Kybernetika 52 no. 5, 735-756, 2016

On convergence of kernel density estimates in particle filtering

David CoufalDOI: 10.14736/kyb-2016-5-0735

Abstract:

The paper deals with kernel density estimates of filtering densities in the particle filter. The convergence of the estimates is investigated by means of Fourier analysis. It is shown that the estimates converge to the theoretical filtering densities in the mean integrated squared error. An upper bound on the convergence rate is given. The result is provided under a certain assumption on the Sobolev character of the filtering densities. A sufficient condition is presented for the persistence of this Sobolev character over time.

Keywords:

particle filter, kernel methods, Fourier analysis

Classification:

65C35

paper.pdf

References:

  1. J. Brabec and B. Hrůza: Matematická analýza II (Mathematical Analysis II, in Czech). SNTL/ALFA, 1986.   CrossRef
  2. D. Crisan and A. Doucet: A survey of convergence results on particle filtering methods for practitioners. IEEE Trans. Signal Processing 50 (2002), 3, 736-746.   DOI:10.1109/78.984773
  3. D. Crisan and J. Míguez: Particle-kernel estimation of the filter density in state-space models. Bernoulli 20 (2014), 4, 1879-1929.   DOI:10.3150/13-bej545
  4. A. Doucet, N. de Freitas and N. Gordon (Eds.): Sequential Monte Carlo Methods in Practice. Springer-Verlag, New York 2001.   DOI:10.1007/978-1-4757-3437-9
  5. A. Doucet and A. M. Johansen: A tutorial on particle filtering and smoothing: fifteen years later. In: The Oxford Handbook of Nonlinear Filtering (D. Crisan and B. Rozovskii, eds.), Oxford University Press, 2011.   CrossRef
  6. B. Fristedt, N. Jain and N. Krylov: Filtering and Prediction: A Primer. American Mathematical Society, 2007.   DOI:10.1090/stml/038
  7. G. H. Givens: Consistency of the local kernel density estimator. Statist. Probab. Lett. 25 (1995), 55-61.   DOI:10.1016/0167-7152(94)00205-m
  8. K. Heine and D. Crisan: Uniform approximations of discrete-time filters. Adv. Appl. Probab. 40 (2008), 4, 979-1001.   DOI:10.1239/aap/1231340161
  9. M. Hürzeler and H. R. Künsch: Monte Carlo approximations for general state-space models. J. Comput. Graph. Statist. 7 (1998), 2, 175-193.   DOI:10.2307/1390812
  10. H. R. Künsch: Recursive Monte Carlo filters: Algorithms and theoretical bounds. Ann. Statist. 33 (2005), 5, 1983-2021.   DOI:10.1214/009053605000000426
  11. F. Le Gland and N. Oudjane: Stability and uniform approximation of nonlinear filters using the Hilbert metric and application to particle filters. Ann. Appl. Probab. 14 (2004), 1, 144-187.   DOI:10.1214/aoap/1075828050
  12. P. Del Moral and A. Guionnet: On the stability of interacting processes with applications to filtering and genetic algorithms. Annales de l'institut Henri Poincaré (B) Probabilités et Statistiques 37 (2001), 2, 155-194.   DOI:10.1016/s0246-0203(00)01064-5
  13. C. Musso, N. Oudjane and F. Le Gland: Improving regularised particle filters. In: Sequential Monte Carlo Methods in Practice (A. Doucet, N. Freitas, and N. Gordon, eds.), Chapter 12, Springer 2001, pp. 247-272.   DOI:10.1007/978-1-4757-3437-9_12
  14. E. Parzen: On estimation of a probability density function and mode. Ann. Math. Statist. 33 (1962), 3, 1065-1076.   DOI:10.1214/aoms/1177704472
  15. S. Särkkä: Bayesian Filtering and Smoothing. Cambridge University Press, 2013.   CrossRef
  16. B. W. Silverman: Density Estimation for Statistics and Data Analysis. Chapman and Hall/CRC, London, New York 1986.   DOI:10.1007/978-1-4899-3324-9
  17. A. B. Tsybakov: Introduction to Nonparametric Estimation. Springer, 2009.   DOI:10.1007/b13794
  18. M. P. Wand and M. C. Jones: Kernel Smoothing. Chapman and Hall/CRC, London, New York 1995.   DOI:10.1007/978-1-4899-4493-1