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A Lagged Particle Filter for Stable Filtering of Certain High-Dimensional State-Space Models

Ruzayqat, Hamza; Er-raiy, Aimad; Beskos, Alexandros; Crisan, Dan; Jasra, Ajay; Kantas, Nikolas; (2022) A Lagged Particle Filter for Stable Filtering of Certain High-Dimensional State-Space Models. SIAM/ASA Journal on Uncertainty Quantification , 10 (3) pp. 1130-1161. 10.1137/21M1450392. Green open access

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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→∞ the cost to achieve a stable mean square error in estimation, for classes of expectations, is of O(Nd2) 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.

Type: Article
Title: A Lagged Particle Filter for Stable Filtering of Certain High-Dimensional State-Space Models
Open access status: An open access version is available from UCL Discovery
DOI: 10.1137/21M1450392
Publisher version: https://doi.org/10.1137/21M1450392
Language: English
Additional information: This version is the version of record. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: filtering, sequential Monte Carlo, lag approximations, high-dimensional particle filter
UCL classification: UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Statistical Science
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL
URI: https://discovery.ucl.ac.uk/id/eprint/10146058
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