2008/02/17 by Radu Mihaescu, Lior Pachter · 25 citations
Biochemistry, Genetics and Molecular Biology · Earth and Planetary Sciences · Mathematics · #Algorithm #Combinatorial proof #Combinatorics #Discrete mathematics #Evolution and Paleontology Studies #Explained sum of squares #Genetic diversity and population structure #Genomics and Phylogenetic Studies #Least-squares function approximation #Markov chain #Mathematics #Multiplicative function #Non-linear least squares #Property (philosophy) #Residual sum of squares #Statistics #math.CO #math.ST #stat.TH
paper · pdf · doi:10.1073/pnas.0802089105
published in Proceedings of the National Academy of Sciences 105(36), 13206-13211 (National Academy of Sciences)
arxiv created 2008/02/17 · openalex publication_date 2008/09/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 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.