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Consideramos uma base de dados hospitalares constituída por informações sobre vítimas de causas externas atendidas no Pronto Socorro do Hospital Municipal Dr. Arthur Ribeiro de Saboya, no período de 01/01/02 a 11/01/03, e registradas pelo Núcleo de Atenção à Vítima de Violência deste hospital. O conjunto de dados foi avaliado sob duas abordagens: a temporal, onde estudamos o numero de eventos ao longo do tempo, e a espacial, onde consideramos a localização geográfica dos eventos. Utilizamos uma modelagem estatística baseada em processos pontuais e técnicas de ondaletas para estimar a intensidade temporal ou espacial, isto é, o numero esperado de eventos por unidade de área (na abordagem espacial) ou tempo (na abordagem temporal). Fatores como sexo, faixa etária e tipo de evento (acidentes ou agressões) também foram considerados na análise. Na análise temporal, os resultados indicam que o número esperado de ocorrências em homens é significantemente maior do que em mulheres ao longo do período de observação. O mesmo ocorre com o numero esperado de acidentes quando comparado com o de agressões. As faixas etárias que compreendem as idades de 0 a 14 anos, 15 a 29 anos, 30 a 59 anos e 60 anos ou mais também apresentam números esperados de casos significantemente diferentes entre si. Na análise espacial, escolhemos uma região do Município de São Paulo, nas proximidades do Hospital Saboya, e elaboramos mapas onde é possível identificar geograficamente os locais onde as ocorrências são mais frequentes. A intensidade estimada para o total de eventos indica uma distribuição espacial não homogênea, com grande concentração de eventos principalmente nos distritos do Jabaquara e Cidade Ademar, além de valores altos ao longo das avenidas Bandeirantes, Jabaquara e Cupecê. As intensidades espaciais relativas às agressões a homens e a mulheres, separadamente, apresentam distribuições não homogêneas. Os locais com maiores riscos de agressões a mulheres parecem estar localizados em regiões mais afastadas das grandes avenidas da região. Quando consideramos os acidentes de trânsito e de transporte para cada dia da semana, a análise indicou uma distribuição espacial e temporal heterogênea, com intensidades estimadas maiores nos fins de semana e menores na segunda e terça-feira / We consider a data set with information on victims that were assisted at the emergency room of the Dr. Arthur Ribeiro de Saboya Municipal Hospital, S~ao Paulo, Brazil, from January 1, 2002 to January 11, 2003. We analyze the data chronologically (number of events along time) and spatially (geographical location). The statistical modelling is based on point processes and wavelet techniques to estimate both temporal and spatial intensities, that is, the expected numbers of events by unit time or unit area. The results indicate that the expected number of events is greater for men than for women along the whole observation period. The same is true for the expected number of accidents and that of aggressions, the former being consistently greater than the latter. The expected numbers of events for different age groups (0 to 14, 15 to 29, 30 to 59 and 60 or more) are significantly different. A neighborhood of Saboya Hospital was considered for spatial analysis, according to which it is possible to identify regions where occurrences are most frequent. The spatial distribution of the number of events is non homogeneous with high concentration mostly on Jabaquara and Cidade Ademar districts and along some big avenues (Bandeirantes, Jabaquara and Cupec^e avenues). Spatial non homogeneity of intensities is also observed for both aggressions to men and to women. The regions with the highest risks of aggression to women seam to be located away from the big avenues. When considering traffic and transport accidents separately by each day of the week, the analysis has shown both time and spatial non homogeneous distributions of events with highest estimated intensities during weekends and lowest ones on Monday and Tuesday

