1. Particle Filters

2. Outline Introduction to particle filters Bayesian Importance sampling
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. But what if not a gaussian distribution in our problem?

4. Motivation for particle filters

5. Key Idea of Particle Filters
Idea = we try to have more samples where we expect to have the solution

6. Motion Model Reminder
Density of samples represents the expected probability of robot location

8. Global Localization of Robot with Sonar http://www. cs. washington
This is the lost robot problem

10. Particles are used for probability density function Approximation

11. Function Approximation
Particle sets can be used to approximate functions
The more particles fall into an interval, the higher the probability of that interval
How to draw samples from a function/distribution?

12. Importance Sampling Principle

13. Importance Sampling Principle
weight
w = f / g
f is often called target
g is often called proposal
Pre-condition: f(x)>0  g(x)>0

14. Importance sampling: another example of calculating weight samples
How to calculate formally the f/g value?

15. Importance Sampling Formulas for f, g and f/g

16. History of Monte Carlo Idea and especially Particle Filters
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
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.

17. What is the problem that we want to solve?
The problem is 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

18. 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
These lead to various particle filters

19. 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

20. Particle Filter characteristics

21. Approaches to Particle Filters
METAPHORS

22. Particle filters Sequential and Monte Carlo properties
Representing belief by sets of samples or particles
are nonnegative weights called importance factors
Updating procedure is sequential importance sampling with re-sampling

23. Tracking in 1D: the blue trajectory is the target
Tracking in 1D: the blue trajectory is the target. The best of10 particles is in red.

24. Short, more formal, Introduction to Particle Filters and Monte Carlo Localization

25. Proximity Sensor Model Reminder

26. Particle filtering ideas
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

27. Particle filtering ideas
Particle filters are based on recursive generation of random measures that approximate the distributions of the unknowns.
Random measures: particles and importance weights.
As new observations become available, the particles and the weights are propagated by exploiting Bayes theorem.
Sample space
Posterior density

28. Mathematical tools needed for Particle Filters
Recall “law of total probability” and “Bayes’ rule”

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

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

31. Recursive Bayesian estimation (III)
Update:
where:
since:

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

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

34. Assumptions: Markov Process
Predict:
Update:

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

36. Particle Filters: Compare Gaussian and Particle Filters

37. Example 1:
theoretical PDF

38. Example 1: theoretical PDF
Step 0: initialization
Step 1: updating

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

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

41. Robot Motion

42. Example 2: Particle Filter

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

44. Robot Motion

45. Compare Particle Filter with Bayes Filter with Known Distribution
Updating
Example 1
Example 2
Predicting
Example 1
Example 2

46. 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.

47. Monte Carlo Localization

48. Mobile Robot Localization
Each particle is a potential pose of the robot
Proposal distribution is the motion model of the robot (prediction step)
The observation model is used to compute the importance weight (correction step)

49. Monte Carlo Localization
Each particle is a potential pose of the robot
Proposal distribution is the motion model of the robot (prediction step)
The observation model is used to compute the importance weight (correction step)

50. Sample-based Localization (sonar)

51. 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

52. Random samples and the pdf (II)

53. Random samples and the pdf (III)

54. Importance Sampling

55. Importance Sampling where
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

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

57. Sensor Information: Importance Sampling

58. 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

59. 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)

60. Rejection Sampling

61. 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 sample otherwise reject the sample
f(x’) c c’ OK
f(x) x x’

62. Importance Sampling with Resampling: Landmark Detection Example

63. Distributions

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

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

66. Importance sampling with Resampling
After Resampling

67. Particle Filter Algorithm

68. weight =
target distribution / proposal distribution

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

70. Particle Filter Algorithm

71. 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

72. Particle Filter for Localization

73. Particle Filter in Matlab

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

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

76. Sources Longin Jan Latecki Keith Copsey Paul E. Rybski
Cyrill Stachniss
Sebastian Thrun
Alex Teichman
Michael Pfeiffer
J. Hightower
L. Liao
D. Schulz
G. Borriello
Honggang Zhang
Wolfram Burgard
Dieter Fox
Giorgio Grisetti
Maren Bennewitz
Christian Plagemann
Dirk Haehnel
Mike Montemerlo
Nick Roy
Kai Arras
Patrick Pfaff
Miodrag Bolic
Haris Baltzakis

78. 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: