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A stable particle filter for a class of high-dimensional state-space models

Published online by Cambridge University Press:  17 March 2017

Alexandros Beskos*
Affiliation:
University College London
Dan Crisan*
Affiliation:
Imperial College London
Ajay Jasra*
Affiliation:
National University of Singapore
Kengo Kamatani*
Affiliation:
Osaka University
Yan Zhou*
Affiliation:
National University of Singapore
*
* Postal address: Department of Statistical Science, University College London, London WC1E 6BT, UK.
** Postal address: Department of Mathematics, Imperial College London, London SW7 2AZ, UK.
*** Postal address: Department of Statistics and Applied Probability, National University of Singapore, 117546, Singapore.
***** Postal address: Graduate School of Engineering Science, Osaka University, Osaka 565-0871, Japan.
*** Postal address: Department of Statistics and Applied Probability, National University of Singapore, 117546, Singapore.
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Abstract

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We consider the numerical approximation of the filtering problem in high dimensions, that is, when the hidden state lies in ℝd with large d. For low-dimensional problems, one of the most popular numerical procedures for consistent inference is the class of approximations termed particle filters or sequential Monte Carlo methods. However, in high dimensions, standard particle filters (e.g. the bootstrap particle filter) can have a cost that is exponential in d for the algorithm to be stable in an appropriate sense. We develop a new particle filter, called the space‒time particle filter, for a specific family of state-space models in discrete time. This new class of particle filters provides consistent Monte Carlo estimates for any fixed d, as do standard particle filters. Moreover, when there is a spatial mixing element in the dimension of the state vector, the space‒time particle filter will scale much better with d than the standard filter for a class of filtering problems. We illustrate this analytically for a model of a simple independent and identically distributed structure and a model of an L-Markovian structure (L≥ 1, L independent of d) in the d-dimensional space direction, when we show that the algorithm exhibits certain stability properties as d increases at a cost 𝒪(nNd2), where n is the time parameter and N is the number of Monte Carlo samples, which are fixed and independent of d. Our theoretical results are also supported by numerical simulations on practical models of complex structures. The results suggest that it is indeed possible to tackle some high-dimensional filtering problems using the space‒time particle filter that standard particle filters cannot handle.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2017 

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