Math @ Duke

Publications [#212344] of Ezra Miller
Papers Submitted
 with Megan Owen and Scott Provan, Polyhedral computational geometry for averaging metric phylogenetic trees
(2012) (43 pages.) [math.MG/1211.7046]
(last updated on 2013/12/20)
Abstract: 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 is continuously differentiable
within each cell of the corresponding
subdivision. 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
Frechet means and extend our main results to
more general globally nonpositively curved
spaces composed of Euclidean orthants.


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