This dissertation introduces new methodologies for the analysis of neural spike trains. Biological properties of the nervous system, and how they are reflected in neural data, can motivate specific analytic tools. Some of these biological aspects motivate multiscale frameworks, which allow for simultaneous modelling of the local and global behaviour of neurons. Chapter 1 provides the preliminary background on the biology of the nervous system and details the concept of information and randomness in the analysis of the neural spike trains. It also provides the reader with a thorough literature review on the current statistical models in the analysis of neural spike trains. The material presented in the next six chapters (2-7) have been the focus of three papers, which have either already been published or are being prepared for publication. It is demonstrated in Chapters 2 and 3 that the multiscale complexity penalized likelihood method, introduced in Kolaczyk and Nowak (2004), is a powerful model in the simultaneous modelling of spike trains with biological properties from different time scales. To detect the periodic spiking activities of neurons, two periodic models from the literature, Bickel et al. (2007, 2008); Shao and Li (2011), were combined and modified in a multiscale penalized likelihood model. The contributions of these chapters are (1) employinh a powerful visualization tool, inter-spike interval (ISI) plot, (2) combining the multiscale method of Kolaczyk and Nowak (2004) with the periodic models ofBickel et al. (2007, 2008) and Shao and Li (2011), to introduce the so-called additive and multiplicative models for the intensity function of neural spike trains and introducing a cross-validation scheme to estimate their tuning parameters, (3) providing the numerical bootstrap confidence bands for the multiscale estimate of the intensity function, and (4) studying the effect of time-scale on the statistical properties of spike counts. Motivated by neural integration phenomena, as well as the adjustments for the neural refractory period, Chapters 4 and 5 study the Skellam process and introduce the Skellam Process with Resetting (SPR). Introducing SPR and its application in the analysis of neural spike trains is one of the major contributions of this dissertation. This stochastic process is biologically plausible, and unlike the Poisson process, it does not suffer from limited dependency structure. It also has multivariate generalizations for the simultaneous analysis of multiple spike trains. A computationally efficient recursive algorithm for the estimation of the parameters of SPR is introduced in Chapter 5. Except for the literature review at the beginning of Chapter 4, the rest of the material within these two chapters is original. The specific contributions of Chapters 4 and 5 are (1) introducing the Skellam Process with Resetting as a statistical tool to analyze neural spike trains and studying its properties, including all theorems and lemmas provided in Chapter 4, (2) the two fairly standard definitions of the Skellam process (homogeneous and inhomogeneous) and the proof of their equivalency, (3) deriving the likelihood function based on the observable data (spike trains) and developing a computationally efficient recursive algorithm for parameter estimation, and (4) studying the effect of time scales on the SPR model. The challenging problem of multivariate analysis of the neural spike trains is addressed in Chapter 6. As far as we know, the multivariate models which are available in the literature suffer from limited dependency structures. In particular, modelling negative correlation among spike trains is a challenging problem. To address this issue, the multivariate Skellam distribution, as well as the multivariate Skellam process, which both have flexible dependency structures, are developed. Chapter 5 also introduces a multivariate version of Skellam Process with Resetting (MSPR), and a so-called profile-moment likelihood estimation of its parameters. This chapter generalizes the results of Chapter 4 and 5, and therefore, except for the brief literature review provided at the beginning of the chapter, the remainder of the material is original work. In particular, the contributions of this chapter are (1) introducing multivariate Skellam distribution, (2) introducing two definitions of the Multivariate Skellam process in both homogeneous and inhomogeneous cases and proving their equivalence, (3) introducing Multivariate Skellam Process with Resetting (MSPR) to simultaneously model spike trains from an ensemble of neurons, and (4) utilizing the so-called profile-moment likelihood method to compute estimates of the parameters of MSPR. The discussion of the developed methodologies as well as the ``next steps'' are outlined in Chapter 7.

