Journal of Mathematical Biology

, Volume 56, Issue 3, pp 391–412

Counting labeled transitions in continuous-time Markov models of evolution



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.


Counting processes Continuous-time Markov chains Evolution Phylogenetics 

Mathematics Subject Classification (2000)

60J27 92D15 92D20 


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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of BiomathematicsDavid Geffen School of Medicine at UCLALos AngelesUSA
  2. 2.Department of StatisticsUniversity of WashingtonSeattleUSA
  3. 3.Department of BiostatisticsUCLA School of Public HealthLos AngelesUSA
  4. 4.Department of Human GeneticsDavid Geffen School of Medicine at UCLALos AngelesUSA

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