often be the case for a large fraction of sensors that will form the {em digital skin} of an Internet of Things (IoT) ecosystem. Motivated by these factors, this work studies fundamental performance limitations that appear due to the inherent unreliability of energy harvesting when it is used as a primary or secondary source of energy by different elements of the wireless network, such as mobile users, IoT sensors, and/or base stations. The first step taken towards this objective is studying the joint uplink and downlink coverage of radio-frequency (RF) powered cellular-based IoT. Modeling the locations of the IoT devices and the base stations (BSs) using two independent Poisson point processes (PPPs), the joint uplink/downlink coverage probability is derived. The resulting expressions characterize how different system parameters impact coverage performance. Both mathematical expressions and simulation results show how these system parameters should be tuned in order to achieve the performance of the regularly powered IoT (IoT devices are powered by regular batteries). The placement of RF-powered devices close to the RF sources, to harvest more energy, imposes some concerns on the security of the signals transmitted by these RF sources to their intended receivers. Studying this problem is the second step taken in this dissertation towards better understanding of energy harvesting wireless networks. While these secrecy concerns have been recently addressed for the point-to-point link, it received less attention for the more general networks with randomly located transmitters (RF sources) and RF-powered devices, which is the main contribution in the second part of this dissertation. In the last part of this dissertation, we study the stability of solar-powered cellular networks. We use tools from percolation theory to study percolation probability of energy-drained BSs. We study the effect of two system parameters on that metric, namely, the energy arrival rate and the user density. Our results show the existence of a critical value for the ratio of the energy arrival rate to the user density, above which the percolation probability is zero. The next step to further improve the accuracy of the stability analysis is to study the effect of correlation between the battery levels at neighboring BSs. We provide an initial study that captures this correlation. The main insight drawn from our analysis is the existence of an optimal overlapping coverage area for neighboring BSs to serve each other's users when they are energy-drained. / Ph. D. / Renewable energy is a strong potential candidate for powering wireless networks, in order to ensure green, environment-friendly, and self-perpetual wireless networks. In particular, renewable energy gains its importance when cellular coverage is required in off-grid areas where there is no stable resource of energy. In that case, it makes sense to use solar-powered base stations to provide cellular coverage. In fact, solar-powered base stations are deployed already in multiple locations around the globe. However, in order to extend this to a large scale deployment, many fundamental aspects of the performance of such networks needs to be studied. One of these aspects is the stability of solar-powered cellular networks. In this dissertation, we study the stability of such networks by applying probabilistic analysis that leads to a set of useful system-level insights. In particular, we show the existence of a critical value for the energy intensity, above which the system stability is ensured. Another type of wireless networks that will greatly benefit from renewable energy is internet of things (IoT). IoT devices usually require several orders of magnitude lower power compared to the base stations. In addition, they are expected to be massively deployed, often in hard-to-reach locations. This makes it impractical or at least cost inefficient to rely on replacing or recharging batteries in these devices. Among many possible resources of renewable energy, radio frequency (RF) energy harvesting is the strongest candidate for powering IoT devices, due to ubiquity of RF signals even at hard-to-reach places. However, relying on RF signals as the sole resource of energy may affect the overall reliability of the IoT. Hence, rigorous performance analysis of RF-powered IoT networks is required. In this dissertation, we study multiple aspects of the performance of such networks, using tools from probability theory and stochastic geometry. In particular, we provide concrete mathematical expressions that can be used to determine the performance drop resulting from using renewable energy as the sole source of power. One more aspect of the performance of RF-powered IoT is the secrecy of the RF signals used by the IoT devices to harvest energy. The placement of RF-powered devices close to the RF sources, to harvest more energy, imposes some concerns on the security of the signals transmitted by these RF sources to their intended receivers. We study the effect of using secrecy enhancing techniques by the RF sources on the amount of energy harvested by the RF-powered devices. We provide performance comparison of three popular secrecy-enhancing techniques. In particular, we study the scenarios under which each of these techniques outperforms the others in terms of secrecy performance and energy harvesting probability. This material is based upon work supported by the U.S. National Science Foundation (Grant CCF1464293). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the NSF.