Johansen, Adam M., Singh, Sumeetpal S. (Sumeetpal Sidhu), Doucet, Arnaud and Vo, Ba-Ngu. (2006) Convergence of the SMC implementation of the PHD filter. Methodology and Computing in Applied Probability, Vol.8 (No.2). pp. 265-291. ISSN 1387-5841

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Abstract

The probability hypothesis density (PHD) filter is a first moment approximation to the evolution of a dynamic point process which can be used to approximate the optimal filtering equations of the multiple-object tracking problem. We show that, under reasonable assumptions, a sequential Monte Carlo (SMC) approximation of the PHD filter converges in mean of order p ≥ 1, and hence almost surely, to the true PHD filter. We also present a central limit theorem for the SMC approximation, show that the variance is finite under similar assumptions and establish a recursion for the asymptotic variance. This provides a theoretical justification for this implementation of a tractable multiple-object filtering methodology and generalises some results from sequential Monte Carlo theory.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH):	Filters (Mathematics), Point processes
Journal or Publication Title:	Methodology and Computing in Applied Probability
Publisher:	Springer
ISSN:	1387-5841
Date:	2006
Volume:	Vol.8
Number:	No.2
Page Range:	pp. 265-291
Identification Number:	10.1007/s11009-006-8552-y
Status:	Peer Reviewed
Access rights to Published version:	Restricted or Subscription Access
References:	1. M. Abramowitz and I. A. Stegun, editors. Handbook of Mathematical Functions. Dover, ninth printing edition, 1970. 2. P. Billingsley. Probability and Measure. Wiley Series in Probability and Mathematical Statistics. John Wiley and Sons, second edition, 1986. 3. N. Chopin. Central limit theorem for sequential Monte Carlo methods and its applications to Bayesian inference. Annals of Statistics, 32(6):2385-2411, December 2004. 4. D. J. Daley and D. Vere-Jones. An Introduction to the Theory of Point Processes, volume I: Elementary Theory and Methods of Probability and Its Applications. Springer, New York, second edition, 2003. 5. P. Del Moral. Feynman-Kac formulae: genealogical and interacting particle systems with applications. Probability and Its Applications. Springer Verlag, New York, 2004. 6. P. Del Moral and A. Guionnet. Central limit theorem for non linear filtering and interacting particle systems. Annals of Applied Probability, 9(2):275-297, 1999. 7. A. Doucet, N. de Freitas, and N. Gordon, editors. Sequential Monte Carlo Methods in Practice. Statistics for Engineering and Information Science. Springer Verlag, New York, 2001. 8. I. Gentil, B. R´emillard, and P. Del Moral. Filtering of images for detecting multiple targets trajectories. In Statistical Modeling and Analysis for Complex Data Problems. Springer, April 2005. To Appear. 9. I. R. Goodman, R. P. S. Mahler, and H. T. Nguyen. Mathematics of Data Fusion. Kluwer Academic Publishers, 1997. 10. H. R. K¨unsch. Recursive Monte Carlo filters: Algorithms and theoretical analysis. Research Report 112, ETH: Seminar F¨ur Statistik, 2003. To appear in Annals of Statistics. 11. R. P. S. Mahler. Multitarget Bayes filtering via first-order multitarget moments. IEEE Transactions on Aerospace and Electronic Systems, 39(4):1152, October 2003. 12. C. P. Robert and G. Casella. Monte Carlo Statistical Methods. Springer Verlag, New York, second edition, 2004. 13. A. N. Shiryaev. Probability. Number 95 in Graduate Texts in Mathematics. Springer Verlag, New York, second edition, 1995. 14. B. Vo, S.S. Singh, and A. Doucet. Sequential Monte Carlo methods for multi-target filtering with random finite sets. IEEE Transactions on Aerospace and Electronic Systems, June 2005. To Appear. http://www2.ee.unimelb.edu.au/staff/bv/vo/phdfilterAESrevfinal2cola.pdf.
URI:	http://wrap.warwick.ac.uk/id/eprint/37284

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