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Comparative survey on non linear filtering methods : the quantization and the particle filtering approaches Afef SELLAMI. Chang Young Kim. Overview. Introduction Bayes filters Quantization based filters Zero order scheme First order schemes Particle filters Sequential importance

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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

overview
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
non linear filter estimators
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

overview4
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
bayes filter
Bayes Filter
  • Bayesian approach: We attempt to construct the πnfof the state given all measurements.
    • Prediction
    • Correction
bayes filter6
Bayes Filter
  • One step transition bayes filter equation
  • By introducint the operaters , sequential definition of the unnormalized filter πn
  • Forward Expression
overview7
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
quantization based filters
Quantization based filters
  • Zero order scheme
  • First order schemes
    • One step recursive first order scheme
    • Two step recursive first order scheme
zero order scheme
Zero order scheme
  • Quantization
  • Sequential definition of the unnormalized filter πn
  • Forward Expression
recalling taylor series
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:  
first order schemes
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.
one step recursive scheme
One step recursive scheme
  • The recursive definition of the differential term estimator
  • Forward Expression
two step recursive scheme
Two step recursive scheme
  • An integration by part formula

where

where

comparisons of convergence rate
Comparisons of convergence rate
  • Zero order scheme
  • First order schemes
    • One step recursive first order scheme
    • Two step recursive first order scheme
overview18
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
particle filtering
Particle filtering
  • Consists of two basic elements:
    • Monte Carlo integration
    • Importance sampling
importance sampling

p

(

x

)

`

wl

=

q

(

x

)

`

Importance sampling

Proposal distribution: easy to sample from

Original distribution: hard to sample from, easy to evaluate

Importance

weights

sequential importance sampling sis filter
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

sis filter algorithm

draw xit-1from Bel(xt-1)

draw xitfrom p(xt | xit-1)

Importance factor for xit:

SIS Filter Algorithm
sampling importance resampling sir
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
sampling importance resampling sir24
Sampling-Importance Resampling(SIR)

Prediction

Resampling

Update

Sensor model

x

overview25
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
elements for a comparison
Elements for a comparison
  • Complexity
  • Numerical performances in three state models:
    • Kalman filter (KF)
    • Canonical stochastic volatility model (SVM)
    • Explicit non linear filter
numerical performances
Numerical performances
  • Three models chosen to make up the benchmark.
    • Kalman filter (KF)
    • Canonical stochastic volatility model (SVM)
    • Explicit non linear filter
kalman filter kf
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)
canonical stochastic volatility model svm
Canonical stochastic volatility model (SVM)
  • The time discretization of a continuous diffusion model.
  • State Model
explicit non linear filter
Explicit non linear filter
  • A non linear non Gaussian state equation
  • Serial Gaussian distributions SG()
  • State Model
numerical performance results
Numerical performance Results
  • Convergence tests
    • three test functions:
    • Kalman filter: d=1
numerical performance results convergence rate improvement
Numerical performance Results : Convergence rate improvement
  • Kalman filter: d=3

<Regression slopes on the log-log scale representation (d=3)>

numerical performance results34
Numerical performance Results
  • Stochastic volatility model
  • <Particle filter for large particle sizes (N = 10000) and quantization filter approximations for SVM as a function of the quantizer size>
numerical performance results35
Numerical performance Results
  • Non linear explicit filter
  • <Explicit filter estimators as function of grid sizes >
conclusions
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