Continuous-time Markov chains are a standard tool in phylogenetic inference.
If homogeneity is assumed, the chain is formulated by specifying
time-independent rates of substitutions between states in the chain. In
applications, there are usually extra constraints on the rates, depending on
the situation. If a model is formulated in this way, it is possible to
generalise it and allow for an inhomogeneous process, with time-dependent rates
satisfying the same constraints. It is then useful to require that there exists
a homogeneous average of this inhomogeneous process within the same model. This
leads to the definition of "Lie Markov models", which are precisely the class
of models where such an average exists. These models form Lie algebras and
hence concepts from Lie group theory are central to their derivation. In this
paper, we concentrate on applications to phylogenetics and nucleotide
evolution, and derive the complete hierarchy of Lie Markov models that respect
the grouping of nucleotides into purines and pyrimidines -- that is, models
with purine/pyrimidine symmetry. We also discuss how to handle the subtleties
of applying Lie group methods, most naturally defined over the complex field,
to the stochastic case of a Markov process, where parameter values are
restricted to be real and positive. In particular, we explore the geometric
embedding of the cone of stochastic rate matrices within the ambient space of
the associated complex Lie algebra.
The whole list of Lie Markov models with purine/pyrimidine symmetry is
available at http://www.pagines.ma1.upc.edu/~jfernandez/LMNR.pdf.Comment: 32 page