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

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Title: A stable particle filter for a class of high-dimensional state-space models
Authors: Beskos, A
Crisan, D
Jasra, A
Kamatani, K
Zhou, Y
Item Type: Journal Article
Abstract: 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(nN d2), 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.
Issue Date: 17-Mar-2017
Date of Acceptance: 1-Mar-2017
URI: http://hdl.handle.net/10044/1/49663
DOI: https://dx.doi.org/10.1017/apr.2016.77
ISSN: 0001-8678
Publisher: Applied Probability Trust
Start Page: 24
End Page: 48
Journal / Book Title: Advances in Applied Probability
Volume: 49
Issue: 1
Copyright Statement: © 2017 Applied Probability Trust. First published in Advances in Applied Probability, 49(1), 24-48. doi:10.1017/apr.2016.77.
Sponsor/Funder: Engineering & Physical Science Research Council (EPSRC)
Funder's Grant Number: EP/H000550/1
Keywords: Science & Technology
Physical Sciences
Statistics & Probability
Mathematics
State-space model
high dimensions
particle filter
MONTE-CARLO METHODS
SAMPLERS
0102 Applied Mathematics
0104 Statistics
Publication Status: Published
Appears in Collections:Pure Mathematics
Mathematics
Faculty of Natural Sciences