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Mathematische Annalen

, Volume 344, Issue 3, pp 619–644 | Cite as

On the ideals of equivariant tree models

  • Jan DraismaEmail author
  • Jochen Kuttler
Open Access
Article

Abstract

We introduce equivariant tree models in algebraic statistics, which unify and generalise existing tree models such as the general Markov model, the strand symmetric model, and group-based models such as the Jukes–Cantor and Kimura models. We focus on the ideals of such models. We show how the ideals for general trees can be determined from the ideals for stars. A corollary of theoretical importance is that the ideal for a general tree is generated by the ideals of its flattenings at vertices. The main novelty is that our results yield generators of the full ideal rather than an ideal which only defines the model set-theoretically.

Keywords

Bilinear Form Toric Variety Root Distribution Equivariant Model Internal Vertex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

The first author thanks Seth Sullivant for his great EIDMA/DIAMANT course on algebraic statistics in Eindhoven. It was Seth who pointed out that a result like the one in Sect. 3 could be used to treat various existing tree models in a unified manner. We also thank the anonymous referees for many valuable suggestions to improve the exposition.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution,and reproduction in any medium, provided the original author(s) and source are credited.

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Copyright information

© The Author(s) 2008

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceTechnische Universiteit EindhovenEindhovenThe Netherlands
  2. 2.Centrum Wiskunde and InformaticaAmsterdamThe Netherlands
  3. 3.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada

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