[Paper Review] Averaging metric phylogenetic trees

Ezra Miller, Megan Owen+1 more|arXiv (Cornell University)|Nov 29, 2012

Morphological variations and asymmetryMathematics49 references20 citations

TL;DR

This paper develops computational methods for calculating the Frechet mean of metric phylogenetic trees using the geometry of tree space, which is a nonpositively curved space composed of Euclidean orthants. It introduces a polyhedral subdivision of tree space that enables continuous differentiability of the variance function and proposes two iterative algorithms—based on Sturm's Law of Large Numbers and descent methods on convex polyhedra—that converge to the Frechet mean.

ABSTRACT

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 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 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.

Motivation & Objective

• To develop computationally feasible methods for calculating the Frechet mean in metric phylogenetic tree space.
• To understand the combinatorial structure of geodesics in tree space with a fixed endpoint.
• To establish conditions under which the variance function is continuously differentiable.
• To extend the theory of Frechet means to general globally nonpositively curved spaces composed of Euclidean orthants.
• To provide biological applications of Frechet means in phylogenetics through computational algorithms.

Proposed method

• The paper uses Sturm's theory of probability measures on nonpositively curved spaces to define the Frechet mean and variance in tree space.
• It constructs a polyhedral subdivision of tree space such that the geodesic structure with a fixed endpoint depends only on the location of the varying endpoint within each cell.
• The variance function is shown to be continuously differentiable within each cell of this subdivision.
• An iterative algorithm based on Sturm's Law of Large Numbers is proposed to converge to the Frechet mean.
• A second algorithm uses descent methods for smooth functions on convex polyhedra to minimize the variance function.
• The results are extended to general globally nonpositively curved spaces made of Euclidean orthants, generalizing the framework beyond phylogenetic trees.

Experimental results

Research questions

• RQ1 How can the Frechet mean of a finite set of metric phylogenetic trees be computed efficiently in tree space?
• RQ2 What determines the combinatorics of geodesics in tree space when one endpoint is fixed?
• RQ3 In which regions of tree space is the variance function continuously differentiable?
• RQ4 Can iterative algorithms be constructed to converge to the Frechet mean using the geometric structure of tree space?
• RQ5 How can the theory of Frechet means be extended to more general nonpositively curved spaces composed of Euclidean orthants?

Key findings

• The geodesic structure in tree space with a fixed endpoint is determined by the location of the varying endpoint within a specific polyhedral subdivision of tree space.
• The variance function associated with a finite subset of tree space is continuously differentiable within each cell of the corresponding polyhedral subdivision.
• Two iterative algorithms are established that converge to the Frechet mean: one based on Sturm's Law of Large Numbers and another using descent on smooth convex functions.
• The Frechet mean is well-defined and computable in tree space due to its global nonpositive curvature and the existence of a unique minimizer for the variance function.
• The framework is extended to general globally nonpositively curved spaces composed of Euclidean orthants, broadening applicability beyond phylogenetic trees.
• Biological applications of Frechet means in phylogenetics are supported by the computational tractability and geometric foundations of the proposed methods.