Consideramos uma base de dados hospitalares constituída por informações sobre vítimas de causas externas atendidas no Pronto Socorro do Hospital Municipal Dr. Arthur Ribeiro de Saboya, no período de 01/01/02 a 11/01/03, e registradas pelo Núcleo de Atenção à Vítima de Violência deste hospital. O conjunto de dados foi avaliado sob duas abordagens: a temporal, onde estudamos o numero de eventos ao longo do tempo, e a espacial, onde consideramos a localização geográfica dos eventos. Utilizamos uma modelagem estatística baseada em processos pontuais e técnicas de ondaletas para estimar a intensidade temporal ou espacial, isto é, o numero esperado de eventos por unidade de área (na abordagem espacial) ou tempo (na abordagem temporal). Fatores como sexo, faixa etária e tipo de evento (acidentes ou agressões) também foram considerados na análise. Na análise temporal, os resultados indicam que o número esperado de ocorrências em homens é significantemente maior do que em mulheres ao longo do período de observação. O mesmo ocorre com o numero esperado de acidentes quando comparado com o de agressões. As faixas etárias que compreendem as idades de 0 a 14 anos, 15 a 29 anos, 30 a 59 anos e 60 anos ou mais também apresentam números esperados de casos significantemente diferentes entre si. Na análise espacial, escolhemos uma região do Município de São Paulo, nas proximidades do Hospital Saboya, e elaboramos mapas onde é possível identificar geograficamente os locais onde as ocorrências são mais frequentes. A intensidade estimada para o total de eventos indica uma distribuição espacial não homogênea, com grande concentração de eventos principalmente nos distritos do Jabaquara e Cidade Ademar, além de valores altos ao longo das avenidas Bandeirantes, Jabaquara e Cupecê. As intensidades espaciais relativas às agressões a homens e a mulheres, separadamente, apresentam distribuições não homogêneas. Os locais com maiores riscos de agressões a mulheres parecem estar localizados em regiões mais afastadas das grandes avenidas da região. Quando consideramos os acidentes de trânsito e de transporte para cada dia da semana, a análise indicou uma distribuição espacial e temporal heterogênea, com intensidades estimadas maiores nos fins de semana e menores na segunda e terça-feira / We consider a data set with information on victims that were assisted at the emergency room of the Dr. Arthur Ribeiro de Saboya Municipal Hospital, S~ao Paulo, Brazil, from January 1, 2002 to January 11, 2003. We analyze the data chronologically (number of events along time) and spatially (geographical location). The statistical modelling is based on point processes and wavelet techniques to estimate both temporal and spatial intensities, that is, the expected numbers of events by unit time or unit area. The results indicate that the expected number of events is greater for men than for women along the whole observation period. The same is true for the expected number of accidents and that of aggressions, the former being consistently greater than the latter. The expected numbers of events for different age groups (0 to 14, 15 to 29, 30 to 59 and 60 or more) are significantly different. A neighborhood of Saboya Hospital was considered for spatial analysis, according to which it is possible to identify regions where occurrences are most frequent. The spatial distribution of the number of events is non homogeneous with high concentration mostly on Jabaquara and Cidade Ademar districts and along some big avenues (Bandeirantes, Jabaquara and Cupec^e avenues). Spatial non homogeneity of intensities is also observed for both aggressions to men and to women. The regions with the highest risks of aggression to women seam to be located away from the big avenues. When considering traffic and transport accidents separately by each day of the week, the analysis has shown both time and spatial non homogeneous distributions of events with highest estimated intensities during weekends and lowest ones on Monday and Tuesday

This thesis consists of an introduction and five papers, of which two contribute to the theory of directed random graphs and three to the theory of greedy walks on point processes.           We consider a directed random graph on a partially ordered vertex set, with an edge between any two comparable vertices present with probability p, independently of all other edges, and each edge is directed from the vertex with smaller label to the vertex with larger label. In Paper I we consider a directed random graph on ℤ2 with the vertices ordered according to the product order and we show that the limiting distribution of the centered and rescaled length of the longest path from (0,0) to (n, [na] ), a<3/14, is the Tracy-Widom distribution. In Paper II we show that, under a suitable rescaling, the closure of vertex 0 of a directed random graph on ℤ with edge probability n−1 converges in distribution to the Poisson-weighted infinite tree. Moreover, we derive limit theorems for the length of the longest path of the Poisson-weighted infinite tree.           The greedy walk is a deterministic walk on a point process that always moves from its current position to the nearest not yet visited point. Since the greedy walk on a homogeneous Poisson process on the real line, starting from 0, almost surely does not visit all points, in Paper III we find the distribution of the number of visited points on the negative half-line and the distribution of the index at which the walk achieves its minimum. In Paper IV we place homogeneous Poisson processes first on two intersecting lines and then on two parallel lines and we study whether the greedy walk visits all points of the processes. In Paper V we consider the greedy walk on an inhomogeneous Poisson process on the real line and we determine sufficient and necessary conditions on the mean measure of the process for the walk to visit all points.

