The Identifiability of Tree Topology for Phylogenetic Models, Including Covarion and Mixture Models

Abstract

For a model of molecular evolution to be useful for phylogenetic inference, the topology of evolutionary trees must be identifiable. That is, from a joint distribution the model predicts, it must be possible to recover the tree parameter. We establish tree identifiability for a number of phylogenetic models, including a covarion model and a variety of mixture models with a limited number of classes. The proof is based on the introduction of a more general model, allowing more states at internal nodes of the tree than at leaves, and the study of the algebraic variety formed by the joint distributions to which it gives rise. Tree identifiability is first established for this general model through the use of certain phylogenetic invariants.

Identifying evolutionary trees and substitution parameters for the general Markov model with invariable sites

“…In [8], the viewpoint of algebraic geometry is used to show the generic identifiability of the tree parameter for the covarion model of [9] and for certain mixture models with a small number of classes. Though this result is far more general than previous identifiability results, it still fails to cover the type of rate-variation models currently in common use for data analysis, and does not address identifiability of numerical parameters at all.…”

“…Although identifiability of the tree parameter for GM+I follows from more general results in [8], that paper did not consider identifiability of numerical parameters. Our arguments here are tailored to GM+I and yield stronger results addressing numerical parameters as well as the tree.…”

“…Our approach is again based on the determination of phylogenetic invariants for the model. While the invariants described in [8] are invariants for more general models than GM+I, the ones given in this paper apply only to GM+I and its submodels, and are of much lower degree. As a byproduct of the development of these GM+I invariants, we are led to rational formulas for recovering all the parameters related to the invariable sites from a joint distribution.…”

Toric geometry of the 3-Kimura model for any tree

“…Much more is known about the identifiability of different models. For the precise results the reader is advised to look in [AR06] or [APRS] and the references therein.…”

Tying Up Loose Strands: Defining Equations of the Strand Symmetric Model

“…Their study was originally proposed as a method for reconstructing phylogenetic trees [4,10], but they have also been useful theoretical tools in proving identifiability results (see e.g. [2]). Results in [6] imply that to determine generators of the ideal of phylogenetic invariants for any trivalent tree, it suffices to determine them for the claw tree, K 1,3 .…”