Nonlinear and Non-Gaussian Estimation with A Focus on Particle Filters

1. Nonlinear and Non-Gaussian Estimation with A Focus on Particle Filters Prasanth Jeevan Mary Knox May 12, 2006

2. Background • Optimal linear filters • Wiener  Stationary • Kalman  Gaussian Posterior, p(x|y) • Filters for nonlinear systems • Extended Kalman • Particle

3. Extended Kalman Filter (EKF) • Locally linearize the non-linear functions • Assume p(xk|y1,…,k) is Gaussian

4. Particle Filter (PF) • Weighted point mass or “particle” representation of possibly intractable posterior probability density functions, p(x|y) • Estimates recursively in time allowing for online calculations • Attempts to place particles in important regions of the posterior pdf • O(N) complexity on number of particles

5. Particle Filter Background [Ristic et. al. 2004] • Monte Carlo Estimation • Pick N>>1 “particles” with distribution p(x) Assumption: xi is independent

6. Importance Sampling • Cannot sample directly from p(x) • Instead sample from known importance density, q(x), where: • Estimate I from samples and importance weights where

7. Sequential Importance Sampling (SIS) • Iteratively represent posterior density function by random samples with associated weights Assumptions: xk Hidden Markov process, yk conditionally independent given xk

8. Degeneracy • Variance of sample weights increases with time if importance density not optimal [Doucet 2000] • In a few cycles all but one particle will have negligible weights • PF will updating particles that contribute little in approximating the posterior • Neff, estimate of effective sample size [Kong et. al. 1994]:

9. Optimal Importance Density [Doucet et. al. 2000] • Minimizes variance of importance weights to prevent degeneracy • Rarely possible to obtain, instead often use

10. Resampling • Generate new set of samples from: • Weights are equal after i.i.d. sampling • O(N) complexity • Coupled with SIS, these are the two key components of a PF

11. Sample Impoverishment • Set of particles with low diversity • Particles with high weights are selected more often

12. Sampling Importance Resampling (SIR)[Gordon et. al. 1993] • Importance density is the transitional prior • Resampling at every time step

13. SIR Pros and Cons • Pro: importance density and weight updates are easy to evaluate • Con: Observations not used when transitioning state to next time step

14. A Cycle of SIR

15. Auxiliary SIR - Motivation[Pitt and Shephard 1999] • Want to use observation when exploring the state space ( ’s) • To have particles in regions of high likelihood • Incorporate into resampling at time k-1 • Looking one step ahead to choose particles

16. ASIR - from SIR • From SIR we had • If we move the likelihood inside we get: • We don’t have though • Use , a characterization of given • such as

17. ASIR continued • So then we get: • And the new importance weight becomes:

18. ASIR Pros & Cons • Pro • Can be less sensitive to peaked likelihoods and outliers by using observation • Outliers - Model-improbable states that can result in a dramatic loss of high-weight particles • Cons • Added computation per cycle • If is a bad characterization of (ie. large process noise), then resampling suffers, and performance can degrade

19. Simulation Linear • System Equations: where v ~ N(0,6) and w ~ N(0,5)

20. Simulation Linear10 Samples

21. Simulation Linear50 Samples

22. Simulation Linear Table 1: Mean Squared Error Per Time Step

23. Simulation Nonlinear • System Equations: where v ~ N(0,6) and w ~ N(0,5)

24. Simulation Nonlinear10 Samples

25. Simulation Nonlinear50 Samples

26. Simulation Nonlinear100 Samples

27. Simulation Nonlinear1000 Samples

28. Simulation Nonlinear Table 2: Mean Squared Error Per Time Step

29. Conclusion • PF approaches KF optimal estimates as N   • PF better than EKF for nonlinear systems • ASIR generates ‘better particles’ in certain conditions by incorporating the observation • PF is applicable to a broad class of system dynamics • Simulation approaches have their own limitations • Degeneracy and sample impoverishment

30. Conclusion (2) • Particle filters composed of SIS and resampling • Many variations to improve efficiency (both computationally and for getting ‘better’ particles) • Other PFs: Regularized PF, (EKF/UKF)+PF, etc.