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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 

References

[1] Ball, F. and Yeo, G. F. (1993). Lumpability and marginalisability for continuous-time Markov chains. J. Appl. Prob. 30, 518–528.Google Scholar
[2] Ewens, W. J. and Grant, G. R. (2005). Statistical Methods in Bioinformatics: An Introduction, 2nd edn. Springer, New York.Google Scholar
[3] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
[4] Felsenstein, J. (1981). Evolutionary trees from DNA sequences: a maximum likelihood approach. J. Molec. Evol. 17, 368–376.Google Scholar
[5] Felsenstein, J. (2004). Inferring Phylogenies. Sinauer, New York.Google Scholar
[6] Graur, D. and Li, W.-H. (2000). Fundamentals of Molecular Evolution. Sinauer, Sunderland, MA.Google Scholar
[7] Hasegawa, M. and Fujiwara, M. (1993). Relative efficiencies of the maximum likelihood, maximum parsimony, and neighbor-joining methods for estimating protein phylogeny. Molec. Phylogenet. Evol. 2, 1–5.Google Scholar
[8] Hasegawa, M., Kishino, H. and Yano, T. (1985). Phylogenetic relationships among eukaryotic kingdoms inferred from ribosomal RNA sequences. J. Molec. Evol. 22, 32–38.Google Scholar
[9] Jukes, T. H. and Cantor, C. (1969). Evolution of protein molecules. In Mammalian Protein Metabolism. Academic Press, New York, pp. 21–132.Google Scholar
[10] Keilson, J. (1979). Markov Chain Models—Rarity and Exponentiality (Appl. Math. Sci. 28). Springer, New York.Google Scholar
[11] Kelly, F. P. (1979). Reversibility and Stochastic Networks. John Wiley, Chichester.Google Scholar
[12] Kimura, M. (1980). A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences. J. Molec. Evol. 16, 111–120.Google Scholar
[13] Kishino, H. and Hasegawa, M. (1989). Evaluation of the maximum likelihood estimate of the evolutionary tree topologies from DNA sequence data, and the branching order in Hominoidea. J. Molec. Evol. 29, 170–179.Google Scholar
[14] Norris, J. R. (1997). Markov Chains. Cambridge University Press.Google Scholar
[15] Schadt, E. E., Sinsheimer, J. S. and Lange, K. (1989). Computational advances in maximum likelihood methods for molecular phylogeny. Genome Res. 8, 222–233.Google Scholar
[16] Tamura, K. (1992). Estimation of the number of nucleotide substitutions when there are strong transition-transversion and G+C content biases. Molec. Biol. Evol. 9, 678–687.Google Scholar
[17] Tamura, K. and Nei, M. (1993). Estimation of the number of nucleotide substitutions in the control region of mitochondrial DNA in humans and chimpanzees. Molec. Biol. Evol. 10, 512–526.Google Scholar