Models and Estimation for Phylogenetic Trees

Abstract

In this thesis, we consider Markov models for matched sequences. De¯ne fij(t) = P(X(t) = i; Y (t) = jjX(0) = Y (0)); where fij is the joint probability that, for a given site, the ¯rst and second sequences have the values i and j at a given site, given that they were the same at time 0. This can generalized to several sequences. The sequences (taxa) are then arranged in an evolutionary tree (phylogenetic tree) depicting how taxa diverge from their common ancestors. We develop tests and estimation methods for the parameters of di®erent models. Standard phylogenetic methods assume stationarity, homogeneity and reversibility for the Markov processes, and often impose further restrictions on the parameters. The parameters in these cases are estimated using many popular packages, including PHYLIP and PAUP*. We describe a new and more general method for calculating the joint probability distribution under stationary and homogeneous models for the more general models with some weakening of the stationarity and homogeneity assumptions. We describe the method for a two edged tree and then extend it to the case for a K tipped tree. We discuss the case of a ¯ve edged tree for a set of bacterial sequences for which stationarity and homogeneity are not present. This data set is very similar to that of Galtier and Gouy (1995), and the search for methods appropriate for its analysis has provided the raison d'etre for this work. The extension we propose is to allow non-stationarity, so that from the root of the tree we permit di®erent Markov processes to operate along different descendant lineages; furthermore, we permit non-homogeneous Markov processes to operate across the tree. We obtain methods that