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(x k |y 1,…,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: x i 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: x k Hidden Markov process, y k conditionally independent given x k

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 N eff, 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 Linear 10 Samples

21. Simulation Linear 50 Samples

22. Simulation Linear Table 1: Mean Squared Error Per Time Step Number of Particles Filter10501001000 KF0.03490.0351 0.0350 0.0352 ASIR0.77920.08860.04170.0350 SIR0.90530.09770.04960.0354

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

24. Simulation Nonlinear 10 Samples

25. Simulation Nonlinear 50 Samples

26. Simulation Nonlinear 100 Samples

27. Simulation Nonlinear 1000 Samples

28. Simulation Nonlinear Table 2: Mean Squared Error Per Time Step Number of Particles Filter10501001000 EKF 812.08826.20827.94838.75 ASIR 30.14 20.1518.8117.86 SIR37.9722.62 21.49 19.78

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.