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On the Relationships Between Lumpability and Filtering of Finite Stochastic Systems

Part of: Markov processes Stochastic systems and control

Published online by Cambridge University Press:  14 July 2016

James Ledoux ,
Langford B. White  and
Gary D. Brushe
James Ledoux*
Affiliation:
Université de Poitiers and CNRS UMR 6086
Langford B. White*
Affiliation:
University of Adelaide
Gary D. Brushe*
Affiliation:
Defence Science and Technology Organisation
*
∗Postal address: Département de Mathématiques, Université de Poitiers, BP 30179, Téléport 2, Boulevard Marie et Pierre Curie, F-86962 Futuroscope-Chasseneuil cedex, France.
∗∗Postal address: School of Electrical and Electronic Engineering, University of Adelaide, SA 5005, Australia.
∗∗∗Postal address: Signals Analysis Discipline, C3I Division, Defence Science and Technology Organisation, PO Box 1500, Edinburgh, SA 5111, Australia.
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Abstract

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The aim of this paper is to provide the conditions necessary to reduce the complexity of state filtering for finite stochastic systems (FSSs). A concept of lumpability for FSSs is introduced. In this paper we assert that the unnormalised filter for a lumped FSS has linear dynamics. Two sufficient conditions for such a lumpability property to hold are discussed. We show that the first condition is also necessary for the lumped FSS to have linear dynamics. Next, we prove that the second condition allows the filter of the original FSS to be obtained directly from the filter for the lumped FSS. Finally, we generalise an earlier published result for the approximation of a general FSS by a lumpable FSS.

Keywords

Optimal filteringmodel reductionhidden Markov chaindiscrete Markovian arrival process

MSC classification

Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 93E11: Filtering
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 3 , September 2008 , pp. 650 - 669
Copyright
Copyright © Applied Probability Trust 2008 

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