The Gaussian geostatistical model has been widely used in Bayesian modeling of spatial data. A core difficulty for this model is at inverting the n x n covariance matrix, where n is a sample size. The computational complexity of matrix inversion increases as O(n3). This difficulty is involved in almost all statistical inferences approaches of the model, such as Kriging and Bayesian modeling. In Bayesian inference, the inverse of covariance matrix needs to be evaluated at each iteration in posterior simulations, so Bayesian approach is infeasible for large sample size n due to the current computational power limit. In this dissertation, we propose two approaches to address this computational issue, namely, the auxiliary lattice model (ALM) approach and the Bayesian site selection (BSS) approach. The key feature of ALM is to introduce a latent regular lattice which links Gaussian Markov Random Field (GMRF) with Gaussian Field (GF) of the observations. The GMRF on the auxiliary lattice represents an approximation to the Gaussian process. The distinctive feature of ALM from other approximations lies in that ALM avoids completely the problem of the matrix inversion by using analytical likelihood of GMRF. The computational complexity of ALM is rather attractive, which increase linearly with sample size. The second approach, Bayesian site selection (BSS), attempts to reduce the dimension of data through a smart selection of a representative subset of the observations. The BSS method first split the observations into two parts, the observations near the target prediction sites (part I) and their remaining (part II). Then, by treating the observations in part I as response variable and those in part II as explanatory variables, BSS forms a regression model which relates all observations through a conditional likelihood derived from the original model. The dimension of the data can then be reduced by applying a stochastic variable selection procedure to the regression model, which selects only a subset of the part II data as explanatory data. BSS can provide us more understanding to the underlying true Gaussian process, as it directly works on the original process without any approximations involved. The practical performance of ALM and BSS will be illustrated with simulated data and real data sets.

L’objet de cette thèse est d’étudier la probabilité de survie d’un processus de branchement en environnement aléatoire markovien et d’étendre dans ce cadre les résultats connus en milieu aléatoire i.i.d.. le cœur de l’étude repose sur l’utilisation des théorèmes limites locaux pour une marche aléatoire centrée (Sn)n≥0 sur R à pas markoviens et pour (rnn)n≥0, où mn = min (0, S1,... , Sn). Pour traiter le cas d’un environnement aléatoire markovien, nous développons dans un premier temps une étude des théorèmes locaux pour une chaîne semi-markovienne à valeurs réelles en améliorant certains résultats déjà connus et développés initialement par E. L. Presman (voir aussi [21]). Nous utilisons ensuite ces résultats pour l’étude du comportement asymptotique de la probabilité de survie d’un processus de branchement critique en environnement aléatoire markovien. Les résultats principaux de cette thèse (théorème limite local et son application au processus de branchement critique eu milieu aléatoire) ont été acceptés et publiés dans le Comptes Rendus de l‘Académie des Sciences ([20]). Le texte principal de cette mémoire de thèse consisite les détails des preuves. / The purpose of this thesis is to study the survival probability of a branching process in markovian random environment and expand in this framework some known results which have been developed for a branching processus in i.i.d. random environment, the core of the study is based on the use of the local limit theorem for a centered random walk (Sn)n≥o on R with markovian increasements and for (mn)n≥0. where mn = min (O. S1,……. , Sn). In order to treat the case of a markovian random environment, we establish firstly a local limit theorem for a semi-markovian chain on R. which improves certain results developed initially by E. P. Presman (see also [21]). And then we use these results to study the asymptotic behavior of a critical branching process in markovian environment. The main results et this thesis (local limit theorem and its application to the critical branching process in random environment) are accepted and published in Comptes Rendus de l’Académie des Sciences ([20]). The principal text et this thesis contains the details of the proofs.

Organisms have movements that are usually modeled by particles’ random walks. Under some mathematical technical assumptions the movements are described by diffusion equations. However, empirical data often show that the movements are not simple random walks. Instead, they are correlated random walks and are described by telegraph equations. This thesis considers telegraph equations with and without bias corresponding to correlated random walks with and without bias. Analytical solutions to the equations with absorbing boundary conditions and their mean passage times are obtained. Numerical simulations of the corresponding correlated random walks are also performed. The simulation results show that the solutions are approximated very well by the corresponding correlated random walks and the mean first passage times are highly consistent with those from simulations on the corresponding random walks. This suggests that telegraph equations can be a good model for organisms with the movement pattern of correlated random walks. Furthermore, utilizing the consistency of mean first passage times, we can estimate the parameters of telegraph equations through the mean first passage time, which can be estimated through experimental observation. This provides biologists an easy way to obtain parameter values. Finally, this thesis analyzes the velocity distribution and correlations of movement steps of amoebas, leaving fitting the movement data to telegraph equations as future work.

O presente trabalho estuda o modelo de Blume-Capel na rede aleatória e também analisa a inclusão de um termo de campo cristalino aleatório e de um termo de campo local aleatório. Ao resolver o modelo na rede aleatória, uma técnica de conectividade finita foi utilizada, na qual cada spin é conectado a um número finito de outros spins. Os spins foram conectados de acordo com uma distribuição de Poisson, os termos de campo aleatório seguiram uma distribuição bimodal e as interações entre os spins foram consideradas uniformes. Desse modo, só há desordem nas conexões entre os spins. O foco desse trabalho foi determinar como a natureza da transição de fase é alterada com a conectividade e se há um comportamento reentrante das linhas de transição de fase. A técnica de réplicas é usada para obter equações de ponto de sela para a distribuição de campos locais. Um Ansatz de simetria de réplicas foi utilizado para a função de ordem e esse foi escrito em termos de uma distribuição bidimensional de campos efetivos, onde uma das componentes é associada com um termo linear dos spins e a outra com o termo de campo cristalino. Com isso, equações para as funções de ordem e a energia livre podem ser obtidas. Uma técnica de dinâmica populacional é usada para resolver numericamente a equação auto-consistente para a distribuição de campos locais e outros parâmetros, como a magnetização, a atividade da rede e a energia livre. Os resultados indicam que a natureza da transição ferromagnética-paramagnética, a posição do ponto tricrítico e a existência de reentrância dependem fortemente do valor da conectividade e, nos casos com um termo de campo aleatório, dependem da intensidade dos campos aleatórios. No caso em que o campo cristalino é aleatório, o ponto tricrítico é suprimido para valores acima de um certo valor de aleatoriedade. / The present work studies the Blume-Capel model in a random network and also analyses the inclusion of a random crystal-field term and a random field term. To solve the model in a random network a finite connectivity technique is used, in which each spin is connected to a finite number of other spins. The spins were connected according a Poisson distribution, the random field terms followed a bimodal distribution and the bonds between the spins were considered uniform. Thus, there is only a connection disorder. The focus of this work was on determining how the nature of the phase transition changes with the connectivity and the random fields and if there is a reentrant behavior of the phase boundaries. The replica technique is used to obtain saddle-point equations for the effective local-field distribution. The replica symmetric Ansatz for the order function is written in terms of a two-dimensional effective-field distribution, where one of the components is associated with a linear form in the spins and the other with the crystal-field term. This allows one to derive equations for the order function and for the free-energy. A population dynamics procedure is used to solve numerically a self-consistency equation for the distribution of the local field and with it some physical parameters, like magnetization and free-energy. The results obtained indicate that the nature of the F-P transition, the location of the tricritical point and the presence of a reentrant phase depend strongly on the connectivity. In the cases with a random field term, those are also dependent on the intensity of the fields. For the case with a random crystal-field term, the tricritical point is supressed above a certain value of randomness.

abstract: Optimal experimental design for generalized linear models is often done using a pseudo-Bayesian approach that integrates the design criterion across a prior distribution on the parameter values. This approach ignores the lack of utility of certain models contained in the prior, and a case is demonstrated where the heavy focus on such hopeless models results in a design with poor performance and with wild swings in coverage probabilities for Wald-type confidence intervals. Design construction using a utility-based approach is shown to result in much more stable coverage probabilities in the area of greatest concern. The pseudo-Bayesian approach can be applied to the problem of optimal design construction under dependent observations. Often, correlation between observations exists due to restrictions on randomization. Several techniques for optimal design construction are proposed in the case of the conditional response distribution being a natural exponential family member but with a normally distributed block effect . The reviewed pseudo-Bayesian approach is compared to an approach based on substituting the marginal likelihood with the joint likelihood and an approach based on projections of the score function (often called quasi-likelihood). These approaches are compared for several models with normal, Poisson, and binomial conditional response distributions via the true determinant of the expected Fisher information matrix where the dispersion of the random blocks is considered a nuisance parameter. A case study using the developed methods is performed. The joint and quasi-likelihood methods are then extended to address the case when the magnitude of random block dispersion is of concern. Again, a simulation study over several models is performed, followed by a case study when the conditional response distribution is a Poisson distribution. / Dissertation/Thesis / Doctoral Dissertation Industrial Engineering 2015

