Particle Filters

1. Particle Filters

2. Importance Sampling

3. Importance Sampling • Unfortunately it is often not possible to sample directly from the posterior distribution, but we can use importance sampling. • Let p(x) be a pdf from which it is difficult to draw samples. • Let xi ~ q(x), i=1, …, N, be samples that are easily generated from a proposal pdf q, which is called an importance density. • Then approximation to the density p is given by • where

4. Bayesian Importance Sampling • By drawing samples from a known easy to sample proposal distribution we obtain: • where • are normalized weights.

5. Sensor Information: Importance Sampling

6. Sequential Importance Sampling (I) • Factorizing the proposal distribution: • and remembering that the state evolution is modeled as a Markov process • we obtain a recursive estimate of the importance weights: • Factorizing is obtained by recursively applying

7. Sequential Importance Sampling (SIS) Particle Filter • SIS Particle Filter Algorithm • for i=1:N • Draw a particle • Assign a weight • end • (k is index over time and i is the particle index)

8. Rejection Sampling

9. Rejection Sampling • Let us assume that f(x)<1 for all x • Sample x from a uniform distribution • Sample c from [0,1] • if f(x) > c keep the sampleotherwise reject the sample • f(x’) • c • c’ • OK • f(x) • x • x’

10. Importance Sampling with Resampling:Landmark Detection Example

11. Distributions

12. Distributions • Wanted: samples distributed according to p(x| z1, z2, z3)

13. This is Easy! • We can draw samples from p(x|zl) by adding noise to the detection parameters.

14. Importance sampling with Resampling After Resampling

15. Particle Filter Algorithm

16. weight = target distribution / proposal distribution

17. draw xit-1from Bel(xt-1) • draw xitfrom p(xt | xit-1,ut-1) • Importance factor for xit: Particle Filter Algorithm

18. Particle Filter Algorithm • Algorithm particle_filter( St-1, ut-1 zt): • For Generate new samples • Sample index j(i) from the discrete distribution given by wt-1 • Sample from using and • Compute importance weight • Update normalization factor • Insert • For • Normalize weights

19. Particle Filter Algorithm

20. Particle Filter for Localization

21. Particle Filter in Matlab

22. Matlab code: truex is a vector of 100 positions to be tracked.

23. Application: Particle Filter for Localization (Known Map)

24. Resampling

25. Resampling

26. Resampling

27. Resampling Algorithm • Algorithm systematic_resampling(S,n): • For Generate cdf • Initialize threshold • For Draw samples … • While ( ) Skip until next threshold reached • Insert • Increment threshold • ReturnS’ • Also called stochastic universal sampling

28. Low Variance Resampling

29. SIS weights

30. Derivation of SIS weights (I) • The main idea is Factorizing : • and • Our goal is to expand p and q in time t

31. Derivation of SIS weights (II)

32. Derivation of SIS weights (II) • and under Markov assumptions

33. SIS Particle Filter Foundation • At each time step k • Random samples are drawn from the proposal distribution for i=1, …, N • They represent posterior distribution using a set of samples or particles • Since the weights are given by • and q factorizes as

34. Sequential Importance Sampling (II) • Choice of the proposal distribution: • Choose proposal function to minimize variance of (Doucet et al. 1999): • Although common choice is the prior distribution: We obtain then

35. Sequential Importance Sampling (III) • Illustration of SIS: • Degeneracy problems: • variance of importance ratios increases stochastically over time (Kong et al. 1994; Doucet et al. 1999). • In most cases then after a few iterations, all but one particle will have negligible weight

36. Sequential Importance Sampling (IV) • Illustration of degeneracy:

37. SIS - why variance increase • Suppose we want to sample from the posterior • choose a proposal density to be very close to the posterior density • Then • and • So we expect the variance to be close to 0 to obtain reasonable estimates • thus a variance increase has a harmful effect on accuracy

39. Sampling-Importance Resampling

40. Sampling-Importance Resampling • SIS suffers from degeneracy problems so we don’t want to do that! • Introduce a selection (resampling) step to eliminate samples with low importance ratios and multiply samples with high importance ratios. • Resampling maps the weighted random measure on to the equally weighted random measure • by sampling uniformly with replacement from with probabilities • Scheme generates children such that and satisfies:

41. Basic SIR Particle Filter - Schematic • Initialisation • measurement • Resampling • step • Importance • sampling step • Extract estimate,

42. Basic SIR Particle Filter algorithm (I) • Initialisation • For sample • and set • Importance Sampling step • For sample • For compute the importance weights wik • Normalise the importance weights, • and set

43. Basic SIR Particle Filter algorithm (II) • Resampling step • Resample with replacement particles: • from the set: • according to the normalised importance weights, • Set • proceed to the Importance Sampling step, as the next measurement arrives.

44. Resampling • x

45. Generic SIR Particle Filter algorithm • M. S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, “A tutorial on particle filters …,” IEEE Trans. on Signal Processing, 50( 2), 2002.

46. Improvements to SIR (I) • Variety of resampling schemes with varying performance in terms of the variance of the particles : • Residual sampling (Liu & Chen, 1998). • Systematic sampling (Carpenter et al., 1999). • Mixture of SIS and SIR, only resample when necessary (Liu & Chen, 1995; Doucet et al., 1999). • Degeneracy may still be a problem: • During resampling a sample with high importance weight may be duplicated many times. • Samples may eventually collapse to a single point.

47. Improvements to SIR (II) • To alleviate numerical degeneracy problems, sample smoothing methods may be adopted. • Roughening (Gordon et al., 1993). • Adds an independent jitter to the resampled particles • Prior boosting (Gordon et al., 1993). • Increase the number of samples from the proposal distribution to M>N, • but in the resampling stage only draw N particles.

48. Improvements to SIR (III) • Local Monte Carlo methods for alleviating degeneracy: • Local linearisation - using an EKF (Doucet, 1999; Pitt & Shephard, 1999) or UKF (Doucet et al, 2000) to estimate the importance distribution. • Rejection methods (Müller, 1991; Doucet, 1999; Pitt & Shephard, 1999). • Auxiliary particle filters (Pitt & Shephard, 1999) • Kernel smoothing (Gordon, 1994; Hürzeler & Künsch, 1998; Liu & West, 2000; Musso et al., 2000). • MCMC methods (Müller, 1992; Gordon & Whitby, 1995; Berzuini et al., 1997; Gilks & Berzuini, 1998; Andrieu et al., 1999).

49. Improvements to SIR (IV) • Illustration of SIR with sample smoothing:

50. Ingredients for SMC • Importance sampling function • Gordon et al • Optimal  • UKF  pdf from UKF at • Redistribution scheme • Gordon et al SIR • Liu & Chen  Residual • Carpenter et al  Systematic • Liu & Chen, Doucet et al  Resample when necessary • Careful initialisation procedure (for efficiency)