A fractal stochastic point process (FSPP) is used to model molecular evolution in agreement with the relationship between the variance and mean numbers of synonymous and nonsynonymous substitutions in mammals. Like other episodic models such as the doubly stochastic Poisson process, this model accounts for the large variances observed in amino acid substitution rates, but unlike other models, it also accounts for the results of Ohta's (1995) analysis of synonymous and nonsynonymous substitutions in mammalian genes. That analysis yields a power-law increase in the index of dispersion and an inverse power-law decrease in the coefficient of variation with the mean number of substitutions, as predicted by the FSPP model but not by the doubly stochastic Poisson model. This result is compatible with the selection theory of evolution and the nearly-neutral theory of evolution.

The use of spatial statistics to investigate ecological processes in plant communities is becoming increasingly widespread. In diverse communities such as tropical rainforests, analysis of spatial structure may help to unravel the various processes that act and interact to maintain high levels of diversity. In particular, a number of contrasting mechanisms have been suggested to explain species coexistence, and these differ greatly in their practical implications for the ecology and conservation of tropical forests. Traditional first-order measures of community structure have proved unable to distinguish these mechanisms in practice, but statistics that describe spatial structure may be able to do so. This is of great interest and relevance as spatially explicit data become available for a range of ecological communities and analysis methods for these data become more accessible. This thesis investigates the potential for inference about underlying ecological processes in plant communities using spatial statistics. Current methodologies for spatial analysis are reviewed and extended, and are used to characterise the spatial signals of the principal theorised mechanisms of coexistence. The sensitivity of a range of spatial statistics to these signals is assessed, and the strength of such signals in natural communities is investigated. The spatial signals of the processes considered here are found to be strong and robust to modelled stochastic variation. Several new and existing spatial statistics are found to be sensitive to these signals, and offer great promise for inference about underlying processes from empirical data. The relative strengths of particular processes are found to vary between natural communities, with any one theory being insufficient to explain observed patterns. This thesis extends both understanding of species coexistence in diverse plant communities and the methodology for assessing underlying process in particular cases. It demonstrates that the potential of spatial statistics in ecology is great and largely unexplored.

This thesis deals with the estimation of model parameters of the interacting segments process in plane. The motivation is application on the system of stress fibers in human mesenchymal stem cells, which are detected by fluorescent microscopy. The model of segments is defined as a spatial Gibbs point process with marks. We use two methods for parameter estimation: moment method and Takacs-Fiksel method. Further, we implement algorithm for these estimation methods in software Mathematica. Also we are able to simulate the model structure by Markov Chain Monte Carlo, using birth-death process. Numerical results are presented for real and simulated data. Match of model and data is considered by descriptive statistics. Powered by TCPDF (www.tcpdf.org)

In this thesis we are concerned with representation of random closed sets in Rd with values concentrated on a space UX of locally finite unions of sets from a given class X ⊂ F. We examine existence of their repre- sentations with particle processes on the same space X, which keep invariance to rigid motions, which the initial random set was invariant to. We discuss existence of such representations for selected practically applicable spaces X: we go through the known results for convex sets and introduce new proofs for cases of sets with positive reach and for smooth k-dimensional submanifolds. Beside that we present series of general results related to representation of random UX sets. 1

<div>Networks and point processes provide flexible tools for representing and modeling complex dependencies in data arising from various social and physical domains. Graphs, or networks, encode relational dependencies between entities, while point processes characterize temporal or spatial interactions among events.</div><div><br></div><div>In the first part of this dissertation, we consider dynamic network data (such as communication networks) in which links connecting pairs of nodes appear continuously over time. We propose latent space point process models to capture two different aspects of the data: (i) communication occurs at a higher rate between individuals with similar latent attributes (i.e., homophily); and (ii) individuals tend to reciprocate communications from others, but in a varied manner. Our framework marries ideas from point process models, including Poisson and Hawkes processes, with ideas from latent space models of static networks. We evaluate our models on several real-world datasets and show that a dual latent space model, which accounts for heterogeneity in both homophily and reciprocity, significantly improves performance in various link prediction and network embedding tasks.</div><div><br></div><div>In the second part of this dissertation, we develop nonparametric goodness-of-fit tests for discrete distributions and point processes that contain intractable normalization constants, providing the first generally applicable and computationally feasible approaches under those circumstances. Specifically, we propose and characterize Stein operators for discrete distributions, and construct a general Stein operator for point processes using the Papangelou conditional intensity function. Based on the proposed Stein operators, we establish kernelized Stein discrepancy measures for discrete distributions and point processes, which enable us to develop nonparametric goodness-of-fit tests for un-normalized density/intensity functions. We apply the kernelized Stein discrepancy tests to discrete distributions (including network models) as well as temporal and spatial point processes. Our experiments demonstrate that the proposed tests typically outperform two-sample tests based on the maximum mean discrepancy, which, unlike our goodness-of-fit tests, assume the availability of exact samples from the null model.</div><div><br></div>

