Esta tese propõe duas abordagens para estimar a seqüência oculta de uma cadeia de Markov oculta: blocos de consenso e blocos de regeneração. Em ambos os casos os algoritmos resultantes dependem de um número de operações que cresce polinomialmente com o tamanho da seqüência. Na primeira abordagem, quebramos a seqüência visível em blocos e estimamos a seqüência oculta de acordo com a maioria de símbolos que enxergamos na seqüência visível. Na segunda abordagem, utilizamos a estrutura regenerativa da cadeia para decompor em blocos independentes. Obtivemos limites superiores para a probabilidade de erro de estimação com os dois métodos. Na segunda abordagem, utilizamos o método de Monte Carlo markoviano e o algoritmo de Metrópolis para construir iterativamente a seqüência de instantes de regeneração e os blocos correspondentes de estados ocultos, dada a seqüência visível da cadeia. Na demonstração dos resultados foram utilizados resultados de esquemas regenerativos, o método de Chernofi e a desigualdade de Hoefiding. Esta tese tem também uma componente computacional. Com efeito, desenvolvemos rotinas em R que implementam os diversos algoritmos propostos. Também fizemos simulações que ilustram a funcionalidade dos algoritmos. / This dissertation introduces two new approaches to the problem of the estimation of the hidden chain in a hidden Markov model. In both approaches the number of steps necessary to estimate the hidden chain increase polinomially with the size of the sample (the observable chain). In the rst approach, through consensus substrings, we split the observable chain in substrings and estimate the hidden values as the one with appears in the most of the steps of the observable substring. In the second approach, through regenerative substrings, we split the chain using its regenerative structure. In both cases we obtain upper bounds for the error probability of the algorithms. In the second approach, we use the Monte Carlo Markov chains and the Metropolis algorithm to construct iteratively the sequence of the regeneration times and the corresponding substrings of hidden states, given the observable chain. The main tools use in the proofs of the theorems were the regenerative construction of the Markov chain, Chernoff's method and Hoe ding's inequality. This dissertation has also a computational aspect. In e aspect, all the algorithms introduced here have been implemented in R and the corresponding codes are available in the appendix. We also present several simulations to illustrate the applicability of the algorithms.

Esta dissertação trata de um teorema de limite hidrodinâmico para a densidade de partículas no processo de exclusão simples com taxa lenta no bordo. Nesse processo as partículas realizam passeios aleatórios a tempo contínuo independentes no espaço {0,1,..., N}, condicionados ao evento de que duas partículas nunca ocupam o mesmo sítio simultaneamente. Além disso, partículas podem ser injetadas ou retiradas do sistema nos sítios 0 e N, com taxas proporcionais a 1/N. O teorema do limite hidrodinâmico diz que reescalando o espaço por 1/N e o tempo por N² a evolução da densidade de partículas converge para a solução de uma equação do calor com certas condições de fronteira. Esta dissertação se concentra na caracterização dos pontos limites das medidas empíricas, um dos principais passos da demonstração. Além disso são apresentados resultados básicos de Processos de Markov relacionados ao problema do limite hidrodinâmico. / This text is about an hydrodynamic limit theorem for the particle density in the symmetric simple exclusion process with slow boundaries. In this process particles execute independent continuous time simple random walks in {0,1,...,N} conditioned to the event that at any time there is at most one particle per site. Moreover, particles can be injected or ejected from the system at sites 0 and N, with rates proportional to 1/N. The hydrodynamic limit theorem asserts that, rescaling space by 1/N and time by N², the particle density converges to the solution of a heat equation with certain boundary conditions. This text focuses on the characterization of limit points of empirical measures, a crucial step in the proof of the hydrodynamic limit. We also present some basic material on Markov Process related to the hydrodynamic limit problem.

In this thesis, we investigate two di erent approaches to animal movement modelling; nite mixture models, and di usion processes. These models are considered in two di erent contexts, rstly for analysis of data obtained in home range studies, and then, on a much smaller scale, modelling the movements of larvae. We consider the application of mixture models to home range movement data, and compare their performance with kernel density estimators commonly used for this purpose. Mixtures of bivariate normal distributions and bivariate t distributions are considered, and the latter are found to be good models for simulated and real movement data. The mixtures of bivariate t distributions are shown to provide a robust parametric approach. Subsequently, we investigate several measures of overlap for assessing site delity in home range data. Di usion processes for home range data are considered to model the tracks of animals. In particular, we apply models based on a bivariate Ornstein-Uhlenbeck process to recorded coyote movements. We then study modelling in a di erent application area involving tracks. Di usion models for the movements of larvae are used to investigate their behaviour when exposed to chemical compounds in a scienti c study. We nd that the tted models represent the movements of the larvae well, and correctly distinguish between the behaviour of larvae exposed to attractant and repellent compounds. Mixtures of di usion processes and Hidden Markov models provide more exible alternatives to single di usion processes, and are found to improve upon them considerably. A Hidden Markov model with 4 states is determined to be optimal, with states accounting for directed movement, localized movement and stationary observations. Models incorporating higherorder dependence are investigated, but are found to be less e ective than the use of multiple states for modelling the larvae movements.

M.Sc. (Mathematics) / In chapter 1, we give the reader some background concerning digraphs that are used in the discussion of Markov chains; namely, their Markov digraphs. Warshall’s Algorithm for reachability is also introduced as this is used to define terms such as transient states and irreducibility. Some initial theory and definitions concerning Markov chains and their corresponding Markov digraphs are given in chapter 2. Furthermore, we discuss l–step transitions (walks of length l) for homogeneous and inhomogeneous Markov chains and other generalizations. In chapter 3, we define terms such as communication, intercommunication, recurrence and transience. We also prove some results regarding the irreducibility of some Markov chains through the use of the reachability matrix. Furthermore, periodicity and aperiodicity are also investigated and the existence of walks of any length greater than some specified integer N is also considered. A discussion on random walks on an undirected torus is also contained in this chapter. In chapter 4, we explore stationary distributions and what it means for a Markov chain to be reversible. Furthermore, the hitting time and the mean hitting time in a Markov digraph are also defined and the proof of the theorems regarding them are done. The demonstrations of the theorems concerning the existence and uniqueness of stationary distributions and the Markov Chain Convergence Theorem are carried out. Later in this chapter, we define the Markov digraph of undirected graphs, which are Markov chains as well. The existing theory is then applied to these. In chapter 5, we explore and see how to simulate Markov chains on a computer by using Markov Chain Monte Carlo Methods. We also show how these apply to random q–colourings of undirected graphs. Finally, in chapter 6, we consider a physical application of these Graph Theoretical concepts—the simulation of the Ising model. Initially, the relevant concepts of the Potts model are given and then the Gibbs sampler algorithm in chapter 5 is modified and used to simulate the Ising model. A relation between the chromatic polynomial and the partition function of the Potts model is also demonstrated.

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M.Ing. / The main contribution of this thesis can be summarized as follows. Firstly, we implemented a high speed error gap recording system, which can run with at most with a slight adaptation, on any personal computer, doing gap recording for digital communication systems, achieving a data rate of up to 200 kbits/sec. Secondly, we extended previous experimental investigations of the Fritchman channel model to three other channel models, namely the Gilbert, Gilbert-Elliott, and Aldridge- Ghanbari models, and implemented these models experimentally by a set of computer programs. Thirdly, we investigated the statistical modeling of two analog channels, employed for digital transmission. These are an analog audio cassette, magnetic tape recorder and an analog cordless telephone operating in the 46/49 MHz. band, in the CT1 system. No evidence could be found in the literature of modeling error distributions on these two channels.

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