Tutorial on Particle Filters assembled and extended by Longin Jan Latecki

1. Tutorial on Particle Filters assembled and extended by Longin Jan Latecki Temple University, latecki@temple.edu using slides from Keith Copsey, Pattern and Information Processing Group, DERA Malvern; D. Fox, J. Hightower, L. Liao, D. Schulz, and G. Borriello, Univ. of Washington, Seattle Honggang Zhang, Univ. of Maryland, College Park Miodrag Bolic, University of Ottawa, Canada Michael Pfeiffer, TU Gratz, Austria

2. Outline • Introduction to particle filters • Recursive Bayesian estimation • Bayesian Importance sampling • Sequential Importance sampling (SIS) • Sampling Importance resampling (SIR) • Improvements to SIR • On-line Markov chain Monte Carlo • Basic Particle Filter algorithm • Example for robot localization • Conclusions

3. Particle Filters • Sequential Monte Carlo methods for on-line learning within a Bayesian framework. • Known as • Particle filters • Sequential sampling-importance resampling (SIR) • Bootstrap filters • Condensation trackers • Interacting particle approximations • Survival of the fittest

4. History • First attempts – simulations of growing polymers • M. N. Rosenbluth and A.W. Rosenbluth, “Monte Carlo calculation of the average extension of molecular chains,” Journal of Chemical Physics, vol. 23, no. 2, pp. 356–359, 1956. • First application in signal processing - 1993 • N. J. Gordon, D. J. Salmond, and A. F. M. Smith, “Novel approach to nonlinear/non-Gaussian Bayesian state estimation,” IEE Proceedings-F, vol. 140, no. 2, pp. 107–113, 1993. • Books • A. Doucet, N. de Freitas, and N. Gordon, Eds., Sequential Monte Carlo Methods in Practice, Springer, 2001. • B. Ristic, S. Arulampalam, N. Gordon, Beyond the Kalman Filter: Particle Filters for Tracking Applications, Artech House Publishers, 2004. • Tutorials • M. S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, “A tutorial on particle filters for online nonlinear/non-gaussian Bayesian tracking,” IEEE Transactions on Signal Processing, vol. 50, no. 2, pp. 174–188, 2002.

5. Problem Statement • Tracking the state of a system as it evolves over time • Sequentially arriving (noisy or ambiguous) observations • We want to know: Best possible estimate of the hidden variables

6. Solution: Sequential Update • Storing and processing all incoming measurements is inconvenient and may be impossible • Recursive filtering: –Predict next state pdf from current estimate –Update the prediction using sequentially arriving new measurements • Optimal Bayesian solution: recursively calculating exact posterior density

7. Particle filtering ideas • Particle filter is a technique for implementing recursive Bayesian filter by Monte Carlo sampling • The idea: represent the posterior density by a set of random particles with associated weights. • Compute estimates based on these samples and weights Posterior density Sample space

8. Recall “law of total probability” and “Bayes’ rule” Tools needed

9. Recursive Bayesian estimation (I) • Recursive filter: • System model: • Measurement model: • Information available:

10. Recursive Bayesian estimation (II) • Seek: • i = 0: filtering. • i > 0: prediction. • i<0: smoothing. • Prediction: • since:

11. Recursive Bayesian estimation (III) • Update: • where: • since:

12. System state dynamics Observation dynamics We are interested in: Belief or posterior density Bayes Filters (second pass) Estimating system state from noisy observations

13. From above, constructing two steps of Bayes Filters Predict: Update:

14. Assumptions: Markov Process Predict: Update:

15. Bayes Filter How to use it? What else to know? Motion Model Perceptual Model Start from:

16. Step 0: initialization Step 1: updating Example 1

17. Step 3: updating Step 4: predicting Step 2: predicting Example 1 (continue) 1

18. Classical approximations • Analytical methods: • Extended Kalman filter, • Gaussian sums… (Alspach et al. 1971) • Perform poorly in numerous cases of interest • Numerical methods: • point masses approximations, • splines. (Bucy 1971, de Figueiro 1974…) • Very complex to implement, not flexible.

19. Perfect Monte Carlo simulation • Recall that • Random samples are drawn from the posterior distribution. • Represent posterior distribution using a set of samples or particles. • Easy to approximate expectations of the form: • by:

20. Random samples and the pdf (I) • Take p(x)=Gamma(4,1) • Generate some random samples • Plot histogram and basic approximation to pdf 200 samples

21. Random samples and the pdf (II) 500 samples 1000 samples

22. Random samples and the pdf (III) 200000 samples 5000 samples

23. 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

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

26. 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

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

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

29. Derivation of SIS weights (II)

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

31. 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

32. 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

33. 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

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

35. 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

37. 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:

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

39. 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

40. 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.

41. Resampling x

42. 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.

43. 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.

44. 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.

45. 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).

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

47. 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)

48. Particle filters • Also known as Sequential Monte Carlo Methods • Representing belief by sets of samples or particles • are nonnegative weights called importance factors • Updating procedure is sequential importance sampling with re-sampling

49. Step 0: initialization Each particle has the same weight Step 1: updating weights. Weights are proportional to p(z|x) Example 2: Particle Filter

50. Step 3: updating weights. Weights are proportional to p(z|x) Step 4: predicting. Predict the new locations of particles. Step 2: predicting. Predict the new locations of particles. Example 2: Particle Filter Particles are more concentrated in the region where the person is more likely to be