Phylogenetic ideals and varieties for the general Markov model

Allman, Elizabeth S.; Rhodes, John A.

doi:10.1016/j.aam.2006.10.002

Cited by 125 publications

(220 citation statements)

References 18 publications

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“…Similar to the observations in [1], as a consequence of the construction of CV EM (T ), it is a closed cone (i.e. invariant under scalar multiplication in L(T )) and therefore uniquely defines a projective variety in P(L(T )), denoted P(CV EM (T )), and defined by the same ideal as CV EM (T ).…”

Section: Remark 22mentioning

confidence: 63%

“…The main missing ingredients for successful applications are equations for star models. These are very hard to come by: [9] posed several conjectures concerning these for the general Markov model, and special cases of these conjectures were proved in [1,13,14]. For certain important equivariant models equations were found in [4,16].…”

Section: Proof Of Theorem 17 Recall the Statement Of The Theorem: Fomentioning

confidence: 99%

“…In particular [1] contains set-theoretic versions of our theorems for the general Markov model, and poses Theorem 1.7 for the general Markov model as Conjecture 5. In [16] the ideals of equivariant models with G abelian and all V p equal to the regular representation K G are determined, following ideas from [8].…”

Section: Theorem 17 For Any G-spaced Tree T We Havementioning

confidence: 99%

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On the ideals of equivariant tree models

Draisma

Kuttler

2008

Math. Ann.

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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. Set-up and theoremsIn phylogenetics, tree models have been introduced to describe the evolution of a number of species from a distant common ancestor. Given suitably aligned strings of nucleotides of n species alive today, one assumes that the individual positions in J. Draisma has been supported by DIAMANT, an NWO mathematics cluster and J. Kuttler by an NSERC Discovery Grant.

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Section: Remark 22mentioning

confidence: 63%

Section: Proof Of Theorem 17 Recall the Statement Of The Theorem: Fomentioning

confidence: 99%

Section: Theorem 17 For Any G-spaced Tree T We Havementioning

confidence: 99%

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On the ideals of equivariant tree models

Draisma

Kuttler

2008

Math. Ann.

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“…Under mild additional restrictions on model parameters, changing the root location corresponds to a simple invertible change of variables in the parameterization. (See [13], [14], or [15] for details.) This justifies our slight abuse of language in referring to the GM or GM+I model on T , rather than on T r , and we omit future references to root location.…”

Section: The Gm+i Modelmentioning

confidence: 99%

“…For the star tree, the 2-state GM ideal is known from [15]. Thus elimination can be used to find GM+I invariants.…”

Section: A3 Computation Of 2-state Gm+i Ideal 4-taxon Trees Using mentioning

confidence: 99%

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

Allman

Rhodes

2008

Mathematical Biosciences

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The general Markov plus invariable sites (GM+I) model of biological sequence evolution is a two-class model in which an unknown proportion of sites are not allowed to change, while the remainder undergo substitutions according to a Markov process on a tree. For statistical use it is important to know if the model is identifiable; can both the tree topology and the numerical parameters be determined from a joint distribution describing sequences only at the leaves of the tree? We establish that for generic parameters both the tree and all numerical parameter values can be recovered, up to clearly understood issues of 'label swapping.' The method of analysis is algebraic, using phylogenetic invariants to study the variety defined by the model. Simple rational formulas, expressed in terms of determinantal ratios, are found for recovering numerical parameters describing the invariable sites.

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Graphical Latent Structure Testing

Evans

2014

Advances in Latent Variables

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Phylogenetic ideals and varieties for the general Markov model

Cited by 125 publications

References 18 publications

On the ideals of equivariant tree models

On the ideals of equivariant tree models

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

Graphical Latent Structure Testing

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