Um alisador de máxima verosimilhança local (MVL) com o modelo de regressão de Poisson, para análise de regressão de dados de contagem

Dissertação apresentada como requisito parcial para obtenção do grau de Mestre em Estatística e Gestão de Informação.

Use of Poisson spatiotemporal regression models for the Brazilian Amazon Forest: malaria count data

INTRODUCTION: Malaria is a serious problem in the Brazilian Amazon region, and the detection of possible risk factors could be of great interest for public health authorities. The objective of this article was to investigate the association between environmental variables and the yearly registers of malaria in the Amazon region using Bayesian spatiotemporal methods. METHODS: We used Poisson spatiotemporal regression models to analyze the Brazilian Amazon forest malaria count for the period from 1999 to 2008. In this study, we included some covariates that could be important in the yearly prediction of malaria, such as deforestation rate. We obtained the inferences using a Bayesian approach and Markov Chain Monte Carlo (MCMC) methods to simulate samples for the joint posterior distribution of interest. The discrimination of different models was also discussed. RESULTS: The model proposed here suggests that deforestation rate, the number of inhabitants per km², and the human development index (HDI) are important in the prediction of malaria cases. CONCLUSIONS: It is possible to conclude that human development, population growth, deforestation, and their associated ecological alterations are conducive to increasing malaria risk. We conclude that the use of Poisson regression models that capture the spatial and temporal effects under the Bayesian paradigm is a good strategy for modeling malaria counts.

Optimal designs for mixed effects poisson regression models

Magdeburg, Univ., Fak. für Mathematik, Diss., 2010

Análise estatística da distribuição de Poisson

The general properties of POISSON distributions and their relations to the binomial distribuitions are discussed. Two methods of statistical analysis are dealt with in detail: X2-test. In order to carry out the X2-test, the mean frequency and the theoretical frequencies for all classes are calculated. Than the observed and the calculated frequencies are compared, using the well nown formula: f(obs) - f(esp) 2; i(esp). When the expected frequencies are small, one must not forget that the value of X2 may only be calculated, if the expected frequencies are biger than 5. If smaller values should occur, the frequencies of neighboroughing classes must ge pooled. As a second test reintroduced by BRIEGER, consists in comparing the observed and expected error standard of the series. The observed error is calculated by the general formula: δ + Σ f . VK n-1 where n represents the number of cases. The theoretical error of a POISSON series with mean frequency m is always ± Vm. These two values may be compared either by dividing the observed by the theoretical error and using BRIEGER's tables for # or by dividing the respective variances and using SNEDECOR's tables for F. The degree of freedom for the observed error is one less the number of cases studied, and that of the theoretical error is always infinite. In carrying out these tests, one important point must never be overlloked. The values for the first class, even if no concrete cases of the type were observed, must always be zero, an dthe value of the subsequent classes must be 1, 2, 3, etc.. This is easily seen in some of the classical experiments. For instance in BORKEWITZ example of accidents in Prussian armee corps, the classes are: no, one, two, etc., accidents. When counting the frequency of bacteria, these values are: no, one, two, etc., bacteria or cultures of bacteria. Ins studies of plant diseases equally the frequencies are : no, one, two, etc., plants deseased. Howewer more complicated cases may occur. For instance, when analising the degree of polyembriony, frequently the case of "no polyembryony" corresponds to the occurrence of one embryo per each seed. Thus the classes are not: no, one, etc., embryo per seed, but they are: no additional embryo, one additional embryo, etc., per seed with at least one embryo. Another interestin case was found by BRIEGER in genetic studies on the number os rows in maize. Here the minimum number is of course not: no rows, but: no additional beyond eight rows. The next class is not: nine rows, but: 10 rows, since the row number varies always in pairs of rows. Thus the value of successive classes are: no additional pair of rows beyond 8, one additional pair (or 10 rows), two additional pairs (or 12 rows) etc.. The application of the methods is finally shown on the hand of three examples : the number of seeds per fruit in the oranges M Natal" and "Coco" and in "Calamondin". As shown in the text and the tables, the agreement with a POISSON series is very satisfactory in the first two cases. In the third case BRIEGER's error test indicated a significant reduction of variability, and the X2 test showed that there were two many fruits with 4 or 5 seeds and too few with more or with less seeds. Howewer the fact that no fruit was found without seed, may be taken to indicate that in Calamondin fruits are not fully parthenocarpic and may develop only with one seed at the least. Thus a new analysis was carried out, on another class basis. As value for the first class the following value was accepted: no additional seed beyond the indispensable minimum number of one seed, and for the later classes the values were: one, two, etc., additional seeds. Using this new basis for all calculations, a complete agreement of the observed and expected frequencies, of the correspondig POISSON series was obtained, thus proving that our hypothesis of the impossibility of obtaining fruits without any seed was correct for Calamondin while the other two oranges were completely parthenocarpic and fruits without seeds did occur.

An extension of the Marsden-Ratiu reduction for Poisson manifolds

We propose a generalization of the reduction of Poisson manifolds by distributions introduced by Marsden and Ratiu. Our proposal overcomes some of the restrictions of the original procedure, and makes the reduced Poisson structure effectively dependent on the distribution. Different applications are discussed, as well as the algebraic interpretation of the procedure and its formulation in terms of Dirac structures.

