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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
Special Issue: Algebraic Methods in Phylogenetics

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 \(n \ge 2r+5\). We also give improved bounds for claw trees and give computational evidence that 2-tree and 3-tree mixtures are nondefective for small n.

Notes

Acknowledgements

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.

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Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  • Hector Baños
    • 1
  • Nathaniel Bushek
    • 2
  • Ruth Davidson
    • 3
  • Elizabeth Gross
    • 4
  • Pamela E. Harris
    • 5
  • Robert Krone
    • 6
  • Colby Long
    • 7
  • Allen Stewart
    • 8
  • Robert Walker
    • 9
  1. 1.Department of Mathematics and StatisticsUniversity of Alaska FairbanksFairbanksUS
  2. 2.Department of Mathematics and StatisticsUniversity of Minnesota DuluthDuluthUSA
  3. 3.Department of MathematicsUniversity of Illinois Urbana-ChampaignUrbanaUSA
  4. 4.Department of MathematicsUniversity of Hawai’i at MānoaHonoluluUSA
  5. 5.Department of Mathematics and StatisticsWilliams CollegeWilliamstownUSA
  6. 6.Department of MathematicsUniversity of California DavisDavisUSA
  7. 7.Mathematical Biosciences InstituteThe Ohio State UniversityColumbusUSA
  8. 8.Department of MathematicsSeattle UniversitySeattleUSA
  9. 9.Department of MathematicsUniversity of MichiganAnn ArborUSA

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