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.

Thesis. 1976. Ph.D.--Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science. / Microfiche copy available in Archives and Engineering. / Vita. / Includes bibliographical references. / by Samuel Joseph Dolinar, Jr. / Ph.D.

Processing user requests quickly requires not only fast servers, but also demands methods to quickly locate idle servers to process those requests. Methods of finding idle servers are analogous to open addressing in hash tables, but with the key difference that servers may return to an idle state after having been busy rather than staying busy. Probing sequences for open addressing are well-studied, but algorithms for locating idle servers are less understood. We investigate sequential probing with a random start as a method for finding idle servers, especially in cases of heavy traffic. We present a procedure for finding the distribution of the number of probes required for finding an idle server by using a Markov chain and ideas from enumerative combinatorics, then present numerical simulation results in lieu of a general analytic solution.

Interaction event times observed on a social network provide valuable information for social scientists to gain insight into complex social dynamics that are challenging to understand. However, it can be difficult to accurately represent the heterogeneity in the data and to model the dependence structure in the network system. This requires flexible models that can capture the complicated dynamics and complex patterns. Point process models offer an elegant framework for modeling event time data. This dissertation concentrates on developing point process models and related diagnostic tools, with a real data application involving an animal behavior network. In this dissertation, we first propose a Markov-modulated Hawkes process (MMHP) model to capture the sporadic and bursty patterns often observed in event time data. A Bayesian inference procedure is developed to evaluate the likelihood by using a variational approximation and the forward-backward algorithm. The validity of the proposed model and associated estimation algorithms is demonstrated using synthetic data and the animal behavior data. Facilitated by the power of the MMHP model, we construct network point process models that can capture a social hierarchy structure by embedding nodes in a latent space that can represent the underlying social ranks. Our model provides a ranking method for social hierarchy studies and describes the dynamics of social hierarchy formation from a novel perspective – taking advantage of the detailed information available in event time data. We show that the network point process models appropriately captures the temporal dynamics and heterogeneity in the network event time data, by providing meaningful inferred rankings and by calibrating the accuracy of predictions with relevant measures of uncertainty. In addition to developing a sensible and flexible model for network event time data, the last part of this dissertation provides essential tools for diagnosing lack of fit issues for such models. We develop a systematic set of diagnostic tools and visualizations for point process models fitted to data in the dynamic network setting. By inspecting the structure of the residual process and Pearson residual on the network, we can validate whether a model adequately captures the temporal and network dependence structures in the observed data.

optimizationover a discrete number of random variables. In this paperwe extend this theory from the discrete to the continuous case, andconsider the limiting distribution of the location of the best offer asthe number of offers tends to infinity.Given a set   Rd of possible offers we seek a distribution over ,the argmax measure of the best offer. It depends on , the samplingdistribution of offer locations, and a measure index , which assignsto each point x 2  a probability distribution of offers.This problem is closely related to argmax theory of marked pointprocesses, altough we consider deterministic sequences of points inspace, to allow for greater generality. We first define a finite sampleargmax measure and then give conditions under which it converges asthe number of offers tends to infinity.To this end, we introduce a max-field of best offers and use continuityproperties of this field to calculate the argmax measure. Wedemonstrate the usefulness of the method by giving explicit formulasfor the limiting argmax distribution for a large class of models, includingexponential independent offers with a deterministic, additivedisturbance term. Finally, we illustrate the theory by simulations.