Polyhedral computational geometry for averaging metric phylogenetic trees


Journal Article

© 2015 Elsevier Inc. All rights reserved. This paper investigates the computational geometry relevant to calculations of the Fréchet mean and variance for probability distributions on the phylogenetic tree space of Billera, Holmes and Vogtmann, using the theory of probability measures on spaces of nonpositive curvature developed by Sturm. We show that the combinatorics of geodesics with a specified fixed endpoint in tree space are determined by the location of the varying endpoint in a certain polyhedral subdivision of tree space. The variance function associated to a finite subset of tree space has a fixed C algebraic formula within each cell of the corresponding subdivision, and is continuously differentiable in the interior of each orthant of tree space. We use this subdivision to establish two iterative methods for producing sequences that converge to the Fréchet mean: one based on Sturm's Law of Large Numbers, and another based on descent algorithms for finding optima of smooth functions on convex polyhedra. We present properties and biological applications of Fréchet means and extend our main results to more general globally nonpositively curved spaces composed of Euclidean orthants.

Full Text

Duke Authors

Cited Authors

  • Miller, E; Owen, M; Provan, JS

Published Date

  • January 1, 2015

Published In

Volume / Issue

  • 68 /

Start / End Page

  • 51 - 91

Electronic International Standard Serial Number (EISSN)

  • 1090-2074

International Standard Serial Number (ISSN)

  • 0196-8858

Digital Object Identifier (DOI)

  • 10.1016/j.aam.2015.04.002

Citation Source

  • Scopus