O padrão espacial de espécies em florestas nativas pode fornecer evidências sobre a estrutura da comunidade vegetal. Fatores ambientais podem influenciar o padrão espacial das espécies, como as características edáficas e processos que dependem da densidade, como competição intra e interespecífica. Desse modo, a pesquisa da relação entre as características ambientais e o padrão espacial de espécies florestais pode ajudar a entender a dinâmica de florestas. O objetivo deste estudo foi empregar técnicas da análise de processos pontuais para verificar o efeito de fatores ambientais sobre a ocorrência de espécies florestais. A área de estudo foi a Estação Ecológica de Assis (EEA), da unidade de Conservação do Estado de São Paulo em parcelas permanentes, dentro do projeto \"Diversidade, dinâmica e conservação em florestas do Estado de São Paulo: 40 ha de parcelas permanentes\" do programa Biota da FAPESP. A descrição do padrão espacial das espécies mais abundantes na área de estudo foi avaliada pela função K proposta por Ripley e suas extensões para processo não homogêneos, por meio das coordenadas geográficas das espécies com circunferência na altura do peito igual ou superior a 15 cm. Modelos do Processo Poisson Homogêneo, Processo Poisson Não Homogêneos e do Processo Log Gaussiano de Cox foram ajustados para cada espécie. Foi utilizado o critério de AIC para selecionar o modelo que melhor se ajusta aos dados. Testes de diagnósticos dos modelos foram feitos utilizando a função K não homogênea sob a hipótese de Completa Aleatoriedade Espacial. Os resultados indicaram que as espécies mais abundantes na EEA apresentam um padrão de distribuição agregado, ou seja, o número esperado de indivíduos próximos de um evento qualquer é maior do que esperado para uma distribuição aleatória. Conforme esperado, os fatores ambientais desempenharam um importante papel para explicar a distribuição espacial das espécies, porém, os resultados indicaram que existe uma variação espacialmente estruturada que não foi incluída na análise que é imprescindível para um bom ajuste dos modelos. Portanto os resultados sugerem que outros fatores não incluídos nos modelos e dados disponíveis podem estar determinando os padrões espaciais além das (co)variáveis medidas. / The spatial pattern of species in native forests may provide evidence on the structure of the plant community. Environmental factors may influence the species\' spatial patterns, as well as soil characteristics and processes which depend on the density, as intraspecific and interspecific competition. Therefore, researching the relationship among the environmental features and the spatial pattern of the forest species may aid in understanding forest dynamics. The goal of this study was to apply point process techniques to verify the effect of environmental factors on the occurence of forest species. The study area was the \"Assis\'s Ecological Station\" (AES), of the \"Unit of conservation of the state of São Paulo in permanent plots\". The data was collected as part of the project entitled \"Diversity, dynamics and conservation in forests of São Paulo state: 40 ha of permanent plots\", from FAPESP\'s Biota program. The description of the spatial pattern of the most abundant species in the study area was assessed using Ripley\'s K function, using the species\' geographic coordinates with circumference at chest height equal or larger than 15 cm. Homogeneous and Non-Homogeneous Poisson Process models, as well as Cox Log Gaussian Process models were fitted to each species. Model selection was made using the Akaike information criterion. Diagnostics tests were made using the non-homogeneous K function under the hypothesis of complete spatial randomness. Results suggested that the most abundant species in the AES present an aggregate distribution pattern, i.e., the expected number of individuals next to any event is larger than the expected by a random distribution. As it was expected, environmental factors played a major role in explaining the spatial distribution of the species. However, results suggested that there is a spatially structured variation that was not included in the analysis and is needed to a good model fit. Therefore, further studies are needed to assess which environmental feature which was not considered in this study presents an effect on the occurence of these forest species

