Year of Publication

2015

Degree Name

Doctor of Philosophy (PhD)

Document Type

Doctoral Dissertation

College

Arts and Sciences

Department

Statistics

First Advisor

Dr. Ruriko Yoshida

Abstract

This dissertation is an effort to adapt two classical non-parametric statistical techniques, kernel density estimation (KDE) and principal components analysis (PCA), to the Billera-Holmes-Vogtmann (BHV) metric space for phylogenetic trees. This adaption gives a more general framework for developing and testing various hypotheses about apparent differences or similarities between sets of phylogenetic trees than currently exists.

For example, while the majority of gene histories found in a clade of organisms are expected to be generated by a common evolutionary process, numerous other coexisting processes (e.g. horizontal gene transfers, gene duplication and subsequent neofunctionalization) will cause some genes to exhibit a history quite distinct from the histories of the majority of genes. Such “outlying” gene trees are considered to be biologically interesting and identifying these genes has become an important problem in phylogenetics.

The R sofware package kdetrees, developed in Chapter 2, contains an implementation of the kernel density estimation method. The primary theoretical difficulty involved in this adaptation concerns the normalizion of the kernel functions in the BHV metric space. This problem is addressed in Chapter 3. In both chapters, the software package is applied to both simulated and empirical datasets to demonstrate the properties of the method.

A few first theoretical steps in adaption of principal components analysis to the BHV space are presented in Chapter 4. It becomes necessary to generalize the notion of a set of perpendicular vectors in Euclidean space to the BHV metric space, but there some ambiguity about how to best proceed. We show that convex hulls are one reasonable approach to the problem. The Nye-PCA- algorithm provides a method of projecting onto arbitrary convex hulls in BHV space, providing the core of a modified PCA-type method.

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