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Optimal SIR algorithm vs. fully adapted auxiliary particle filter: a non asymptotic analysis

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Abstract

Particle filters (PF) and auxiliary particle filters (APF) are widely used sequential Monte Carlo (SMC) techniques. In this paper we comparatively analyse, from a non asymptotic point of view, the Sampling Importance Resampling (SIR) PF with optimal conditional importance distribution (CID) and the fully adapted APF (FA). We compute the (finite samples) conditional second order moments of Monte Carlo (MC) estimators of a moment of interest of the filtering pdf, and analyse under which circumstances the FA-based estimator outperforms (or not) the optimal Sequential Importance Sampling (SIS)-based one. Our analysis is local, in the sense that we compare the estimators produced by one time step of the different SMC algorithms, starting from a common set of weighted points. This analysis enables us to propose a hybrid SIS/FA algorithm which automatically switches at each time step from one loop to the other. We finally validate our results via computer simulations.

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Notes

  1. In view of the analysis of Sects. 3 and 4 we consider the case where the number of samples N′ at time n is possibly different to that N at time n−1, even though in SMC techniques we often take N′=N.

  2. The bounds α n and β n given in Corollary 1 come from simple sufficient conditions, and no conclusion holds if N eff(n)∈(α n ,β n ); so for simplicity we choose in the algorithm threshold \(\frac{\alpha_{n} + \beta_{n}}{2}\). Also in practice, the time varying and difficult to compute thresholds α n and β n can be replaced by a common fixed value which nevertheless takes into account the parameters of the model; this will be our choice in the simulations section.

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Acknowledgements

The authors would like to thank the French MOD DGA/MRIS for financial support of the Ph.D. of Y. Petetin, and Dr. E. Monfrini for fruitful discussions and comments.

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  1. Telecom Institute/Telecom SudParis/CITI Department, CNRS UMR 5157, 9 rue Charles Fourier, 91011, Evry, France

    Yohan Petetin & François Desbouvries

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  1. Yohan Petetin
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  2. François Desbouvries
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Correspondence to François Desbouvries.

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Petetin, Y., Desbouvries, F. Optimal SIR algorithm vs. fully adapted auxiliary particle filter: a non asymptotic analysis. Stat Comput 23, 759–775 (2013). https://doi.org/10.1007/s11222-012-9345-5

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