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Lumpability and marginalisability for continuous-time Markov chains

Part of: Physiological, cellular and medical topics Markov processes

Published online by Cambridge University Press:  14 July 2016

Frank Ball  and
Geoffrey F. Yeo
Frank Ball*
Affiliation:
University of Nottingham
Geoffrey F. Yeo*
Affiliation:
Murdoch University
*
∗ Postal address: Department of Mathematics, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
∗∗ Postal address: School of Mathematical and Physical Sciences, Murdoch University, Murdoch, WA 6150, Australia.
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Abstract

We consider lumpability for continuous-time Markov chains and provide a simple probabilistic proof of necessary and sufficient conditions for strong lumpability, valid in circumstances not covered by known theory. We also consider the following marginalisability problem. Let {X{t)} = {(X1(t), X2(t), · ··, Xm(t))} be a continuous-time Markov chain. Under what conditions are the marginal processes {X1(t)}, {X2(t)}, · ··, {Xm(t)} also continuous-time Markov chains? We show that this is related to lumpability and, if no two of the marginal processes can jump simultaneously, then they are continuous-time Markov chains if and only if they are mutually independent. Applications to ion channel modelling and birth–death processes are discussed briefly.

Keywords

STRONG AND WEAK LUMPABILITYTIME REVERSIBLE PROCESSION CHANNEL MODELLINGBIRTH–DEATH PROCESSES

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 92C05: Biophysics 92C30: Physiology (general)
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 3 , September 1993 , pp. 518 - 528
Copyright
Copyright © Applied Probability Trust 1993 

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References

Ball, F. G. and Sansom, M. S. P. (1988) Aggregated Markov processes incorporating time interval omission. Adv. Appl. Prob. 20, 546–572.Google Scholar
Ball, F. G., Milne, R. K. and Yeo, G. F. (1991) Aggregated semi-Markov processes incorporating time interval omission. Adv. Appl. Prob. 23, 772–797.Google Scholar
Burke, C. J. and Rosenblatt, M. (1958) A Markovian function of a Markov chain. Ann. Math. Statist. 29, 1112–1122.Google Scholar
Colquhoun, D. and Hawkes, A. G. (1977) Relaxation and fluctuations of membrane currents that flow through drug-operated channels. Proc. R. Soc. London B 199, 231–262.Google Scholar
Colquhoun, D. and Hawkes, A. G. (1982) On the stochastic properties of bursts of single ion channel openings and of clusters of bursts. Phil. Trans. R. Soc. London B 300, 1–59.Google Scholar
Colquhoun, D. and Hawkes, A. G. (1983) The principles of stochastic interpretation of ionchannel mechanisms. In Single-Channel Recording, ed. Sakmann, B. and Neher, E., Plenum Press, New York, 135–175.Google Scholar
Dabrowski, A. R., Mcdonald, D. and Rösler, U. (1990) Renewal properties of ion channels. Ann. Statist. 18, 1091–1115.Google Scholar
Disney, R. L. and Kiessler, P. C. (1987) Traffic Processes in Queueing Networks. A Markov Renewal Approach. Johns Hopkins University Press, Baltimore, MD.Google Scholar
Fredkin, D. R. and Rice, J. A. (1986) On aggregated Markov processes. J. Appl. Prob. 23, 208–214.Google Scholar
Fredkin, D. R. and Rice, J. A. (1991) On the superposition of currents from ion channels. Phil. Trans. R. Soc. London B334, 357–384.Google Scholar
Fredkin, D. R., Montal, M. and Rice, J. A. (1985) Identification of aggregated Markovian models: application to the nicotinic acetylcholine receptor. In Proc. Berkeley Conf. in Honor of Jerzy Neyman and Jack Kiefer 1, ed. Le Cam, L. and Olshen, R., Wadsworth, Belmont, CA, 269–290.Google Scholar
Hachigan, J. (1963) Collapsed Markov chains and the Chapman-Kolmogorov equation. Ann. Math. Statist. 34, 233–237.Google Scholar
Hachigan, J. and Rosenblatt, M. (1962) Functions of reversible Markov processes that are Markovian. J. Math. Mech. 11, 951–960.Google Scholar
Keilson, J. (1979) Markov Chain Models-Rarity and Exponentiality. Springer-Verlag, New York.Google Scholar
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, Chichester.Google Scholar
Kemeny, J. and Snell, J. (1976) Finite Markov Chains, 2nd edn. Springer-Verlag, New York.Google Scholar
Kijima, S. and Kijima, H. (1987) Statistical analysis of channel current from a membrane patch II. A stochastic theory of a multi-channel system in the steady state. J. Theoret. Biol. 128, 435–455.Google Scholar
Leysieffer, F. W. (1967) Functions of finite Markov chains. Ann. Math. Statist. 38, 206–212.Google Scholar
Renshaw, E. (1986) A survey of stepping-stone models in population dynamics. Adv. Appl. Prob. 18, 581–627.Google Scholar
Rosenblatt, M. (1959) Functions of a Markov process that are Markovian. J. Math. Mech. 8, 585–596.Google Scholar
Sumita, U. and Rieders, M. (1989) Lumpability and time reversibility in the aggregationdisaggregation method for large Markov chains. Commun. Statist. - Stoch. Models 5, 63–81.Google Scholar
Yeo, G. F., Edeson, R. O., Milne, R. K. and Madsen, R. W. (1989) Superposition properties of independent ion channels. Proc. R. Soc. London B 238, 155–170.Google Scholar