Uniformization stable Markov models and their Jordan algebraic structure

Abstract

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.

On equal-input and monotone Markov matrices

“…A mild extension of the equal-input class is provided by the Tamura-Nei (TN) model from [43]; see also [11]. Here, one considers Markov matrices of the form…”

“…Similarly, when the row sums are all 0, we shall speak of a TN generator. The algebraic structure of TN generators is such that their exponential is always a Markov matrix of TN type, see [11], while it is not clear that a real logarithm should preserve the structure. TN matrices occur within the often-used hierarchy of time-reversible models, like those implemented in popular computational phylogenetics software such as [16].…”

“…with κ = max(c 1 , c 2 , c 3 ), which holds for both choices w = ±|w|. The corresponding two generators read(11) …”