O padrão espacial de espécies em florestas nativas pode fornecer evidências sobre a estrutura da comunidade vegetal. Fatores ambientais podem influenciar o padrão espacial das espécies, como as características edáficas e processos que dependem da densidade, como competição intra e interespecífica. Desse modo, a pesquisa da relação entre as características ambientais e o padrão espacial de espécies florestais pode ajudar a entender a dinâmica de florestas. O objetivo deste estudo foi empregar técnicas da análise de processos pontuais para verificar o efeito de fatores ambientais sobre a ocorrência de espécies florestais. A área de estudo foi a Estação Ecológica de Assis (EEA), da unidade de Conservação do Estado de São Paulo em parcelas permanentes, dentro do projeto \"Diversidade, dinâmica e conservação em florestas do Estado de São Paulo: 40 ha de parcelas permanentes\" do programa Biota da FAPESP. A descrição do padrão espacial das espécies mais abundantes na área de estudo foi avaliada pela função K proposta por Ripley e suas extensões para processo não homogêneos, por meio das coordenadas geográficas das espécies com circunferência na altura do peito igual ou superior a 15 cm. Modelos do Processo Poisson Homogêneo, Processo Poisson Não Homogêneos e do Processo Log Gaussiano de Cox foram ajustados para cada espécie. Foi utilizado o critério de AIC para selecionar o modelo que melhor se ajusta aos dados. Testes de diagnósticos dos modelos foram feitos utilizando a função K não homogênea sob a hipótese de Completa Aleatoriedade Espacial. Os resultados indicaram que as espécies mais abundantes na EEA apresentam um padrão de distribuição agregado, ou seja, o número esperado de indivíduos próximos de um evento qualquer é maior do que esperado para uma distribuição aleatória. Conforme esperado, os fatores ambientais desempenharam um importante papel para explicar a distribuição espacial das espécies, porém, os resultados indicaram que existe uma variação espacialmente estruturada que não foi incluída na análise que é imprescindível para um bom ajuste dos modelos. Portanto os resultados sugerem que outros fatores não incluídos nos modelos e dados disponíveis podem estar determinando os padrões espaciais além das (co)variáveis medidas. / The spatial pattern of species in native forests may provide evidence on the structure of the plant community. Environmental factors may influence the species\' spatial patterns, as well as soil characteristics and processes which depend on the density, as intraspecific and interspecific competition. Therefore, researching the relationship among the environmental features and the spatial pattern of the forest species may aid in understanding forest dynamics. The goal of this study was to apply point process techniques to verify the effect of environmental factors on the occurence of forest species. The study area was the \"Assis\'s Ecological Station\" (AES), of the \"Unit of conservation of the state of São Paulo in permanent plots\". The data was collected as part of the project entitled \"Diversity, dynamics and conservation in forests of São Paulo state: 40 ha of permanent plots\", from FAPESP\'s Biota program. The description of the spatial pattern of the most abundant species in the study area was assessed using Ripley\'s K function, using the species\' geographic coordinates with circumference at chest height equal or larger than 15 cm. Homogeneous and Non-Homogeneous Poisson Process models, as well as Cox Log Gaussian Process models were fitted to each species. Model selection was made using the Akaike information criterion. Diagnostics tests were made using the non-homogeneous K function under the hypothesis of complete spatial randomness. Results suggested that the most abundant species in the AES present an aggregate distribution pattern, i.e., the expected number of individuals next to any event is larger than the expected by a random distribution. As it was expected, environmental factors played a major role in explaining the spatial distribution of the species. However, results suggested that there is a spatially structured variation that was not included in the analysis and is needed to a good model fit. Therefore, further studies are needed to assess which environmental feature which was not considered in this study presents an effect on the occurence of these forest species

Ce travail est consacré aux problèmes de testd’hypothèses pour les processus de Poisson nonhomogènes.L’objectif principal de ce travail est l’étude decomportement des différents tests dans le cas desmodèles statistiques singuliers. L’évolution de lasingularité de la fonction d'intensité est comme suit :régulière (l'information de Fisher finie), continue maisnon différentiable (singularité de type “cusp”),discontinue (singularité de type saut) et discontinueavec un saut de taille variable. Dans tous les cas ondécrit analytiquement les tests. Dans le cas d’un saut detaille variable, on présente également les propriétésasymptotiques des estimateurs.En particulier, on décrit les statistiques de tests, le choixdes seuils et le comportement des fonctions depuissance sous les alternatives locales. Le problèmeinitial est toujours le test d’une hypothèse simple contreune alternative unilatérale. La méthode principale est lathéorie de la convergence faible dans l’espace desfonctions discontinues. Cette théorie est appliquée àl’étude des processus de rapport de vraisemblancenormalisé dans