Particle Filters : Monte Carlo approach to signal processing

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Particle Filters Monte Carlo approach to signal
processing

• Saikat Saha

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Stochastic System and Signal Theory Group (SST)

• Project Particle Filters and Their Applications
  to
• Target Tracking and Detection
• Financed by THALES Nederland B.V.

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Outline

• Historic development of filtering problems
• Basic algorithm of a generic PF
• Advantages of PF
• Some successful applications
• Ongoing research

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What is a filter ?

• X(t) true signal
• Y(t) measurement of X(t) corrupted by noise
  N(t)
• Objective
• Estimate X(t) based on the knowledge of Y( ) up
  to time t

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Solved independently by Kolmogorov and Wiener in
early 40s

• Kolmogorovs Approach
• Based on
• Projection Theorem
• on Hilbert Space

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Wieners approach Based on (cross) correlation
of signal observation

• Key assumptions
• Scalar processes
• Joint stationarity of signal
  noise processes
• Leads to Wiener Hopf equation
• Wiener Hopf equation now has numerous
  applications to diverse fields of Applied
  Mathematics

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Extensions

• Non stationary not obvious in the Wiener set
  up
• Vector case analysis highly complicated

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Breakthrough Kalman Filter ( 1960 )

• Model for signal a dramatic paradigm shift
  !!!
• Dynamic State Space formulation
• Naturally multi dimensional

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• Immediate sensational success in space missions
• Today, there is not a single branch in science
  or engineering where KF is not used !!!!!!
• Also routinely used in economics and finance

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Kalman Filter Discrete Version
In MMSE sense
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Note

filter density / posterior density
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Remarks

• KF is restricted to linear-Gaussian model
• Optimal filter density is Gaussian
• characterized by conditional mean and
• conditional covariance

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Nonlinear Filter

• filter density non Gaussian
• - work directly with filter density
• Question How to (recursively) update the filter
• density ?

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Emergence of Particle Filter (Early 90s)

• Represent the filter density by a large set of
  (weighted) particles
• For example

Each with weight 1/N, N Nr. Of particles
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Importance Sampling

• Ideally generate particles from

filter density
Not known !!!

• Sample from Proposal Density ( ?() )
• Known as Importance sampling density in Monte
  Carlo
• literature

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Principle of a PF

• Propose new particles
• Weight update

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One Cycleof the basic algorithm
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One Cycleof the basic algorithm
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Working principle of a PF Propose, weight,
resample

• Propose new particles
• Weight update
• Resample

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Advantages

• Linear-Gaussian assumption not required
• Leads to an (approximate) estimate of the
  complete probability distribution
• Approximation of pdf rather than approximating
  state space model as in ad-hoc extensions of
  Kalman filter
• Basic version is very easy to code
• - simple core algorithms, modularity

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Some Applications

• (Radar) Target Tracking and Detection
• Financial Engineering
• Robotics and Computer Visions
• Mobile Communications, Image, Audio Signals
• Rare Event Simulation
• Electricity Load Forecasting

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PF in action ( Example 2 ).Courtsey Yvo Boers
( THALES Nederland )
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Ongoing research
Active field of research with a lot of open
problems (both theoretical practical) including

• Number of particles
• Choice of proposal
• Optimal
• Sampling in large dimensions

• Number of particles
• Choice of proposal
• Optimal
• Sampling in large dimensions

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Question ??....