Particle filters (continued

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Particle filters (continued)
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Recall

• Particle filters
• Track state sequence xi given the measurements
  (y0, y1, ., yi)
• Non-linear dynamics
• Non-linear measurements

Non-Gaussian
Non-Gaussian
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Recall

• Maintain a representation of
• Two stages
• Prediction
• Correction (Bayesian)

Dynamic model (Markov)
Likelihood
Prior
Posterior
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3 Useful tools

• Importance sampling
• Tool 1 Representing a distribution
• Tool 2 Marginalizing
• Tool 3 Transforming prior to posterior

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Tool 1 Representing a distribution

• Have a set of samples ui with weights wi
• (ui, wi ) Sampled representation of f(u)
• Expectation under f(u)
• Samples used only as a means to evaluate
  expectations (Not true samples!)

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Tool 2 Marginalization

• Marginalization
• Sampled representation
• Just retain the required components and ignore
  the rest!

Drop ni
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Tool 3 From Prior to Posterior

• Modify the weights to transform from one
  distribution to another
• Similarly for going from prior to posterior

?
To
From
To
From
Scale factor is the same for all the samples
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Simple Particle filter

• Prediction
• 2 steps
• Sampling from joint distribution
• Marginalization

Dynamic model (Markov)
(Notation Chapter 2)
Drop
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Simple Particle filter

• Correction
• Modify weights

Likelihood
Prior
Posterior
Let
Likelihood
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Improved Particle filter

• Simple Particle filter
• Many samples have small weights
• Number of samples increases at every step
• Lots of samples wasted
• Resample (Sampling-Importance -Resampling)
• Prior
• Predictions
• Resampling also takes care of increasing number
  of samples

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Tracking interacting targets

• Using partilce filters to track multiple
  interacting targets (ants)

Khan et al., MCMC-Based Particle Filtering for
Tracking a Variable Number of Interacting
Targets, PAMI, 2005.
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Independent Particle filters

• Targets lose identity
• Identical appearance
• Multiple peaks in the likelihood
• Best peak hijacks all the nearby targets

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Alternate view of Particle filters

• Notation

Marginalization
Posterior
Prior
Likelihood
State at time t
Measurement at time t
All measurements upto time t
Khan et al., MCMC-Based Particle Filtering for
Tracking a Variable Number of Interacting
Targets, PAMI, 2005.
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Alternate view of Particle filters

• Sampled representation of prior
• Monte-Carlo approximation

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Alternate view of Particle filters

• Sequential Importance Resampling (SIR)
• Particles at time t
• Weights (easy to verify!)
• Prediction and correction in one step

Particles sampled from a mixture distribution
formed by previous particle set
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Independent vs. Joint filters

• Multiple targets
• Joint state space Union of individual state
  spaces
• Independent targets
• Predictions are made independently from
  respective spaces
• Interacting targets
• Predictions are from the joint state space
• High dimensionality MCMC better than Importance
  sampling?

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Interacting targets

• Targets influence the dynamics of others
• Particles cannot be propagated independently
• Model interactions between targets using Markov
  Random Fields (MRF)

Individual dynamics
Pair wise interactions
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MRF
Edges are formed only when templates overlap

• Interaction potential
• g(Xit , Xjt) penalizes overlap between targets
• Takes care of hijacking

Overlap is penalized by the interaction potential
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Joint MRF Particle filter

• Sequential Importance Resampling
• Particles at time t
• Weights
• Interactions affect only the weights

Equivalent to independent particle filters
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Target overlap

• Targets overlap on each other and then segregate
• Overlapped target state hijacked
• Probably hard to model this?

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Why MCMC?

• Joint MRF Particle filter
• Importance sampling in high dimensional spaces
• Weights of most particles go to zero
• MCMC is used to sample particles directly from
  the posterior distribution

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MCMC Joint MRF Particle filter

• True samples (no weights) at each step
• Stationary distribution for MCMC
• Proposal density for Metropolis Hastings (MH)
• Select a target randomly
• Sample from the single target state proposal
  density

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MCMC Joint MRF Particle filter

• MCMC-MH iterations are run every time step to
  obtain particles
• One target at a time proposal has advantages
• Acceptance probability is simplified
• One likelihood evaluation for every MH iteration
• Computationally efficient
• Requires fewer samples compared to SIR

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Variable number of targets

• Target identifiers kt is a state variable
• Each kt determines a corresponding state space
• State space is the union of state spaces indexed
  by kt
• Particle filtering
• RJMCMC to jump across state spaces

Prediction Correction
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Thank you!