A tensorial approach to the inversion of group-based phylogenetic models

BMC Evolutionary Biology

Barbara R Holland¹  Peter D Jarvis¹  Jeremy G Sumner¹ 

[1] School of Physical Sciences, University of Tasmania, Hobart TAS 7001, Australia

关键词: Markov chains;    Symmetry;    Representation theory;    Groups;   

Others  :  1121794
DOI  :  10.1186/s12862-014-0236-6

 received in 2014-04-15, accepted in 2014-11-06,  发布年份 2014

【 摘 要 】

Background

Hadamard conjugation is part of the standard mathematical armoury in the analysis of molecular phylogenetic methods. For group-based models, the approach provides a one-to-one correspondence between the so-called “edge length” and “sequence” spectrum on a phylogenetic tree. The Hadamard conjugation has been used in diverse phylogenetic applications not only for inference but also as an important conceptual tool for thinking about molecular data leading to generalizations beyond strictly tree-like evolutionary modelling.

Results

For general group-based models of phylogenetic branching processes, we reformulate the problem of constructing a one-one correspondence between pattern probabilities and edge parameters. This takes a classic result previously shown through use of Fourier analysis and presents it in the language of tensors and group representation theory. This derivation makes it clear why the inversion is possible, because, under their usual definition, group-based models are defined for abelian groups only.

Conclusion

We provide an inversion of group-based phylogenetic models that can implemented using matrix multiplication between rectangular matrices indexed by ordered-partitions of varying sizes. Our approach provides additional context for the construction of phylogenetic probability distributions on network structures, and highlights the potential limitations of restricting to group-based models in this setting.

【 授权许可】

   
2014 Sumner et al.; licensee BioMed Central Ltd.

【 图 表 】

Figure 1.

Figure 2.

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