In the first part of this thesis, we study critical points of random polynomials. We choose two deterministic sequences of complex numbers, whose empirical measures converge to the same probability measure in complex plane. We make a sequence of polynomials whose zeros are chosen from either of sequences at random. We show that the limiting empirical measure of zeros and critical points agree for these polynomials. As a consequence we show that when we randomly perturb the zeros of a deterministic sequence of polynomials, the limiting empirical measures of zeros and critical points agree. This result can be interpreted as an extension of earlier results where randomness is reduced. Pemantle and Rivin initiated the study of critical points of random polynomials. Kabluchko proved the result considering the zeros to be i.i.d. random variables. In the second part we deal with the spectrum of products of Ginibre matrices. Exact eigenvalue density is known for a very few matrix ensembles. For the known ones they often lead to determinantal point process. Let X1, X2,..., Xk be i.i.d Ginibre matrices of size n ×n whose entries are standard complex Gaussian random variables. We derive eigenvalue density for matrices of the form X1 ε1 X2 ε2 ... Xk εk , where εi = ±1 for i =1,2,..., k. We show that the eigenvalues form a determinantal point process. The case where k =2, ε1 +ε2 =0 was derived earlier by Krishnapur. In the case where εi =1 for i =1,2,...,n was derived by Akemann and Burda. These two known cases can be obtained as special cases of our result.

O presente trabalho estuda o modelo de Blume-Capel na rede aleatória e também analisa a inclusão de um termo de campo cristalino aleatório e de um termo de campo local aleatório. Ao resolver o modelo na rede aleatória, uma técnica de conectividade finita foi utilizada, na qual cada spin é conectado a um número finito de outros spins. Os spins foram conectados de acordo com uma distribuição de Poisson, os termos de campo aleatório seguiram uma distribuição bimodal e as interações entre os spins foram consideradas uniformes. Desse modo, só há desordem nas conexões entre os spins. O foco desse trabalho foi determinar como a natureza da transição de fase é alterada com a conectividade e se há um comportamento reentrante das linhas de transição de fase. A técnica de réplicas é usada para obter equações de ponto de sela para a distribuição de campos locais. Um Ansatz de simetria de réplicas foi utilizado para a função de ordem e esse foi escrito em termos de uma distribuição bidimensional de campos efetivos, onde uma das componentes é associada com um termo linear dos spins e a outra com o termo de campo cristalino. Com isso, equações para as funções de ordem e a energia livre podem ser obtidas. Uma técnica de dinâmica populacional é usada para resolver numericamente a equação auto-consistente para a distribuição de campos locais e outros parâmetros, como a magnetização, a atividade da rede e a energia livre. Os resultados indicam que a natureza da transição ferromagnética-paramagnética, a posição do ponto tricrítico e a existência de reentrância dependem fortemente do valor da conectividade e, nos casos com um termo de campo aleatório, dependem da intensidade dos campos aleatórios. No caso em que o campo cristalino é aleatório, o ponto tricrítico é suprimido para valores acima de um certo valor de aleatoriedade. / The present work studies the Blume-Capel model in a random network and also analyses the inclusion of a random crystal-field term and a random field term. To solve the model in a random network a finite connectivity technique is used, in which each spin is connected to a finite number of other spins. The spins were connected according a Poisson distribution, the random field terms followed a bimodal distribution and the bonds between the spins were considered uniform. Thus, there is only a connection disorder. The focus of this work was on determining how the nature of the phase transition changes with the connectivity and the random fields and if there is a reentrant behavior of the phase boundaries. The replica technique is used to obtain saddle-point equations for the effective local-field distribution. The replica symmetric Ansatz for the order function is written in terms of a two-dimensional effective-field distribution, where one of the components is associated with a linear form in the spins and the other with the crystal-field term. This allows one to derive equations for the order function and for the free-energy. A population dynamics procedure is used to solve numerically a self-consistency equation for the distribution of the local field and with it some physical parameters, like magnetization and free-energy. The results obtained indicate that the nature of the F-P transition, the location of the tricritical point and the presence of a reentrant phase depend strongly on the connectivity. In the cases with a random field term, those are also dependent on the intensity of the fields. For the case with a random crystal-field term, the tricritical point is supressed above a certain value of randomness.

