Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-19T01:57:10.767Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Inference from stochastic processes Stochastic processes

Published online by Cambridge University Press:  17 March 2017

Alexandros Beskos ,
Dan Crisan ,
Ajay Jasra ,
Kengo Kamatani  and
Yan Zhou
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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

State-space modelhigh dimensionsparticle filter

MSC classification

Primary: 62M20: Prediction
Secondary: 60G35: Signal detection and filtering
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 1 , March 2017 , pp. 24 - 48
Copyright
Copyright © Applied Probability Trust 2017 

References

[1] Bérard, J.,Del Moral, P. and Doucet, A. (2014).A lognormal central limit theorem for particle approximations of normalizing constants.Electron. J. Prob. 19,1–28.CrossRefGoogle Scholar
[2] Beskos, A.,Crisan, D. and Jasra, A. (2014).On the stability of sequential Monte Carlo methods in high dimensions.Ann. Appl. Prob. 24,1396–1445.Google Scholar
[3] Beskos, A.,Crisan, D.,Jasra, A. and Whiteley, N. P. (2014).Error bounds and normalising constants for sequential Monte Carlo samplers in high dimensions.Adv. Appl. Prob. 46,279–306.Google Scholar
[4] Bickel, P.,Li, B. and Bengtsson, T. (2008).Sharp failure rates for the bootstrap particle filter in high dimensions.In Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh(Inst. Math. Statist. (IMS) Collect. 3),eds B. Clarke and S. Ghosal,Institute of Mathematical Statistics,Beachwood, OH,pp.318–329.Google Scholar
[5] Cérou, F.,Del Moral, P. and Guyader, A. (2011).A nonasymptotic theorem for unnormalized Feynman‒Kac particle models.Ann. Inst. H. Poincaré Prob. Statist. 47,629–649.CrossRefGoogle Scholar
[6] Del Moral, P. (2004).Feynman‒Kac Formulae: Genealogical and Interacting Particle Systems with Applications.Springer,New York.Google Scholar
[7] Del Moral, P. (2013).Mean Field Simulation for Monte Carlo Integration(Monogr. Statist. Appl. Prob. 126).CRC Press,Boca Raton, FL.Google Scholar
[8] Del Moral, P.,Doucet, A. and Jasra, A. (2006).Sequential Monte Carlo samplers.J. R. Statist. Soc. B 68,411–436.Google Scholar
[9] Del Moral, P.,Doucet, A. and Jasra, A. (2012).On adaptive resampling procedures for sequential Monte Carlo methods.Bernoulli 18,252–278.Google Scholar
[10] Doucet, A. and Johansen, A. M. (2011).A tutorial on particle filtering and smoothing: fifteen years later.In The Oxford Handbook of Nonlinear Filtering,eds D. Crisan and B. Rozovsky,Oxford University Press,pp.656–704.Google Scholar
[11] Johansen, A. M.,Whiteley, N. and Doucet, A. (2012).Exact approximation of Rao‒Blackwellised particle filters.In Proc. 16th IFAC Symp. on System Identification The International Federation of Automatic Control,Brussels,pp.488–493.Google Scholar
[12] Kantas, N.,Beskos, A. and Jasra, A. (2014).Sequential Monte Carlo methods for high-dimensional inverse problems: a case study for the Navier‒Stokes equations.SIAM/ASA J. Uncertain. Quantif. 2,464–489.CrossRefGoogle Scholar
[13] Naesseth, C. A.,Lindsten, F. and Schön, T. B. (2015).Nested sequential Monte Carlo methods.In Proc. 32nd Internat. Conf. on Machine Learning,pp.1292–1301.Google Scholar
[14] Poyiadjis, G.,Doucet, A. and Singh, S. S. (2011).Particle approximations of the score and observed information matrix in state space models with application to parameter estimation.Biometrika 98,65–80.CrossRefGoogle Scholar
[15] Rebeschini, P. and Van Handel, R. (2015).Can local particle filters beat the curse of dimensionality?Ann. Appl. Prob. 25,2809–2866.Google Scholar
[16] Rubin, D. (1988).Using the SIR algorithm to simulate posterior distributions.In Bayesian Statistics 3,eds J. M. Bernado et al.,Oxford University Press,pp.395–402.Google Scholar
[17] Vergé, C.,Dubarry, C.,Del Moral, P. and Moulines, E. (2015).On parallel implementation of Sequential Monte Carlo methods: the island particle model.Statist. Comput. 25,243–260.Google Scholar