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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: asger@birc.au.dk
∗∗ Postal address: Department of Mathematical Sciences, Aarhus University, Ny Munkegade Bldg 1530, DK-8000 Aarhus C, Denmark. Email address: jlj@imf.au.dk
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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 

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