Principal components analysis in the space of phylogenetic trees

TL;DR

This paper introduces a novel geometrical method for applying Principal Components Analysis to collections of phylogenetic trees, enabling the analysis of variation in tree topology and branch lengths.

Contribution

It develops a new approach to PCA in tree-space using geodesic paths, addressing the challenge of non-vector space structure in phylogenetic trees.

Findings

Identifies principal paths that explain main variation in tree data

Demonstrates method on simulated and real gene trees

Reveals sources of variation in topology and branch lengths

Abstract

Phylogenetic analysis of DNA or other data commonly gives rise to a collection or sample of inferred evolutionary trees. Principal Components Analysis (PCA) cannot be applied directly to collections of trees since the space of evolutionary trees on a fixed set of taxa is not a vector space. This paper describes a novel geometrical approach to PCA in tree-space that constructs the first principal path in an analogous way to standard linear Euclidean PCA. Given a data set of phylogenetic trees, a geodesic principal path is sought that maximizes the variance of the data under a form of projection onto the path. Due to the high dimensionality of tree-space and the nonlinear nature of this problem, the computational complexity is potentially very high, so approximate optimization algorithms are used to search for the optimal path. Principal paths identified in this way reveal and quantify the main sources of variation in the original collection of trees in terms of both topology and branch lengths. The approach is illustrated by application to simulated sets of trees and to a set of gene trees from metazoan (animal) species.