title: Counting labeled transitions in continuous-time Markov models of evolution.
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.
authors.affiliation: Department of Biomathematics, David Geffen School of Medicine at UCLA, Los Angeles, CA 90095, USA. vminin@stat.washington.edu
authors.forename: Vladimir N
authors.initials: VN
authors.lastname: Minin
authors.forename: Marc A
authors.initials: MA
authors.lastname: Suchard
mesh_terms: D000465:Algorithms; D005075:Biological Evolution; D003198:Computer Simulation; D019143:Evolution, Molecular; D008390:Markov Chains; D008957:Models, Genetic; D009154:Mutation / Q000235:genetics; D010641:Phenotype; D010802:Phylogeny