Em econometria um dos tópicos que vem se tornando ao longo dos anos primordial e a análise de ultra-frequência, ou seja, a análise da transação negócio a negócio. Ela tem se mostrado fundamental na modelagem da microestrutura do mercado intraday. Ainda assim temos uma teoria escassa que vem crescendo de forma humilde a cerca deste tema. Buscamos desenvolver um teste de hipótese para verificar se os dados de ultra-frequência apresentam incrementos independentes e estacionários, pois neste cenário saber disso é de grande importância, ja que muitos trabalhos tem como base essa hipótese. Além disso Grimshaw et. al. (2005)[6] mostrou que ao utilizarmos uma distribuição de probabilidade contínua para modelarmos dados econômicos, em geral, estimamos uma função de intensidade crescente, devido a resultados viciados obtidos como consequência do arredondamento, em nosso trabalho buscamos trabalhar com distribuições discretas para que contornar esse problema acarretado pelo uso de distribuições contínuas / In econometrics a topic that is becoming primordial over the years is the ultra frequency analysis, or analysis of the trades to trades transaction. This topic is shown to be fundamental in modeling the microstructure of the market intraday. Nevertheless we have a little theory that is growing so lowly about this topic. We seek to develop a hypothesis test to verify that the data ultrasonic frequency have independent and stationary increments, for this scenario the knowledge of it great importance, since many jobs is based on this hypothesis. In general Grimshaw et. al. (2005)[6] showed that when we use a continuous probability distribution to model ecomomic data, we estimate a function of increasing intensity due to addicts results obtained as a result of rounding. In our research we seek to work with discrete distributions to circumvent this problem entailed by the use of continuous distributions

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Atualmente, a tecnologia celular està presente em todos os aspectos da vida cotidiana: lares, escritÃrios, indÃstrias, etc. Tal tecnologia teve um rÃpido crescimento durante as duas Ãltimas dÃcadas tentando acompanhar o aumento do volume de trÃfego nas redes de comunicaÃÃo sem-fio. Naturalmente, ao propor modelos mais realistas possÃveis, com o propÃsito de caracterizar fenÃmenos que afetam a qualidade do sinal ou o desempenho do sistema, novas ideias, concepÃÃes e outras ferramentas surgem para descrever tais situaÃÃes. Este à o caso da Geometria EstocÃstica ou, particularmente, o processo pontual de Poisson, o qual vem sendo frequentemente utilizado como um modelo de rede celular, a partir da localizaÃÃo aleatÃria dos nÃs na rede. Diante desta ferramenta matemÃtica, à possÃvel implantar estaÃÃes rÃdio base na rede externa celular, bem como pontos de acesso baseados em picocÃlulas, femtocÃlulas, etc. AlÃm disso, permite-se quantificar a interferÃncia, Ãrea de cobertura, probabilidade de outage, dentre outros. Estes resultados tambÃm levam em consideraÃÃo o impacto de mobilidade no desempenho de tais redes. Nesse contexto, este trabalho analisarà redes ad-hoc sem-fio propondo expressÃes analÃticas para as seguintes mÃtricas de caracterizaÃÃo de desempenho: interferÃncia e conectividade de transmissÃo. Essas mÃtricas levam em consideraÃÃo tanto a razÃo sinal-ruÃdo mais interferÃncia (signal-to-interference-plus-noise ratio (SINR)) como a razÃo sinal-interferÃnca (signal-to-interference ratio (SIR)), em que neste caso, a potÃncia de ruÃdo à considerada nula. Especificamente, o fenÃmeno interferÃncia serà caracterizado via modelo shot-noise segundo um processo pontual chamado de processo pontual marcado (marked point process (MPP)), sendo este mais realista do que o tradicional modelo de Poisson. AlÃm disso, este tipo de modelo incorpora os efeitos de propagaÃÃo de rÃdio de pequena e larga escala e sobretudo as diferentes tecnologias de detecÃÃo e tratamento de sinal. Paralelamente, adotaremos um canal de rÃdio com desvanecimento Nakagami-m. Por fim, o tratamento matemÃtico para o modelo proposto torna-se um fator desafiador deste trabalho, visto que, tais resultados generalizam alguns jà publicados na literatura, os quais adotam alguns parÃmetros menos realistas. / Currently, cellular technology is present in all aspects of everyday life: homes, offices, industries, etc. Such technology had grown rapidly over the last two decades trying to follow up with the increased traffic volume on the networks of wireless communication. Naturally, to propose possible more realistic models, with the purpose of characterizing phenomena that affect the signal quality or performance system, new ideas, concepts and other tools to describe such situations arise. This is the case of Stochastic Geometry or, particularly, the point process Poisson, which has been often used as a model for cellular network from the random node locations in the network. Faced with this mathematical tool, it is possible deploy base stations in cellular external network and access points based picocells, femtocells, etc. Moreover, it allows to quantify the interference, coverage area, outage probability, among others. These results also consider the impact of mobility on the performance of such networks. In this context, this thesis will analyze ad-hoc wireless networks offering analytical expressions for the following metrics of performance characterization: interference and transmission connections. These metrics take into account both signal-to-interference-plus-noise ratio (SINR) and signal-to-interference ratio (SIR), in which case, the noise power is considered null. Specifically, the interference phenomena will be characterized via shot-noise model according to a point process called marked point process (MPP), this being more realistic than the traditional Poisson model. Furthermore, this type of model incorporates effects of radio propagation small and large scale, mainly the different technologies for the detection and signal processing. In parallel, we will adopt a radio channel with Nakagami-m fading. Finally, the mathematical treatment for the proposed model becomes a challenging factor in this work, since such results generalize some already published in the literature, which adopt some less realistic parameters.

