Comparative survey on non linear filtering methods : the quantization and the particle filtering approaches Afef SELLAMI

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Comparative survey on non linear filtering
methods thequantization and the particle
filtering approachesAfef SELLAMI

• Chang Young Kim

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Overview

• Introduction
• Bayes filters
• Quantization based filters
• Zero order scheme
• First order schemes
• Particle filters
• Sequential importance
• sampling (SIS) filter
• Sampling-Importance
• Resampling(SIR) filter
• Comparison of two approaches
• Summary

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Non linear filter estimators

• Quantization based filters
• Zero order scheme
• First order schemes
• Particle filtering algorithms
• Sequential importance
• sampling (SIS) filter
• Sampling-Importance
• Resampling(SIR) filter

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Overview

• Introduction
• Bayes filters
• Quantization based filters
• Zero order scheme
• First order schemes
• Particle filters
• Sequential importance
• sampling (SIS) filter
• Sampling-Importance
• Resampling(SIR) filter
• Comparison of two approaches
• Summary

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Bayes Filter

• Bayesian approach We attempt to construct the
  pnf of the state given all measurements.
• Prediction
• Correction

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Bayes Filter

• One step transition bayes filter equation
• By introducint the operaters ,
  sequential definition of the unnormalized filter
  pn
• Forward Expression

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Overview

• Introduction
• Bayes filters
• Quantization based filters
• Zero order scheme
• First order schemes
• Particle filters
• Sequential importance
• sampling (SIS) filter
• Sampling-Importance
• Resampling(SIR) filter
• Comparison of two approaches
• Summary

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Quantization based filters

• Zero order scheme
• First order schemes
• One step recursive first order scheme
• Two step recursive first order scheme

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Zero order scheme

• Quantization
• Sequential definition of the unnormalized filter
  pn
• Forward Expression

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Zero order scheme
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Recalling Taylor Series

• Let's call our point  x0 and let's define a new
  variable that simply measures how far we are
  from x0 call the variable h x x0.
• Taylor Series formula
• First Order Approximation  

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First order schemes

• Introduce first order schemes to improve the
  convergence rate of the zero order schemes.
• Rewriting the sequential definition by mimicking
  some first order Taylor expansion
• Two schemes based on the different approximation
  by
• One step recursive scheme based on a recursive
  definition of the differential term estimator.
• Two step recursive scheme based on an
  integration by part transformation of conditional
  expectation derivative.

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One step recursive scheme

• The recursive definition of the differential term
  estimator
• Forward Expression

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Two step recursive scheme

• An integration by part formula
• where
• where

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Comparisons of convergence rate

• Zero order scheme
• First order schemes
• One step recursive first order scheme
• Two step recursive first order scheme

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Overview

• Introduction
• Bayes filters
• Quantization based filters
• Zero order scheme
• First order schemes
• Particle filters
• Sequential importance
• sampling (SIS) filter
• Sampling-Importance
• Resampling(SIR) filter
• Comparison of two approaches
• Summary

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Particle filtering

• Consists of two basic elements
• Monte Carlo integration
• Importance sampling

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Importance sampling
Proposal distribution easy to sample from
Original distribution hard to sample from, easy
to evaluate
Importance weights
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Sequential importance sampling (SIS) filter

• we want samples from
• and make the following importance sampling
  identifications

Proposal distribution
Distribution from which we want to sample
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SIS Filter Algorithm
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Sampling-Importance Resampling(SIR)

• Problems of SIS
• Weight Degeneration
• Solution ? RESAMPLING
• Resampling eliminates samples with low importance
  weights and multiply samples with high importance
  weights
• Replicate particles when the effective number of
  particles is below a threshold

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Sampling-Importance Resampling(SIR)
Prediction
Resampling
Update
Sensor model
x
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Overview

• Introduction
• Bayes filters
• Quantization based filters
• Zero order scheme
• First order schemes
• Particle filters
• Sequential importance
• sampling (SIS) filter
• Sampling-Importance
• Resampling(SIR) filter
• Comparison of two approaches
• Summary

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Elements for a comparison

• Complexity
• Numerical performances in three state models
• Kalman filter (KF)
• Canonical stochastic volatility model (SVM)
• Explicit non linear filter

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Complexity comparison
Zero order scheme C0N2
One step recursive first order scheme C1N2d3
Two step recursive first order scheme C2N2d
SIS particle filter C3N
SIR particle filter C4N
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Numerical performances

• Three models chosen to make up the benchmark.
• Kalman filter (KF)
• Canonical stochastic volatility model (SVM)
• Explicit non linear filter

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Kalman filter (KF)

• Both signal and observation equations are linear
  with Gaussian independent noises.
• Gaussian process which parameters (the two first
  moments) can be computed sequentially by a
  deterministic algorithm (KF)

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Canonical stochastic volatility model (SVM)

• The time discretization of a continuous diffusion
  model.
• State Model

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Explicit non linear filter

• A non linear non Gaussian state equation
• Serial Gaussian distributions SG()
• State Model

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Numerical performance Results

• Convergence tests
• three test functions
• Kalman filter d1

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Numerical performance Results Convergence rate
improvement

• Kalman filter d3

• ltRegression slopes on the log-log scale
  representation (d3)gt

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Numerical performance Results

• Stochastic volatility model

ltParticle filter for large particle sizes (N
10000) and quantization filter approximations for
SVM as a function of the quantizer sizegt
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Numerical performance Results

• Non linear explicit filter

ltExplicit filter estimators as function of grid
sizes gt
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Conclusions

• Particle methods do not suffer from dimension
  dependency when considering their theoretical
  convergence rate, whereas quantization based
  methods do depend on the dimension of the state
  space.
• Considering the theoretical convergence results,
  quantization methods are still competitive till
  dimension 2 for zero order schemes and till
  dimension 4 for first order ones.
• Quantization methods need smaller grid sizes than
  Monte Carlo methods to attain convergence regions