Processo de Cox marcado modulado por processos Gaussianos para configurações pontuais unidimensionais

Abstract

The theory of point processes is a very important Statistics area to describe the behavior of a certain random phenomenon whose realization results in a set of random points that represent occurrences of a point nature. These points, when indexed by the onedimensional set, they can represent the exact moment of occurrence. However, it can be defined in any indexing set, whether it is time or not. One of the ways to study a point process is through the intensity function, which describes an average rate of occurrences. It have been proposed several models to describe the behavior of the intensity of a point process in the literature, including the recent contribution of Lloyd et al. (2015), based on the Cox processes’ class in which the intensity function is described as a function of a stochastic Gaussian process . Lloyd et al. (2015) approach is based on a variational estimation method with the inclusion of a sparse method, which allows the model to handle a large number of observations. In addition, additional information associated with the occurrences of the point process can be incorporated into the model, which is called by marks. Thus, this thesis aimed to propose a modeling scheme to describe the intensity of a marked point processes, in which the mark is a qualitative variable, with two categories. The proposal was an extension of the Lloyd et al. (2015) model, in which the marked intensity function, based on two categories, was modeled as a function of a sparse bivariate Gaussian process. Following Lloyd et al. (2015), the estimation process was based on the Bayesian variational method, which allowed that the intensity function could be estimated for any point in the index set. As a way of exemplifying the proposal of this thesis, it was made an application from a set of real data based on the occurrence of accidents on Brazilian federal highways. The proposed model proved to be promising, suggesting that other extensions can be made so that the model can describe a much larger set of stochastic phenomena of a point nature.