Journal
ADVANCES IN APPLIED MATHEMATICS
Volume 68, Issue -, Pages 51-91Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.aam.2015.04.002
Keywords
Tree space; Frechet mean; Polyhedral subdivision; Descent method
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Funding
- NSF [DMS-0449102 = DMS-1014112, DMS-1001437]
- desJardins Postdoctoral Fellowship in Mathematical Biology at University of California Berkeley
- U.S. National Science Foundation [DMS-0635449]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1001437] Funding Source: National Science Foundation
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This paper investigates the computational geometry relevant to calculations of the Frechet 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-infinity 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 Frechet 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 Frechet means and extend our main results to more general globally nonpositively curved spaces composed of Euclidean orthants. (C) 2015 Elsevier Inc. All rights reserved.
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