, Volume 56, Issue 3, pp 391-412
Date: 14 Sep 2007

Counting labeled transitions in continuous-time Markov models of evolution

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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.

V.N.M. was supported by a Dissertation Year Fellowship from the UCLA Graduate Division. M.A.S. is an Alfred P. Sloan Research Fellow.