Tutorial on Particle Filters - Jan 2001

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Tutorial on Particle Filters assembled and
extended by Longin Jan Latecki Temple University,
latecki_at_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
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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

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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

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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. 356359, 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. 107113, 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. 174188, 2002.

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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

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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

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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
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Global Localization of Robot with
Sonarhttp//www.cs.washington.edu/ai/Mobile_Robot
ics/mcl/animations/global-floor.gif
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Tools needed
Recall law of total probability (or
marginalization) and Bayes rule
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Recursive Bayesian estimation (I)

• Recursive filter
• System model
• Measurement model
• Information available

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Recursive Bayesian estimation (II)

• Seek
• i 0 filtering.
• i gt 0 prediction.
• ilt0 smoothing.
• Prediction
• since

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Recursive Bayesian estimation (III)

• Update
• where
• since

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Bayes Filters (second pass)
Estimating system state from noisy observations
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Assumptions Markov Process
Predict
Update
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Example 1
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Example 1 (continue)
1
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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.

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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

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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
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Random samples and the pdf (II)
500 samples
1000 samples
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Random samples and the pdf (III)
200000 samples
5000 samples
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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), i1, , 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
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Bayesian Importance Sampling

• By drawing samples from a known easy to
  sample proposal distribution
  we obtain

where
are normalized weights.
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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

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Sequential Importance Sampling (SIS) Particle
Filter
SIS Particle Filter Algorithm
for i1N Draw a particle Assign a weight end
(k is index over time and i is the particle index)
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Derivation of SIS weights (I)

• The main idea is Factorizing

and
Our goal is to expand p and q in time t
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Derivation of SIS weights (II)
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Derivation of SIS weights (II)
and under Markov assumptions
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SIS Particle Filter Foundation

• At each time step k
• Random samples are drawn from the
  proposal distribution for i1, , N
• They represent posterior distribution using a set
  of samples or particles
• Since the weights are given by
• and q factorizes as

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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

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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

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Sequential Importance Sampling (IV)

• Illustration of degeneracy

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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

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Sampling-Importance Resampling

• SIS suffers from degeneracy problems so we dont
  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

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

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Resampling
x
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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.
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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.

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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 MgtN,
• but in the resampling stage only draw N particles.

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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).

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Improvements to SIR (IV)

• Illustration of SIR with sample smoothing

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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)

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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

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Example 2 Particle Filter
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Example 2 Particle Filter
Particles are more concentrated in the region
where the person is more likely to be
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Compare Particle Filter with Bayes Filter with
Known Distribution
Updating
Example 1
Example 2
Predicting
Example 1
Example 2
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Particle Filters
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Sensor Information Importance Sampling
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Robot Motion

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Sensor Information Importance Sampling
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Robot Motion
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Application Examples

• Robot localization
• Robot mapping
• Visual Tracking, e.g., human motion (body parts)
• Prediction of (financial) time series, e.g.,
  mapping gold price to stock price
• Target recognition from single or multiple images
• Guidance of missiles
• Contour grouping http//www.cis.temple.edu/latec
  ki/Papers/PFgrouping_ICCV09_final.pdf
• Matlab PF software for tracking
  http//www.gatsby.ucl.ac.uk/fwood/pf_tutorial/
• Nice video demos
• http//www.cs.washington.edu/ai/Mobile_Robotics/m
  cl/