Phylogenetic comparative methods are a set of statistical methods that model the evolutionary history of species, especially in the context where one has data on certain traits of related extant species that have evolved over a phylogenetic tree in accordance to an underlying stochastic process. 

This thesis presents a Hessian-based closed-form asymptotic confidence region that covers a wide family of Gaussian continuous-trait evolution models; the result has been implemented in an R package. Also, some analyses have been done on the simpler Brownian Motion and Ornstein-Uhlenbeck process cases; and this leads to novel exact confidence regions for the Brownian Motion’s parameters and a closed-form ’partial’ unbiased estimator for the Ornstein-Uhlenbeck process’ varaince-covariance matrix when other parameters are given. 

The thesis contains two papers. Paper I is an applied work that uses discrete-trait speciation and extinction model to investigate early spread of COVID-19; it shows that it is possible to detect statistical signals of inter-continental spread of the virus from a very noisy world-wide phylogeny. Paper II is a more mathematical work that derived the closed-form formulae for the Hessian matrix of a wide family of Gaussian-process-based multivariate continuous-trait PCM models; accompanying with the Paper I have developed an R package called glinvci, publicly available on The Comprehensive R Archive Network (CRAN), that can compute Hessian-based confidence regions for these models while at the same time allowing users to have missing data and multiple evolutionary regimes.