Geometry of the Space of Phylogenetic Trees

Abstract

We consider a continuous space which models the set of all phylogenetic trees having a fixed set of leaves. This space has a natural metric of nonpositive curvature, giving a way of measuring distance between phylogenetic trees and providing some procedures for averaging or combining several trees whose leaves are identical. This geometry also shows which trees appear within a fixed distance of a given tree and enables construction of convex hulls of a set of trees. This geometric model of tree space provides a setting in which questions that have been posed by biologists and statisticians over the last decade can be approached in a systematic fashion. For example, it provides a justification for disregarding portions of a collection of trees that agree, thus simplifying the space in which comparisons are to be made.  2001 Elsevier Science

A Space of Phylogenetic Networks

“…Instead of studying individual network structures, it is sometimes advantageous to consider relationships between them. In 2001, Billera, Holmes, and Vogtmann [3] constructed a model for a space BHV n of metric trees, making it useful for geometric methods such as the calculation of geodesics and centroids [25]. We construct an analogous model CSN n for circular split networks by beginning with a review of BHV n .…”

“…Tree space BHV 5 consists of (2 · 5 − 5)!! = 15 quadrants [0, ∞) 2 glued together; a visualization of this space is provided in Figure 14 in [3]. Its subspace T 5 is the Peterson graph with 15 edges, as displayed in Figure 7(b), with the 10 vertices corresponding to the distinct labeled trees with five leaves and one internal edge (of length one).…”

“…Figure 9 shows S 4 ⊂ CSN 4 , the 1-dimensional simplicial complex formed by gluing together three edges, creating a triangle, parametrizing circular networks whose internal edge lengths sum to 1. It was shown in [3] that BHV n is a CAT(0) space, ensuring that any two points have a unique geodesic between them. This is not the case for networks, however, due to the underlying geometry of S n .…”

“…The rest of this paper is concerned with the relationships among trees, networks, and the real moduli space of curves M 0,n (R), a variety from algebraic geometry [19] appearing in areas such as ζ-motives [13], geometric group theory [6], and Lagrangian Floer theory [11]. There is a substantial relationship between this moduli space and the space of phylogenetic trees: A hint of this relationship was originally discussed in [3], where M 0,n (R) was considered but deemed unsuitable for describing tree space. Recently, Devadoss and Morava [9] formulated an explicit relationship by constructing a smooth blowup Θ : M 0,n (R) → BHV n .…”

Horoball Hulls and Extents in Positive Definite Space

“…In diffusion tensor imaging [3], matrices in P(3) model the flow of water at each voxel of a brain scan, and a goal is to cluster these matrices into groups that capture common flow patterns along fiber tracts. In mechanical engineering [11], stress tensors are modeled as elements of P (6), and identifying groups of similar tensors helps locate homogeneous regions in a material from samples. Kernel matrices in machine learning are elements of P(n) [26], and motivated by the problems of learning and approximating kernels for machine learning tasks, there has been recent interest in studying the geometry of P(n) and related spaces [20,7,28].…”