Formal deformations of Poisson structures in low dimensions

In this paper, we study formal deformations of Poisson structures, especially for three families of Poisson varieties in dimensions two and three. For these families of Poisson structures, using an explicit basis of the second Poisson cohomology space, we solve the deformation equations at each step and obtain a large family of formal deformations for each Poisson structure which we consider. With the help of an explicit formula, we show that this family contains, modulo equivalence, all possible formal eformations. We show moreover that, when the Poisson structure is generic, all members of the family are non-equivalent.

A priori ratemaking using bivariate poisson regression models

In automobile insurance, it is useful to achieve a priori ratemaking by resorting to gene- ralized linear models, and here the Poisson regression model constitutes the most widely accepted basis. However, insurance companies distinguish between claims with or without bodily injuries, or claims with full or partial liability of the insured driver. This paper exa- mines an a priori ratemaking procedure when including two di®erent types of claim. When assuming independence between claim types, the premium can be obtained by summing the premiums for each type of guarantee and is dependent on the rating factors chosen. If the independence assumption is relaxed, then it is unclear as to how the tari® system might be a®ected. In order to answer this question, bivariate Poisson regression models, suitable for paired count data exhibiting correlation, are introduced. It is shown that the usual independence assumption is unrealistic here. These models are applied to an automobile insurance claims database containing 80,994 contracts belonging to a Spanish insurance company. Finally, the consequences for pure and loaded premiums when the independence assumption is relaxed by using a bivariate Poisson regression model are analysed.

Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system

We construct a new family of semi-discrete numerical schemes for the approximation of the one-dimensional periodic Vlasov-Poisson system. The methods are based on the coupling of discontinuous Galerkin approximation to the Vlasov equation and several finite element (conforming, non-conforming and mixed) approximations for the Poisson problem. We show optimal error estimates for the all proposed methods in the case of smooth compactly supported initial data. The issue of energy conservation is also analyzed for some of the methods.

Modelling dependence in a ratemaking procedure with multivariate Poisson regression models

When actuaries face with the problem of pricing an insurance contract that contains different types of coverage, such as a motor insurance or homeowner's insurance policy, they usually assume that types of claim are independent. However, this assumption may not be realistic: several studies have shown that there is a positive correlation between types of claim. Here we introduce different regression models in order to relax the independence assumption, including zero-inflated models to account for excess of zeros and overdispersion. These models have been largely ignored to multivariate Poisson date, mainly because of their computational di±culties. Bayesian inference based on MCMC helps to solve this problem (and also lets us derive, for several quantities of interest, posterior summaries to account for uncertainty). Finally, these models are applied to an automobile insurance claims database with three different types of claims. We analyse the consequences for pure and loaded premiums when the independence assumption is relaxed by using different multivariate Poisson regression models and their zero-inflated versions.

Discontinuous Galerkin methods for the Multi-dimensional Vlasov-Poisson problem

We introduce and analyze two new semi-discrete numerical methods for the multi-dimensional Vlasov-Poisson system. The schemes are constructed by combing a discontinuous Galerkin approximation to the Vlasov equation together with a mixed finite element method for the Poisson problem. We show optimal error estimates in the case of smooth compactly supported initial data. We propose a scheme that preserves the total energy of the system.

Extended truncated Tweedie-Poisson model

It has been argued that by truncating the sample space of the negative binomial and of the inverse Gaussian-Poisson mixture models at zero, one is allowed to extend the parameter space of the model. Here that is proved to be the case for the more general three parameter Tweedie-Poisson mixture model. It is also proved that the distributions in the extended part of the parameter space are not the zero truncation of mixed poisson distributions and that, other than for the negative binomial, they are not mixtures of zero truncated Poisson distributions either. By extending the parameter space one can improve the fit when the frequency of one is larger and the right tail is heavier than is allowed by the unextended model. Considering the extended model also allows one to use the basic maximum likelihood based inference tools when parameter estimates fall in the extended part of the parameter space, and hence when the m.l.e. does not exist under the unextended model. This extended truncated Tweedie-Poisson model is proved to be useful in the analysis of words and species frequency count data.

On left-truncating and mixing Poisson distributions

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Mixture of bivariate Poisson regression models with an application to insurance

In a recent paper Bermúdez [2009] used bivariate Poisson regression models for ratemaking in car insurance, and included zero-inflated models to account for the excess of zeros and the overdispersion in the data set. In the present paper, we revisit this model in order to consider alternatives. We propose a 2-finite mixture of bivariate Poisson regression models to demonstrate that the overdispersion in the data requires more structure if it is to be taken into account, and that a simple zero-inflated bivariate Poisson model does not suffice. At the same time, we show that a finite mixture of bivariate Poisson regression models embraces zero-inflated bivariate Poisson regression models as a special case. Additionally, we describe a model in which the mixing proportions are dependent on covariates when modelling the way in which each individual belongs to a separate cluster. Finally, an EM algorithm is provided in order to ensure the models’ ease-of-fit. These models are applied to the same automobile insurance claims data set as used in Bermúdez [2009] and it is shown that the modelling of the data set can be improved considerably.

An energy preserving discontinouous Galerkin method for Vlasov-Poisson system

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