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

  • Alexandros Beskos (a1), Dan Crisan (a2), Ajay Jasra (a3), Kengo Kamatani (a4) and Yan Zhou (a3)...
Abstract
Abstract

We consider the numerical approximation of the filtering problem in high dimensions, that is, when the hidden state lies in ℝ d 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(nNd 2), 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.

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Corresponding author
* 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.
**** Email address: staja@nus.edu.sg
***** Postal address: Graduate School of Engineering Science, Osaka University, Osaka 565-0871, Japan.
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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1] J. Bérard , P. Del Moral and A. Doucet (2014).A lognormal central limit theorem for particle approximations of normalizing constants.Electron. J. Prob. 19,128.

[2] A. Beskos , D. Crisan and A. Jasra (2014).On the stability of sequential Monte Carlo methods in high dimensions.Ann. Appl. Prob. 24,13961445.

[3] A. Beskos , D. Crisan , A. Jasra and N. P. Whiteley (2014).Error bounds and normalising constants for sequential Monte Carlo samplers in high dimensions.Adv. Appl. Prob. 46,279306.

[5] F. Cérou , P. Del Moral and A. Guyader (2011).A nonasymptotic theorem for unnormalized Feynman‒Kac particle models.Ann. Inst. H. Poincaré Prob. Statist. 47,629649.

[6] P. Del Moral (2004).Feynman‒Kac Formulae: Genealogical and Interacting Particle Systems with Applications.Springer,New York.

[9] P. Del Moral , A. Doucet and A. Jasra (2012).On adaptive resampling procedures for sequential Monte Carlo methods.Bernoulli 18,252278.

[12] N. Kantas , A. Beskos and A. Jasra (2014).Sequential Monte Carlo methods for high-dimensional inverse problems: a case study for the Navier‒Stokes equations.SIAM/ASA J. Uncertain. Quantif. 2,464489.

[14] G. Poyiadjis , A. Doucet and S. S. Singh (2011).Particle approximations of the score and observed information matrix in state space models with application to parameter estimation.Biometrika 98,6580.

[15] P. Rebeschini and R. Van Handel (2015).Can local particle filters beat the curse of dimensionality?Ann. Appl. Prob. 25,28092866.

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Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
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