Foundations of the Wald Space for Phylogenetic Trees

TL;DR

This paper introduces and analyzes the wald space, a geometric framework for phylogenetic trees, establishing its structure, topology, and metric properties to facilitate statistical analysis of evolutionary relationships.

Contribution

It formally defines the wald space, studies its topology and structure, and develops numerical methods for geodesic and statistical computations within this space.

Findings

Wald space has the topology of a disjoint union of open cubes.

It is a contractible and Whitney stratified space.

The space is a geodesic Riemann stratified space with a specific metric.

Abstract

Evolutionary relationships between species are represented by phylogenetic trees, but these relationships are subject to uncertainty due to the random nature of evolution. A geometry for the space of phylogenetic trees is necessary in order to properly quantify this uncertainty during the statistical analysis of collections of possible evolutionary trees inferred from biological data. Recently, the wald space has been introduced: a length space for trees which is a certain subset of the manifold of symmetric positive definite matrices. In this work, the wald space is introduced formally and its topology and structure is studied in detail. In particular, we show that wald space has the topology of a disjoint union of open cubes, it is contractible, and by careful characterization of cube boundaries, we demonstrate that wald space is a Whitney stratified space of type (A). Imposing the metric induced by the affine invariant metric on symmetric positive definite matrices, we prove that wald space is a geodesic Riemann stratified space. A new numerical method is proposed and investigated for construction of geodesics, computation of Fr\'echet means and calculation of curvature in wald space. This work is intended to serve as a mathematical foundation for further geometric and statistical research on this space.