Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-05-01T03:02:24.926Z Has data issue: false hasContentIssue false
  • English
  • Français

A uniformly convergent adaptive particle filter

Part of: Probabilistic methods, simulation and stochastic differential equations Stochastic systems and control Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Anastasia Papavasiliou
Anastasia Papavasiliou*
Affiliation:
Princeton University
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Particle filters are Monte Carlo methods that aim to approximate the optimal filter of a partially observed Markov chain. In this paper, we study the case in which the transition kernel of the Markov chain depends on unknown parameters: we construct a particle filter for the simultaneous estimation of the parameter and the partially observed Markov chain (adaptive estimation) and we prove the convergence of this filter to the correct optimal filter, as time and the number of particles go to infinity. The filter presented here generalizes Del Moral's Monte Carlo particle filter.

Keywords

Nonlinear filteringparticle filterBayesian estimation

MSC classification

Primary: 60G35: Signal detection and filtering
Secondary: 93E10: Estimation and detection 93E11: Filtering 65C50: Other computational problems in probability
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 4 , December 2005 , pp. 1053 - 1068
Copyright
© Applied Probability Trust 2005 

References

Andrieu, C. and Doucet, A. (2002). Particle filtering for partially observed Gaussian state space models. J. R. Statist. Soc. B 64, 827–836.CrossRefGoogle Scholar
Budhiraja, A. and Kushner, H. J. (2000). Approximation and limit results for nonlinear filters over an infinite time interval. II. Random sampling algorithms. SIAM J. Control Optimization 38, 1874–1908.CrossRefGoogle Scholar
Crisan, D., Del Moral, P. and Lyons, T. (1999). Discrete filtering using branching and interacting particle systems. Markov Proc. Relat. Fields 5, 293–318.Google Scholar
Del Moral, P. (1998). A uniform convergence theorem for numerical solving of the nonlinear filtering problem. J. Appl. Prob. 35, 873–884.CrossRefGoogle Scholar
Del Moral, P. and Guionnet, A. (1999). On the stability of measure valued processes with applications to filtering. C. R. Acad. Sci. Paris 329, 429–434.CrossRefGoogle Scholar
Del Moral, P. and Miclo, L. (2000). Branching and interactive particle systems approximations of Feynman–Kac formulae with application to non-linear filtering. In Séminaire de Probabilités XXXIV (Lecture Notes Maths. 1729), eds Azéma, J., Émery, M., Ledoux, M. and Yor, M., Springer, Berlin, pp. 1–145.Google Scholar
Gordon, N., Maskell, S. and Kirubarajan, T. (2002). Efficient particle filters for Joint tracking and classification. In Proc. SPIE Signal Data Process. Small Targets, SPIE, Vol. 4728, pp. 439–449.Google Scholar
Gordon, N. J., Salmond, D. J. and Smith, A. F. M. (1993). Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proc. F 140, 107–113.Google Scholar
Kitagawa, G. (1996). Monte Carlo filter and smoother for non-Gaussian nonlinear state space models. J. Comput. Graphical Statist. 5, 1–25.Google Scholar
Le Gland, F. and Oudjane, N. (2004). Stability and uniform approximation of nonlinear filters using the Hilbert metric, and application to particle filters. Ann. Appl. Prob. 14, 144–187.CrossRefGoogle Scholar
Liu, J. and West, M. (2001). Combined parameter and state estimation in simulation-based filtering. In Sequential Monte Carlo Methods in Practice, eds Doucet, A., de Freitas, N. and Gordon, N., Springer, New York, pp. 197–223.CrossRefGoogle Scholar
Papavasiliou, A. (2005). Parameter estimation and asymptotic stability in stochastic filtering. To appear in Stoch. Process. Appl. Google Scholar
Storvik, G. (2002). Particle filters for state space models with the presence of unknown static parameters. IEEE Trans. Signal Process. 50, 281–289.CrossRefGoogle Scholar