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Similar States in Continuous-Time Markov Chains

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

V. B. Yap
V. B. Yap*
Affiliation:
National University of Singapore
*
∗Postal address: Department of Statistics and Applied Probability, National University of Singapore, Blk S16 Level 7, 6 Science Drive 2, Singapore 117546. Email address: stayapvb@nus.edu.sg
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Abstract

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In a homogeneous continuous-time Markov chain on a finite state space, two states that jump to every other state with the same rate are called similar. By partitioning states into similarity classes, the algebraic derivation of the transition matrix can be simplified, using hidden holding times and lumped Markov chains. When the rate matrix is reversible, the transition matrix is explicitly related in an intuitive way to that of the lumped chain. The theory provides a unified derivation for a whole range of useful DNA base substitution models, and a number of amino acid substitution models.

Keywords

Holding timeuniformisationlumped Markov chainreversibility

MSC classification

Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60J35: Transition functions, generators and resolvents 92D20: Protein sequences, DNA sequences
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 2 , June 2009 , pp. 497 - 506
Copyright
Copyright © Applied Probability Trust 2009 

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