Abstract.

We consider the problem of high-dimensional filtering of state-space models (SSMs) at discrete times. This problem is particularly challenging as analytical solutions are typically not available and many numerical approximation methods can have a cost that scales exponentially with the dimension of the hidden state. Inspired by lag-approximation methods for the smoothing problem [G. Kitagawa and S. Sato, Monte Carlo smoothing and self-organising state-space model, in Sequential Monte Carlo Methods in Practice, Springer, New York, 2001, pp. 178–195; J. Olsson et al., Bernoulli, 14 (2008), pp. 155–179], we introduce a lagged approximation of the smoothing distribution that is necessarily biased. For certain classes of SSMs, particularly those that forget the initial condition exponentially fast in time, the bias of our approximation is shown to be uniformly controlled in the dimension and exponentially small in time. We develop a sequential Monte Carlo (SMC) method to recursively estimate expectations with respect to our biased filtering distributions. Moreover, we prove for a class of SSMs that can contain dependencies amongst coordinates that as the dimension \(d\rightarrow \infty\) the cost to achieve a stable mean square error in estimation, for classes of expectations, is of \(\mathcal{O}(Nd^2)\) per unit time, where \(N\) is the number of simulated samples in the SMC algorithm. Our methodology is implemented on several challenging high-dimensional examples including the conservative shallow-water model.

Keywords

  1. filtering
  2. sequential Monte Carlo
  3. lag approximations
  4. high-dimensional particle filter

MSC codes

  1. 62M20
  2. 60G35
  3. 60J20
  4. 60J10
  5. 94A12
  6. 93E11

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Information & Authors

Information

Published In

cover image SIAM/ASA Journal on Uncertainty Quantification
SIAM/ASA Journal on Uncertainty Quantification
Pages: 1130 - 1161
ISSN (online): 2166-2525

History

Submitted: 4 October 2021
Accepted: 29 March 2022
Published online: 28 September 2022

Keywords

  1. filtering
  2. sequential Monte Carlo
  3. lag approximations
  4. high-dimensional particle filter

MSC codes

  1. 62M20
  2. 60G35
  3. 60J20
  4. 60J10
  5. 94A12
  6. 93E11

Authors

Affiliations

Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal, 23955, KSA ([email protected], [email protected], [email protected])
Aimad Er-raiy
Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal, 23955, KSA ([email protected], [email protected], [email protected])
Department of Statistical Science, University College London, London, WC1E 6BT, UK ([email protected])
Dan Crisan
Department of Mathematics, Imperial College London, London, SW7 2AZ, UK ([email protected], [email protected])
Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal, 23955, KSA ([email protected], [email protected], [email protected])
Nikolas Kantas
Department of Mathematics, Imperial College London, London, SW7 2AZ, UK ([email protected], [email protected])

Funding Information

The work of the first and fifth authors was supported by KAUST baseline funding. The work of the fourth author was partially supported by European Research Council (ERC) Synergy grant STUOD-DLV-8564. The work of the sixth author was supported by a J.P. Morgan A.I. Research Award.

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