Inferência em alguns modelos de processos estocasticamente perturbados

Abstract

In a model of stochastically disturbed processes each observation of the original process can be disturbed at any moment of time by a random noise. Thus the observed process could not be a sample of the original process. In this thesis we present a methodology in order to estimate the parameters of some disturbed stochastically models based on the models proposed by [7] and [12]. We assume that the original hidden process is a variable length Markov chain. This class processes allows many applications since it is parsimonious in relation to the number of parameters and also quite malleable, including the class of xed-order Markov chains. We propose an adaptation in the Baum-Welch algorithm and a bootstrap Bayesian Information Criterion as a way to estimate theparameters of the models analyzed, whose convergence was shown, and show through simulations that the proposed methodology is able to recover very well the real context tree of a stochastically disturbed variable length Markov chain as well as the transition probabilities associated with the tree, within a reasonable range of disturbance levels. We also able to recover the degree of disturbance whatever it has been. We propose a modication to the Viterbi algorithm to nd the most appropriate hidden sequence of a stochastically disturbed variable length Markov chain. We present a model selection criterion to identify the most appropriate model given the observed sample among those analyzed in this thesis. We apply the proposed methodology to a database of neurons activity records of a group of owls in a controlled laboratory experiment. Data were coded in two states, spike and rest. Our goal is to identify the existence of diferent patterns of behavior that neuronal activity according to the estimated probability for the process in relation to the type of visual stimulus that the group of owls was submitted.