slide1 l.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Comparative survey on non linear filtering methods : the quantization and the particle filtering approaches Afef SELLAMI PowerPoint Presentation
Download Presentation
Comparative survey on non linear filtering methods : the quantization and the particle filtering approaches Afef SELLAMI

Loading in 2 Seconds...

play fullscreen
1 / 36

Comparative survey on non linear filtering methods : the quantization and the particle filtering approaches Afef SELLAMI - PowerPoint PPT Presentation


  • 157 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Comparative survey on non linear filtering methods : the quantization and the particle filtering approaches Afef SELLAMI' - ulfah


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
slide1

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