Combinatorics of least squares trees

TL;DR

This paper explores the combinatorial properties of least squares methods in phylogenetics, providing new formulas, characterizations, and an efficient algorithm for estimating evolutionary trees.

Contribution

It introduces a complete characterization of weighted least squares methods satisfying a desirable property, based on a multiplicative four point condition, and offers a time optimal computation algorithm.

Findings

Characterization of methods satisfying the property via a four point condition

Complete generalization of previous least squares and evolution models

Development of a time optimal algorithm for tree estimation

Abstract

A recurring theme in the least squares approach to phylogenetics has been the discovery of elegant combinatorial formulas for the least squares estimates of edge lengths. These formulas have proved useful for the development of efficient algorithms, and have also been important for understanding connections among popular phylogeny algorithms. For example, the selection criterion of the neighbor-joining algorithm is now understood in terms of the combinatorial formulas of Pauplin for estimating tree length. We highlight a phylogenetically desirable property that weighted least squares methods should satisfy, and provide a complete characterization of methods that satisfy the property. The necessary and sufficient condition is a multiplicative four point condition that the the variance matrix needs to satisfy. The proof is based on the observation that the Lagrange multipliers in the proof of the Gauss--Markov theorem are tree-additive. Our results generalize and complete previous work on ordinary least squares, balanced minimum evolution and the taxon weighted variance model. They also provide a time optimal algorithm for computation.