Journal of Mathematical Biology

, Volume 56, Issue 3, pp 391–412

Counting labeled transitions in continuous-time Markov models of evolution

Authors

  • Vladimir N. Minin
    • Department of BiomathematicsDavid Geffen School of Medicine at UCLA
    • Department of StatisticsUniversity of Washington
    • Department of BiomathematicsDavid Geffen School of Medicine at UCLA
    • Department of BiostatisticsUCLA School of Public Health
    • Department of Human GeneticsDavid Geffen School of Medicine at UCLA
Article

DOI: 10.1007/s00285-007-0120-8

Cite this article as:
Minin, V.N. & Suchard, M.A. J. Math. Biol. (2008) 56: 391. doi:10.1007/s00285-007-0120-8

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 processesContinuous-time Markov chainsEvolutionPhylogenetics

Mathematics Subject Classification (2000)

60J2792D1592D20

Copyright information

© Springer-Verlag 2007