Distance to the stochastic part of phylogenetic varieties
| Autor: | Marta Casanellas, Marina Garrote-López, Jesús Fernández-Sánchez |
|---|---|
| Přispěvatelé: | Universitat Politècnica de Catalunya. Departament de Matemàtiques, Universitat Politècnica de Catalunya. GEOMVAP - Geometria de Varietats i Aplicacions |
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Polynomial
14P10 14Q30 62R01 Distribution (number theory) 010103 numerical & computational mathematics 01 natural sciences Phylogenetic variety Hidden markov process FOS: Mathematics Mathematics - Combinatorics 0101 mathematics Quantitative Biology - Populations and Evolution Mathematics Numerical algebraic geometry Algebra and Number Theory Phylogenetic tree 010102 general mathematics Populations and Evolution (q-bio.PE) Substitution (algebra) Matemàtiques i estadística [Àrees temàtiques de la UPC] Algebraic variety Semi-algebraic phylogenetics Algebra Computational Mathematics Data point FOS: Biological sciences Group-based models Quartet topology Euclidean distance degree Long-branch attraction Combinatorics (math.CO) |
| Zdroj: | UPCommons. Portal del coneixement obert de la UPC Universitat Politècnica de Catalunya (UPC) |
| Popis: | Modelling the substitution of nucleotides along a phylogenetic tree is usually done by a hidden Markov process. This allows to define a distribution of characters at the leaves of the trees and one might be able to obtain polynomial relationships among the probabilities of different characters. The study of these polynomials and the geometry of the algebraic varieties defined by them can be used to reconstruct phylogenetic trees. However, not all points in these algebraic varieties have biological sense. In this paper, we explore the extent to which adding semi-algebraic conditions arising from the restriction to parameters with statistical meaning can improve existing methods of phylogenetic reconstruction. To this end, our aim is to compute the distance of data points to algebraic varieties and to the stochastic part of these varieties. Computing these distances involves optimization by nonlinear programming algorithms. We use analytical methods to find some of these distances for quartet trees evolving under the Kimura 3-parameter or the Jukes-Cantor models. Numerical algebraic geometry and computational algebra play also a fundamental role in this paper. Comment: 33 pages; 11 figures; to appear in Journal of Symbolic Computation |
| Databáze: | OpenAIRE |
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