Submitted by Neusa Fagundes (neusa.fagundes@unioeste.br) on 2018-09-06T19:38:50Z No. of bitstreams: 2 Weverton_Verica2018.pdf: 4544186 bytes, checksum: 766200b4dea97433d3d88b08cbe3e548 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-09-06T19:38:50Z (GMT). No. of bitstreams: 2 Weverton_Verica2018.pdf: 4544186 bytes, checksum: 766200b4dea97433d3d88b08cbe3e548 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-02-16 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Knowledge of location and quantity of areas for agriculture or either native or planted forests is relevant for public managers to make their decisions based on reliable data. In addition, part of ICMS revenues from the Municipal Participation Fund (FPM) depends on agricultural production data, number of rural properties and the environmental factor. The objective of this research was to design an objective and semiautomatic methodology to map agricultural areas and targets permanent, and later to identify areas of soybean, corn 1st and 2nd crops, winter crops, semi-perennial agriculture, forests and other permanent targets in the state of Paraná for the harvest years (2013/14 to 2016/17), using temporal series of EVI/Modis vegetation indexes. The proposed methodology follows the steps of the Knowledge Discovery Process in Database – KDD, in which the classification task was performed by the Random Forest algorithm. For the validation of the mappings, samples extracted from Landsat-8 images were used, obtaining the global accuracy indices greater than 84.37% and a kappa index ranging from 0.63 to 0.98, hence considered mappings with good or excellent spatial accuracy. The municipal data of the area of soybean, corn 1st crop, corn 2nd crop and winter crops mapped were confronted with the official statistics obtaining coefficients of linear correlation between 0.61 to 0.9, indicating moderate or strong correlation with the data officials. In this way, the proposed semi-automatic methodology was successful in the mapping, as well as the automation of the process of elaboration of the metrics, thus generating a script in the software R in order to facilitate future mappings with low processing time. / O conhecimento da localização e da quantidade de áreas destinadas a agricultura ou a florestas nativas ou plantadas é relevante para que os gestores públicos tomem suas decisões pautadas em dados fidedignos com a realidade. Além disto, parte das receitas de ICMS advindas do Fundo de Participação aos Municípios (FPM) depende de dados de produção agropecuária, número de propriedades rurais e fator ambiental. Diante disso, esta dissertação teve como objetivo elaborar uma metodologia objetiva e semiautomática para mapear áreas agrícolas e alvos permanente e posteriormente identificar áreas de soja, milho 1ª e 2ª safras, culturas de inverno, agricultura semi-perene, florestas e demais alvos permanentes no estado do Paraná para os anos-safra (2013/14 a 2016/17), utilizando séries temporais de índices de vegetação EVI/Modis. A metodologia proposta segue os passos do Processo de descoberta de conhecimento em base de dados – KDD, sendo que para isso foram elaboradas métricas extraídas do perfil espectro temporal de cada pixel e foi empregada a tarefa de classificação, realizada pelo algoritmo Random Forest. Para a validação dos mapeamentos utilizaram-se amostras extraídas de imagens Landsat-8, obtendo-se os índices de exatidão global maior que 84,37% e um índice kappa variando entre 0,63 e 0,98, sendo, portanto, considerados mapeamentos com boa ou excelente acurácia espacial. Os dados municipais da área de soja, milho 1ª safra, milho 2ª safra e culturas de inverno mapeada foram confrontados com as estatísticas oficiais obtendo-se coeficientes de correlação linear entre 0,61 a 0,9, indicando moderada ou forte correlação com os dados oficiais. Desse modo, a metodologia semiautomática proposta obteve êxito na realização do mapeamento, bem como a automatização do processo de elaboração das métricas, gerando, com isso um script no software R de maneira a facilitar mapeamentos futuros com baixo tempo de processamento.

