Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T11:04:50.406Z Has data issue: false hasContentIssue false
  • English
  • Français

A canonical representation for aggregated Markov processes

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Bret Larget
Bret Larget*
Affiliation:
Duquesne University
*
∗Postal address: Department of Mathematics and Computer Science, Duquesne University, College Hall 440, Pittburgh, PA 15282–1704, USA. E-mail address: larget@mathcs.duq.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

A deterministic function of a Markov process is called an aggregated Markov process. We give necessary and sufficient conditions for the equivalence of continuous-time aggregated Markov processes. For both discrete- and continuous-time, we show that any aggregated Markov process which satisfies mild regularity conditions can be directly converted to a canonical representation which is unique for each class of equivalent models, and furthermore, is a minimal parameterization of all that can be identified about the underlying Markov process. Hidden Markov models on finite state spaces may be framed as aggregated Markov processes by expanding the state space and thus also have canonical representations.

Keywords

Aggregated Markov processcanonical representationhidden Markov modelidentifiabilityequivalenceminimal parameterization

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 2 , June 1998 , pp. 313 - 324
Copyright
Copyright © Applied Probability Trust 1998 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baldi, P., Chauvin, Y., Hunkapiller, T., and McClure, M.A. (1994). Hidden Markov models of biological primary sequence information. Proc. Nat. Acad. Sci. 91, 1059–1063.Google Scholar
Blackwell, D., and Koopmans, L. (1957). On the identifiability problem for functions of finite Markov chains. Ann. Math. Statist. 28, 1011–1015.CrossRefGoogle Scholar
Felsenstein, J., and Churchill, G.A. (1996). A hidden Markov model approach to variation among sites in rate of evolution. Molecular Biology and Evolution 13, 93–104.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. (1985). Identification of aggregated Markovian Models: Application to the nicotinic acetylcholine receptor. Proc. Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, I, eds Le Cam, L.M. and Olshen, R.A. Wadsworth Inc., pp. 269–289.Google Scholar
Gilbert, E.J. (1959). On the identifiability problem for functions of finite Markov chains. Ann. Math. Statist. 30, 688–697.Google Scholar
Ito, H., Amari, S., and Kobayashi, K. (1992). Identifiability of hidden Markov information sources and their minimum degrees of freedom. IEEE Trans. Inf. Theory 38, 324–333.Google Scholar
Kienker, P. (1989). Equivalence of aggregated Markov models of ion-channel gating. Proc. R. Soc. London B. 236, 269–309.Google Scholar
Rabiner, L.R. (1989). A tutorial on hidden Markov models and selected applications in speech recognition. Proc. IEEE 77, 257–286.Google Scholar
Rydén, T. (1993). On identifiability and order of continuous-time hidden Markov chains and Markov modulated Poisson processes. Preprint. Lund University and Lund Institute of Technology.Google Scholar
Zucchini, W., and Guttorp, P. (1991). A hidden Markov model for space-time precipitation. Water Resources Res. 27, 1917–1923.Google Scholar