les modèles singuliers considérés. Laconvergence faible du rapport de vraisemblance sousl’hypothèse et sous les alternatives vers les processuslimites correspondants nous permet de résoudre lesproblèmes mentionnés précédemment.Les résultats asymptotiques sont illustrés par dessimulations numériques contenant la construction destests, le choix des seuils et les fonctions de puissancessous les alternatives locales. / This work is devoted to the hypotheses testing problems for inhomogeneous Poisson processes.The main object of the work is the study of the behaviour of different tests in the case of singular statistical models. The “evolution of singularity” of the intensity function is the following: regular (finite Fisherinformation), continuous but not differentiable (“cusp”type singularity), discontinuous (jump type singularity)and discontinuous with variable jump size. In all thecases we describe analytically the tests. In the case ofvariable jump size we present as well the asymptoticproperties of the estimators.In particular we describe the test statistics, the choice ofthresholds and the form of the power functions for thelocal alternatives. The initial problem is always the testof a simple hypothesis against a one-sided alternative.The main tool is the weak convergence theory in thespace of discontinuous functions. This theory is appliedto the study of the normalized likelihood ratio processesin the considered singular models. The weakconvergence of the likelihood ratio processes underhypothesis and under alternatives to the correspondinglimit processes allows us to solve the mentioned aboveproblems.The asymptotic results are illustrated by numericalsimulations which contain the construction of the tests,the choice of the thresholds, and the power functions forlocal alternatives.

Ce travail est consacré au problème de construction des tests d'ajustement dans le cas des processus stochastiques observés en temps continu. Comme modèles d'observations, nous considérons les processus de diffusion avec « petit bruit » et ergodique et le processus de Poisson non homogène. Sous l'hypothèse nulle, nous traitons le cas où chaque modèle dépend d'un paramètre inconnu unidimensionnel et nous proposons l'estimateur de distance minimale pour ce paramètre. Notre but est la construction des tests d'ajustement « asymptotically distribution free » (ADF) de niveau asymtotique α ϵ (0,1) dans le cas de cette hypothèse paramétrique pour les modèles traités. Nous montrons alors que la limite de chaque statistique étudiée ne dépend ni du modèle ni du paramètre inconnu. Les tests d'ajustement basés sur ces statistiques sont donc ADF. L'objectif principal de ce travail est la construction d'une transformation linéaire spéciale. En particulier, nous résolvons l'équation de Fredholm du second type avec le noyau dégénéré. Sa solution nous permet de construire la transformation linéaire désirée. Ensuite, nous montrons que l'application de cette transformation aux statistiques de base étudiées dans chaque modèle nous aide à introduire des statistiques ayant la même limite (l'intégrale du carrée du processus de Wiener). Cette dernière est « distribution free » vu qu'elle ne dépend ni du modèle ni du paramètre inconnu. Par conséquent, nous proposons des tests d'ajustement ADF en se basant sur cette transformation linéaire pour les processus de diffusion avec « petit bruit » et ergodique et le processus de Poisson non homogène. / This work is devoted to the problem of the construction of several goodness of-fit (GoF) tests in the case of somestochastic processes observed in continuous time. As models of observations, we take "small noise" and ergodic diffusionprocesses and an inhomogeneous Poisson process. Under the null hypothesis, we treat the case where each model depends on an unknown one-dimensional parameter and we consider the minimum distance estimator for this parameter. Our goal is to propose "asymptotically distribution free" (ADF) GoF tests of asymptotic size α ϵ (0,1) in the case of the parametric null hypotheses for the considered models. Indeed, we show that the limit of each studied statistic does not depend on the model and the unknown parameter. Therefore, the tests based on these statistics are ADF.The main purpose of this work is to construct a special linear transformation. In particular, we solve Fredholm equation ofthe second kind with degenerated kernel. Its solution gives us the desired linear transformation. Next, we show that theapplication of this transformation to the basic statistics allows us to introduce statistics with the same limit (the integral of the square of the Wiener process). The latter is "distribution free" because it does not depend on the models and the unknown parameter. Therefore, we construct the ADF GoF tests which are based on this linear transformation for the diffusion ("small noise" and ergodic) and inhomogeneous Poisson processes.