Article

Journal of Mathematical Biology

, Volume 56, Issue 3, pp 391-412

First online:

Counting labeled transitions in continuous-time Markov models of evolution

  • Vladimir N. MininAffiliated withDepartment of Biomathematics, David Geffen School of Medicine at UCLADepartment of Statistics, University of Washington
  • , Marc A. SuchardAffiliated withDepartment of Biomathematics, David Geffen School of Medicine at UCLADepartment of Biostatistics, UCLA School of Public HealthDepartment of Human Genetics, David Geffen School of Medicine at UCLA Email author 

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Abstract

Counting processes that keep track of labeled changes to discrete evolutionary traits play critical roles in evolutionary hypothesis testing. If we assume that trait evolution can be described by a continuous-time Markov chain, then it suffices to study the process that counts labeled transitions of the chain. For a binary trait, we demonstrate that it is possible to obtain closed-form analytic solutions for the probability mass and probability generating functions of this evolutionary counting process. In the general, multi-state case we show how to compute moments of the counting process using an eigen decomposition of the infinitesimal generator, provided the latter is a diagonalizable matrix. We conclude with two examples that demonstrate the utility of our results.

Keywords

Counting processes Continuous-time Markov chains Evolution Phylogenetics

Mathematics Subject Classification (2000)

60J27 92D15 92D20