Les processus ponctuels sont souvent utilisés comme modèles de répartitions spatiales ou spatio-temporelles d'occurrences Dans cette thèse, nous nous intéressons à des processus de Cox dirigés par un processus caché associé à un processus de Dirichlet. Ce modèle correspond à des occurrences cachées influençant l'intensité stochastique des occurrences observées. Nous généralisons la notion de Shot noise Cox process et développons le traitement bayésien. Nous focalisons l'inférence statistique sur l'estimation de la valeur espérée de chaque contribution cachée, leur nombre espéré, degré d'influence spatiale et degré de corrélation L'utilité en épidémiologie et en écologie est démontrée à partir de données de Rubus fruticosa, lbicella lutea et de mortalité dans les cantons de Georgie, USA. En termes de données observées, deux situations sont considérées: d'abord, les positions spatiales des occurrences sont observées entre plusieurs paires de dates consécutives; puis, des comptages sont effectués dans des unités d'échantillonnage spatiales. D'autre part, nous nous intéressons aux processus ponctuels à mémoire introduits par Kagan, Ogata et Vere-Jones. En effet, les processus ponctuels ont une place importante dans l'étude des catalogues sismiques. Nous avons étudié un modèle Epidemie Type Aftershock Sequence avec une intensité d'arrière-plan indépendante du temps et plusieurs fonctions déclenchantes permettant d'intégrer les événements antérieurs récents. Cette approche est utilisée pour étudier la sismicité des Petites Antilles. Une étude comparative des modèles Gamma, Weibull, Log-Normal et loi d'Omori modifiée pour les fonctions déclenchantes est menée. / Point processes are often used as spatial or spatio-temporal distribution models of occurrences. In this Phd dissertation, we focus first on Cox processes driven by a hidden process associated with a Dirichlet process. This model corresponds to hidden occurrences influencing the stochastic intensity of observed occurrences. We generalize the notion of Shot noise Cox process and develop its bayesian analysis. We focus the statistical inference on the estimation of the hidden contribution expected value, the hidden contribution expected number, the spatial influence and correlation parameters. Applications in epidemiology and ecology are shown from Rubus fruticosa data, Ibicella lutea data and death number data in counties of Georgia, USA. Two situations are considered with respect to available data: firstly, the spatial positions of occurrences are observed between several pairs of consecutive dates; secondly, counts are carried out over a fixed time interval in several spatial sampling units. Secondly, we focus on point processes with memory intr oduced by Kagan, Ogata and Vere-Jones. Spatio-temporal point processes play an important role in the studies of earthquake catalogs since they consist of seismic events with their dates and spatial locations. We studied an Epidemic Type Aftershock Sequence model with time independent background intensity and several triggering functions taking into account previous events. We illustrate our approach with a seismicity study of the Lesser Antilles arc. A comparaison study of Gamma, Weibull, Log-Normal and modified Omori law triggering function models is also carried out

The Ginibre gas is a Poisson point process dened on a space of loops related to the Feynman-Kac representation of the ideal Bose gas. Here we study thermodynamic limits of dierent ensembles via Martin-Dynkin boundary technique and show, in which way innitely long loops occur. This effect is the so-called Bose-Einstein condensation.

We are interested in modeling some two-level population dynamics, resulting from the interplay of ecological interactions and phenotypic variation of individuals (or hosts) and the evolution of cells (or parasites) of two types living in these individuals. The ecological parameters of the individual dynamics depend on the number of cells of each type contained by the individual and the cell dynamics depends on the trait of the invaded individual. Our models are rooted in the microscopic description of a random (discrete) population of individuals characterized by one or several adaptive traits and cells characterized by their type. The population is modeled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation and death for individuals and birth and death for cells. The interaction between individuals (resp. between cells) is described by a competition between individual traits (resp. between cell types). We look for tractable large population approximations. By combining various scalings on population size, birth and death rates and mutation step, the single microscopic model is shown to lead to contrasting nonlinear macroscopic limits of different nature: deterministic approximations, in the form of ordinary, integro- or partial differential equations, or probabilistic ones, like stochastic partial differential equations or superprocesses. The study of the long time behavior of these processes seems very hard and we only develop some simple cases enlightening the difficulties involved.