Asymptotics of Ornstein-Uhlenbeck tree models: Theory and computation

Tree models arise in evolutionary biology when sampled species are related to each other according to a phylogenetic tree. Tips of this tree represent contemporary species while internal nodes represent their ancestors. Researchers are often interested in studying the evolution of a phenotype using only observations from the present (at the tips). In this context, the Ornstein-Uhlenbeck (OU) process has been widely used to model trait evolution under the presence of natural selection, whereby a trait is attracted to a selection optimum mu with selection strength alpha. The primary goal of this thesis is to study the asymptotics of OU tree models theoretically and computationally.;In the theoretical part, I report several intrinsic inference difficulties of OU tree models including non-identifiability and non-microergodicity. A well-defined and practical model parametrization is proposed to avoid non-identifiability. Situations where non-microergodicity occurs are also identified. Moreover, I provide a necessary and sufficient condition for the consistency of the maximum likelihood estimator of mu and establish a phase transition on its convergence rate. I also propose a novel estimation method for alpha, which achieves an optimal convergence rate under natural assumptions.;In the computational part, I develop a linear-time algorithm which is applicable to many Gaussian models including OU processes with varying selection parameters along the tree, as well as non-Gaussian models like phylogenetic logistic regression and phylogenetic generalized linear (mixed) models. The traditional calculation methods are computationally heavy, requiring complex and time-consuming matrix calculations, and causing failure or inaccurate results on very large trees. The new algorithm for phylogenetic regression is implemented in the R package phylolm..