Particle Filters

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Particle Filters
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Importance Sampling
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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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Sensor Information Importance Sampling
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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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Rejection Sampling
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Rejection Sampling

• Let us assume that f(x)lt1 for all x
• Sample x from a uniform distribution
• Sample c from 0,1
• if f(x) gt c keep the sampleotherwise reject
  the sample

• f(x)

• c

• c

• OK

• f(x)

• x

• x

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Importance Sampling with ResamplingLandmark
Detection Example
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Distributions
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Distributions

• Wanted samples distributed according to p(x z1,
  z2, z3)

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This is Easy!

• We can draw samples from p(xzl) by adding noise
  to the detection parameters.

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Importance sampling with Resampling

• After Resampling

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Particle Filter Algorithm
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weight target distribution / proposal
distribution
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Particle Filter Algorithm
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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

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Particle Filter Algorithm
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Particle Filter for Localization
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Particle Filter in Matlab
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• Matlab code truex is a vector of 100 positions
  to be tracked.

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Application Particle Filter for Localization
(Known Map)
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Resampling
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Resampling
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Resampling
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Resampling Algorithm

1. Algorithm systematic_resampling(S,n)
3. For Generate cdf
5. Initialize threshold
6. For Draw samples
7. While ( ) Skip until next threshold
   reached
9. Insert
10. 
    Increment threshold
11. Return S

• Also called stochastic universal sampling

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Low Variance Resampling
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SIS weights
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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
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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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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