Lie-Markov Models Derived from Finite Semigroups

Sumner, Jeremy G.; Woodhams, Michael D.

doi:10.1007/s11538-018-0455-x

Cited by 4 publications

(10 citation statements)

References 26 publications

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“…The interested reader may consult [32, Ch. 7.3.2] for how these matrices fit into the setting of 'group-based models', and [36] for an extension to 'semigroup-based models'. The latter classification in particular includes the equal-input models discussed in Section 4.…”

Section: Circulant Matricesmentioning

confidence: 99%

Notes on Markov embedding

Baake

Sumner

2020

Linear Algebra and its Applications

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Section: Circulant Matricesmentioning

confidence: 99%

Notes on Markov embedding

Baake

Sumner

2020

Linear Algebra and its Applications

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“…Putting these lemmas together we obtain the following theorem and the hierarchies of Jordan-Markov models presented in Figure 1. The cases n = 3 and n = 4 include additional cases due to the appearance of the normal subgroups C 3 and V 4 , respectively, and hence fall under the purview of 'group-based' models (see [15], and also [18] for updated perspectives on this class of models).…”

Section: Noting Eimentioning

confidence: 99%

Uniformization stable Markov models and their Jordan algebraic structure

Cooper¹,

Sumner²

2021

Preprint

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We provide a characterisation of the continuous-time Markov models where there exists a elementary relationship between the associated Markov (stochastic) matrices and the rate matrices (generators). We connect this property to the well-known uniformization procedure for continuous-time Markov chains and show that the existence of an underlying Jordan algebra provides a sufficient condition, which becomes necessary for (socalled) linear models. We apply our results to analyse two practically important model hierarchies obtained by assuming (i) time-reversibility and (ii) permutation symmetry, respectively.

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“…We define the stochastic cone of a Lie-Markov model L + of L to be the intersection L + := L ∩ L + GM . If the Lie algebra L of a Lie-Markov model has a stochastic basis {L 1 , L 2 , · · · , L d } , that is, a basis each of whose elements is a linear combination of the {l i j } , with positive real coefficients, then these basis elements can be taken as the extremal rays of the stochastic cone; otherwise the extremal elements (which are necessary for the specification of the model parametrization) must be specified separately from the generators themselves 27 .…”

Section: The Lie-markov Hierarchymentioning

confidence: 99%

“…that is, a basis each of whose elements is a linear combination of the {l i j } , with positive real coefficients, then these basis elements can be taken as the extremal rays of the stochastic cone; otherwise the extremal elements (which are necessary for the specification of the model parametrization) must be specified separately from the generators themselves 27 .…”

Section: The Lie-markov Hierarchymentioning

confidence: 99%

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Systematics and symmetry in molecular phylogenetic modelling: perspectives from physics

Jarvis¹,

Sumner²

2019

J. Phys. A: Math. Theor.

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The aim of this review is to present and analyze the probabilistic models of mathematical phylogenetics which have been intensively used in recent years in biology as the cornerstone of attempts to infer and reconstruct the ancestral relationships between species. We outline the development of theoretical phylogenetics, from the earliest studies based on morphological characters, through to the use of molecular data in a wide variety of forms. We bring the lens of mathematical physics to bear on the formulation of theoretical models, focussing on the applicability of many methods from the toolkit of that tradition -techniques of groups and representations to guide model specification and to exploit the multilinear setting of the models in the presence of underlying symmetries; extensions to coalgebraic properties of the generators associated to rate matrices underlying the models, and possibilities to marry these with the graphical structures (trees and networks) which form the search space for inferring evolutionary trees.Particular aspects which we wish to present to a readership accustomed to thinking from physics, include relating model classes to structural data on relevant matrix Lie algebras, as well as using manipulations with group characters (especially the operation of plethysm, for computing tensor powers) to enumerate various natural polynomial invariants, which can be enormously helpful in tying down robust, low-parameter quantities for use in inference (some of which have only come to light through our perspective). Above all, we wish to emphasize the many features of multipartite entanglement which are shared between descriptions of quantum states on the physics side, and the multi-way tensor probability arrays arising in phylogenetics. In some instances, well-known objects such as the Cayley hyperdeterminant (the 'tangle') can be directly imported into the formalism -in this case, for models with binary character traits, and for providing information about triplets of taxa. In other cases new objects appear, such as the remarkable 'squangle' invariants for quartet tree discrimination, which for DNA data are of quintic degree, with their own unique interpretation in the phylogenetic modelling context. All this hints strongly at the natural and universal presence of entanglement as a phenomenon which reaches across disciplines. We hope that this broad perspective may in turn furnish new insights of use in physics.

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Lie-Markov Models Derived from Finite Semigroups

Cited by 4 publications

References 26 publications

Notes on Markov embedding

Notes on Markov embedding

Uniformization stable Markov models and their Jordan algebraic structure

Systematics and symmetry in molecular phylogenetic modelling: perspectives from physics

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