An introduction to Particle filtering

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An introduction to Particle filtering

• Paul Sundvall
• www.s3.kth.se/pauls
• Presentation in course Optimal filtering
• Signals, Sensors and Systems, KTH
• November 11th 2004

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Outline

• Introduction
• Comparison with the Kalman filter
• Description of the algorithm
• Implementation
• Example

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Introduction

• Particle filtering
• is a method for state estimation
• is a Monte Carlo method
• handles nonlinear models with non-Gaussian noise

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Comparison to the discrete Kalman filter
Any distribution, uni- or multimodal
Gaussian, unimodal
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Significant property

• The particle filter gives an approximate solution
  to an exact model, rather than the optimal
  solution to an approximate model.

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Algorithm

• The propability density function is approximated
  using point weights
• Each point is called a particle
• Each particle has a positive weight

• Basic algorithm
• Initialize
• Time update (move particles)
• Measurement update (change weights)
• Resample (if needed)
• Goto 2 when new measurement arrives
• Each point is called a particle
• Each particle has a positive weight

• Initializew

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Time Update

• One-step prediction of each particle
• Note that a realization of the
• process noise is used for every
• particle.

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Measurement update

• The weights are adjusted using the measurement
• All weights are normalized
• Particles that can explain the measurement gain
  weight
• Particles far off the true state lose weight.
• The density of the cloud changes

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Resampling

• It can be shown that the algorithm degenerates
• Allt particles but one become very light
• Solved by resampling so that all weights become
  equal

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Implementation

• Calculation demand is proportional to the number
  of particles
• The approximation error decreases as the number
  of particles grow
• N can easily be changed during runtime
• One needs to know what to do with p(x)
• is not a good choice for multimodal
  distributions!

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Example

• A boat travels on a one-dimensional sea
• Noisy depth measurements are given
• Given a perfect sea-chart d(x), estimate the
  position!

• Matlab code for the example is available on
  www.s3.kth.se/pauls

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Final comments

• Better the more multimodal, non-linear and
  non-gaussian the system is
• The most basic variants are simple to implement
• It is easy to add model knowledge (saturation,
  limit checking, nonnegativeness...)

• Variants of the particle filters exist
• To reduce the number of particles needed, by
  combining Kalman filters and particle filters
• To ensure that states with low propability but
  high risk are tracked despite few particles
  (fault detection)
• To use discrete states
• To use both discrete and continuous states
  (hybrid)

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References

• A good introduction is given inA tutorial on
  Particle Filters for Online Nonlinear/Non-Gaussian
  Bayesian Tracking M. Sanjeev Arulampalam
  et. al.
• Very good application examples can be found in
  Particle Filters for Positioning, Navigation
  and Tracking Fredrik Gustafsson et. al.
• A description of how particle filters can be used
  for fault detection is found inParticle
  Filters for Rover Fault Diagnosis Vandi
  Verma et. al.