Orientador: Serguei Popov / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T10:22:43Z (GMT). No. of bitstreams: 1 Bernardini_DiegoFernandode_D.pdf: 1410086 bytes, checksum: b77a17aefd06d547f1c5db3c5cc1a8f7 (MD5) Previous issue date: 2014 / Resumo: O modelo de entrelaçamentos aleatórios foi introduzido no sentido de se investigar originalmente o traço deixado por passeios aleatórios em grandes grafos e, basicamente, tal processo é descrito por um processo pontual de Poisson em um espaço de trajetórias duplamente infinitas de passeios aleatórios simples no reticulado d-dimensional, com dimensão d pelo menos igual a três. Neste sentido, o processo é caracterizado por um emaranhado aleatório de trajetórias deste tipo. Tal modelo possui ainda um parâmetro de intensidade, que controla, de certa forma, a quantidade de trajetórias que constituem o processo. Um problema relevante no contexto deste processo, e que tem sido amplamente estudado na literatura, diz respeito à caracterização da relação de dependência (através da covariância) entre os eventos denominados como crescentes neste modelo e suportados em subconjuntos disjuntos do reticulado, e é justamente este o problema no qual nos concentramos. Em uma primeira etapa neste trabalho, determinamos expressões explícitas para a derivada, com respeito ao parâmetro de intensidade, da probabilidade de um evento crescente e suportado em um subconjunto finito do reticulado, estabelecendo assim aquilo que denominamos como a fórmula de Russo para os entrelaçamentos aleatórios. A utilização desta denominação é justificada e motivada pelo amplamente conhecido termo original, que no contexto do modelo usual de percolação estabelece uma expressão para a derivada da probabilidade dos eventos definidos como crescentes naquele modelo. Em seguida, tentamos utilizar este resultado no sentido de estabelecer uma primeira abordagem para o problema da covariância entre os eventos crescentes, e esta investigação é baseada essencialmente em uma observação sobre o número esperado das trajetórias então denominadas como pivotais positivas para o evento de interesse. Por fim, estabelecemos uma nova abordagem para o mesmo problema, utilizando uma construção alternativa do processo de entrelaçamentos baseada na técnica dos soft local times, e investigando uma espécie de pivotalidade conjunta de coleções de excursões das trajetórias dos passeios aleatórios pelos conjuntos nos quais estão suportados os eventos de interesse. Justamente a partir desta abordagem obtemos nosso último resultado sobre a covariância. De forma geral, acreditamos que a investigação e a tentativa de obter uma caracterização cada vez mais precisa para a relação de dependência que mencionamos deve ajudar a entender o processo de entrelaçamentos e suas propriedades de forma cada vez mais clara / Abstract: The random interlacements model was originally introduced in order to investigate the trace left by random walks in large graphs and, basically, such process is described by a Poisson point process in a space of doubly infinite simple random walk trajectories in the d-dimensional lattice, with dimension d at least equal to three. In this sense, the process is characterized by a random tangle of trajectories of this kind. Such model also has an intensity parameter, which controls, in a certain sense, the quantity of trajectories that constitutes the process. A relevant issue in the context of this process, which has been largely studied in the literature, concerns the characterization of the dependence relation (through the covariance) between the so-called increasing events in this model, which are supported on disjoint subsets of the lattice, and this is precisely the issue on which we focus. In a first step in this work, we determine explicit expressions for the derivative, with respect to the intensity parameter, of the probability of an increasing event which is supported in a finite subset of the lattice, thus establishing what we call as Russo¿s formula for random interlacements. The use of this term is justified and motivated by the widely known original term, which, in the context of the usual percolation model, provides an expression for the derivative of the probability of events defined as increasing in that model. Then, we try to use this result to establish a first approach to the problem of the covariance between increasing events, and such investigation is essentially based in a fact about the expected number of the so-called positive pivotal (or plus pivotal) trajectories for the event of interest. Finally, we establish a new approach to the same problem by using an alternative construction of the interlacements process based on the technique of soft local times, and investigating a kind of joint "pivotality" of collections of excursions of the random walk trajectories, through the sets on which the events of interest are supported. From this approach we obtain our last result on the covariance. Overall, we believe that the investigation and the attempt to get an increasingly accurate characterization of the above mentioned dependence relation should help to understand the interlacements process and its properties in an increasingly clear way / Doutorado / Estatistica / Doutor em Estatística

We reinvestigate the validity of mapping the problem of two onsite interacting particles in a random potential onto an effective random matrix model. To this end we first study numerically how the non-interacting basis is coupled by the interaction. Our results indicate that the typical coupling matrix element decreases significantly faster with increasing single-particle localization length than is assumed in the random matrix model. We further show that even for models where the dependency of the coupling matrix element on the single-particle localization length is correctly described by the corresponding random matrix model its predictions for the localization length can be qualitatively incorrect. These results indicate that the mapping of an interacting random system onto an effective random matrix model is potentially dangerous. We also discuss how Imry's block-scaling picture for two interacting particles is influenced by the above arguments.