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Summary Statistics for Endpoint-Conditioned Continuous-Time Markov Chains

Part of: Probability theory and stochastic processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Asger Hobolth  and
Jens Ledet Jensen
Asger Hobolth*
Affiliation:
Aarhus University
Jens Ledet Jensen*
Affiliation:
Aarhus University
*
∗ Postal address: Bioinformatics Research Center, Aarhus University, C. F. Møllers Alle 8, DK-8000 Aarhus C, Denmark. Email address: [email protected]
∗∗ Postal address: Department of Mathematical Sciences, Aarhus University, Ny Munkegade Bldg 1530, DK-8000 Aarhus C, Denmark. Email address: [email protected]
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Abstract

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Continuous-time Markov chains are a widely used modelling tool. Applications include DNA sequence evolution, ion channel gating behaviour, and mathematical finance. We consider the problem of calculating properties of summary statistics (e.g. mean time spent in a state, mean number of jumps between two states, and the distribution of the total number of jumps) for discretely observed continuous-time Markov chains. Three alternative methods for calculating properties of summary statistics are described and the pros and cons of the methods are discussed. The methods are based on (i) an eigenvalue decomposition of the rate matrix, (ii) the uniformization method, and (iii) integrals of matrix exponentials. In particular, we develop a framework that allows for analyses of rather general summary statistics using the uniformization method.

Keywords

Continuous-time Markov chaindwelling timeEM algorithmtransition numberuniformization

MSC classification

Secondary: 60-08: Computational methods (not classified at a more specific level) 60J22: Computational methods in Markov chains 60J25: Continuous-time Markov processes on general state spaces 60J27: Continuous-time Markov processes on discrete state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 4 , December 2011 , pp. 911 - 924
Copyright
Copyright © Applied Probability Trust 2011 

References

Ball, F. and Milne, R. K. (2005). Simple derivations of properties of counting processes associated with Markov renewal processes. J. Appl. Prob. 42, 1031–1043.Google Scholar
Bladt, M. and Sørensen, M. (2005). Statistical inference for discretely observed Markov Jump processes. J. R. Statist. Soc B 67, 395–410.Google Scholar
Bladt, M. and Sørensen, M. (2009). Efficient estimation of transition rates between credit ratings from observations at discrete time points. Quant. Finance 9, 147–160.Google Scholar
Bladt, M., Meini, B., Neuts, M. F. and Sericola, B. (2002). Distributions of reward functions on continuous-time Markov chains. In Matrix-Analytic Methods, ed. Latouche, G., World Scientific, River Edge, NJ, pp. 39–62.Google Scholar
Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. J. R. Statist. Soc B 39, 1–38.Google Scholar
Grassmann, W. K. (1993). Rounding errors in certain algorithms involving Markov chains. ACM Trans. Math. Software 19, 496–508.Google Scholar
Guttorp, P. (1995). Stochastic Modeling of Scientific Data. Chapman and Hall, London.Google Scholar
Hobolth, A. and Jensen, J. L. (2005). Statistical inference in evolutionary models of DNA sequences via the EM algorithm. Statist. Appl. Genet. Molec. Biol. 4, 20pp.Google Scholar
Holmes, I. and Rubin, G. M. (2002). An expectation maximization algorithm for training hidden substitution models. J. Molec. Biol. 317, 753–764.Google Scholar
Jensen, A. (1953). Markoff chains as an aid in the study of Markoff processes. Skand. Aktuarietidskr. 36, 87–91.Google Scholar
Klosterman, P. S. et al. (2006). XRate: a fast prototyping, training and annotation tool for phylo-grammars. BMC Bioinformatics 7, 25pp.Google Scholar
Kosiol, C., Holmes, I. and Goldman, N. (2007). An empirical codon model for protein sequence evolution. Molec. Biol. Evol. 24, 1464–1479.Google Scholar
Metzner, P., Horenko, I. and Schütte, C. (2007). Generator estimation of Markov Jump processes based on incomplete observations nonequidistant in time. Phys. Rev. E 76, 066702, 8 pp.Google Scholar
Minin, V. N. and Suchard, M. A. (2008). Counting labeled transitions in continuous-time Markov models of evolution. J. Math. Biol. 56, 391–412.Google Scholar
Moler, C. and Van Loan, C. (2003). Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45, 3–49.Google Scholar
Narayana, S. and Neuts, M. F. (1992). The first two moment matrices of the counts for the Markovian arrival process. Commun. Statist. Stoch. Models 8, 459–477.Google Scholar
Ross, S. M. (1983). Stochastic Processes. John Wiley, New York.Google Scholar
Siepel, A., Pollard, K. S. and Haussler, D. (2006). New methods for detecting lineage-specific selection. In Research in Computational Molecular Biology (Lecture Notes Bioinformatics 3909), eds Apostolico, A. et al., Springer, Berlin, pp. 190–205.Google Scholar
Van Loan, C. F. (1978). Computing integrals involving the matrix exponential. IEEE Trans. Automatic Control 23, 395–404.Google Scholar