Dimensions of Group-Based Phylogenetic Mixtures

Hector Baños; Nathaniel Bushek; Ruth Davidson; Elizabeth Gross; Pamela E. Harris; Robert Krone; Colby Long; Allen Stewart; Robert Walker

doi:10.1007/s11538-018-0489-0

Dimensions of Group-Based Phylogenetic Mixtures

Hector Baños, Nathaniel Bushek, Ruth Davidson, Elizabeth Gross, Pamela E. Harris, Robert Krone, Colby Long, Allen Stewart, Robert Walker

Mathematics & Statistics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

Mixtures of group-based Markov models of evolution correspond to joins of toric varieties. In this paper, we establish a large number of cases for which these phylogenetic join varieties realize their expected dimension, meaning that they are nondefective. Nondefectiveness is not only interesting from a geometric point-of-view, but has been used to establish combinatorial identifiability for several classes of phylogenetic mixture models. Our focus is on group-based models where the equivalence classes of identified parameters are orbits of a subgroup of the automorphism group of the abelian group defining the model. In particular, we show that for these group-based models, the variety corresponding to the mixture of r trees with n leaves is nondefective when n≥ 2 r+ 5. We also give improved bounds for claw trees and give computational evidence that 2-tree and 3-tree mixtures are nondefective for small n.

Original language	English (US)
Pages (from-to)	316-336
Number of pages	21
Journal	Bulletin of mathematical biology
Volume	81
Issue number	2
DOIs	https://doi.org/10.1007/s11538-018-0489-0
State	Published - Feb 15 2019

Bibliographical note

Funding Information:
This work began at the 2016 AMS Mathematics Research Community on “Algebraic Statistics,” which was supported by the National Science Foundation under Grant number DMS-1321794. RD was supported by NSF DMS-1401591. EG was supported by NSF DMS-1620109. RW was supported by a NSF GRF under Grant number PGF-031543, NSF RTG Grant 0943832, and a Ford Foundation Dissertation Fellowship. HB was supported in part by a research assistantship, funded by the National Institutes of Health Grant R01 GM117590. PEH was partially supported by NSF Grant DMS-1620202.

Publisher Copyright:
© 2018, Society for Mathematical Biology.

PubMed: MeSH publication types

Research Support, Non-U.S. Gov't
Research Support, U.S. Gov't, Non-P.H.S.
Journal Article
Research Support, N.I.H., Extramural

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title = "Dimensions of Group-Based Phylogenetic Mixtures",

abstract = "Mixtures of group-based Markov models of evolution correspond to joins of toric varieties. In this paper, we establish a large number of cases for which these phylogenetic join varieties realize their expected dimension, meaning that they are nondefective. Nondefectiveness is not only interesting from a geometric point-of-view, but has been used to establish combinatorial identifiability for several classes of phylogenetic mixture models. Our focus is on group-based models where the equivalence classes of identified parameters are orbits of a subgroup of the automorphism group of the abelian group defining the model. In particular, we show that for these group-based models, the variety corresponding to the mixture of r trees with n leaves is nondefective when n≥ 2 r+ 5. We also give improved bounds for claw trees and give computational evidence that 2-tree and 3-tree mixtures are nondefective for small n.",

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note = "Funding Information: This work began at the 2016 AMS Mathematics Research Community on “Algebraic Statistics,” which was supported by the National Science Foundation under Grant number DMS-1321794. RD was supported by NSF DMS-1401591. EG was supported by NSF DMS-1620109. RW was supported by a NSF GRF under Grant number PGF-031543, NSF RTG Grant 0943832, and a Ford Foundation Dissertation Fellowship. HB was supported in part by a research assistantship, funded by the National Institutes of Health Grant R01 GM117590. PEH was partially supported by NSF Grant DMS-1620202. Publisher Copyright: {\textcopyright} 2018, Society for Mathematical Biology.",

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AU - Long, Colby

AU - Stewart, Allen

AU - Walker, Robert

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N2 - Mixtures of group-based Markov models of evolution correspond to joins of toric varieties. In this paper, we establish a large number of cases for which these phylogenetic join varieties realize their expected dimension, meaning that they are nondefective. Nondefectiveness is not only interesting from a geometric point-of-view, but has been used to establish combinatorial identifiability for several classes of phylogenetic mixture models. Our focus is on group-based models where the equivalence classes of identified parameters are orbits of a subgroup of the automorphism group of the abelian group defining the model. In particular, we show that for these group-based models, the variety corresponding to the mixture of r trees with n leaves is nondefective when n≥ 2 r+ 5. We also give improved bounds for claw trees and give computational evidence that 2-tree and 3-tree mixtures are nondefective for small n.

AB - Mixtures of group-based Markov models of evolution correspond to joins of toric varieties. In this paper, we establish a large number of cases for which these phylogenetic join varieties realize their expected dimension, meaning that they are nondefective. Nondefectiveness is not only interesting from a geometric point-of-view, but has been used to establish combinatorial identifiability for several classes of phylogenetic mixture models. Our focus is on group-based models where the equivalence classes of identified parameters are orbits of a subgroup of the automorphism group of the abelian group defining the model. In particular, we show that for these group-based models, the variety corresponding to the mixture of r trees with n leaves is nondefective when n≥ 2 r+ 5. We also give improved bounds for claw trees and give computational evidence that 2-tree and 3-tree mixtures are nondefective for small n.

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Dimensions of Group-Based Phylogenetic Mixtures

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