Nudging the particle filter
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Nudging the particle filter Ömer Deniz Akyildiz1,2
· Joaquín Míguez3,4
Received: 24 April 2018 / Accepted: 1 July 2019 © The Author(s) 2019
Abstract We investigate a new sampling scheme aimed at improving the performance of particle filters whenever (a) there is a significant mismatch between the assumed model dynamics and the actual system, or (b) the posterior probability tends to concentrate in relatively small regions of the state space. The proposed scheme pushes some particles toward specific regions where the likelihood is expected to be high, an operation known as nudging in the geophysics literature. We reinterpret nudging in a form applicable to any particle filtering scheme, as it does not involve any changes in the rest of the algorithm. Since the particles are modified, but the importance weights do not account for this modification, the use of nudging leads to additional bias in the resulting estimators. However, we prove analytically that nudged particle filters can still attain asymptotic convergence with the same error rates as conventional particle methods. Simple analysis also yields an alternative interpretation of the nudging operation that explains its robustness to model errors. Finally, we show numerical results that illustrate the improvements that can be attained using the proposed scheme. In particular, we present nonlinear tracking examples with synthetic data and a model inference example using real-world financial data. Keywords Particle filtering · Nudging · Robust filtering · Data assimilation · Model errors · Approximation errors.
1 Introduction 1.1 Background State-space models (SSMs) are ubiquitous in many fields of science and engineering, including weather forecasting, mathematical finance, target tracking, machine learning,
This work was partially supported by Agencia Estatal de Investigación of Spain (TEC2015-69868-C2-1-R ADVENTURE and RTI2018-099655-B-I00 CLARA), the Office of Naval Research (award no. N00014-19-1-2226), and the regional government of Madrid (program CASICAM-CM S2013/ICE-2845). Ö. D. A. is supported by the Lloyds Register Foundation programme on Data Centric Engineering through the London Air Quality project and supported by The Alan Turing Institute for Data Science and AI under EPSRC grant EP/N510129/1).
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Ömer Deniz Akyildiz [email protected]
1
University of Warwick, Coventry, UK
2
The Alan Turing Institute, London, UK
3
Universidad Carlos III de Madrid, Madrid, Spain
4
Instituto de Investigación Sanitaria Gregorio Marañón, Madrid, Spain
population dynamics, etc., where inferring the states of dynamical systems from data plays a key role. A SSM comprises a pair of stochastic processes (xt )t≥0 and (yt )t≥1 called signal process and observation process, respectively. The conditional relations between these processes are defined with a transition and an observation model (also called likelihood model) where observations are conditionally independent given the signal process, and the latter is itself a Markov process. Given an observ
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