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

Chang Young Kim

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 Quantization based filters

Zero order scheme First order schemes

Particle filtering algorithms: Sequential importance sampling (SIS) filter Sampling-Importance Resampling(SIR) filter

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

Bayesian approach: We attempt to construct the πnf of

the state given all measurements.

Prediction

Correction

Bayes Filter

One step transition bayes filter equation

By introducint the operaters , sequential definition of the unnormalized filter πn

Forward Expression

Bayes Filter

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

Zero order scheme First order schemes

One step recursive first order scheme Two step recursive first order scheme

Zero order scheme

Quantization

Sequential definition of the unnormalized filter πn

Forward Expression

Zero order scheme

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:  

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.

First order schemes

One step recursive scheme The recursive definition of the differential term estimator

Forward Expression

Two step recursive scheme

An integration by part formula

where

where

Comparisons of convergence rate

Zero order scheme

First order schemes One step recursive first order scheme

Two step recursive first order scheme

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

Consists of two basic elements:Monte Carlo integration

Importance sampling

limL ! 1

LX

`=1

w`f (x`) =Zf (x)p(x)dx

p(x) ¼LX

`=1

w`±x`

Importance sampling

Proposal distribution:

easy to sample from Original

distribution: hard to

sample from, easy to

evaluate

Ex [f (x)] =Zp(x)f (x)dx

=Z

p(x)q(x)

f (x)q(x)dx

¼1L

LX

`=1

p(x`)q(x`)

f (x`)

Importanceweights

x` » q(¢)

wl=p(x`)q(x`)

we want samples from

and make the following importance sampling identifications

Sequential importance sampling (SIS) filter

Proposal distribution

Distribution from which we want to sample

1 1 1( ) ( | ) ( | ) ( )t t t t t t tBel x p y x p x x Bel x dx

1 1( ) ( | ) ( )t t tq x p x x Bel x ( ) ( )tp x Bel x

draw xit1 from Bel(xt1)

draw xit from p(xt | xi

t1)

Importance factor for xit:

1 1

1 1

( )

( )

( | ) ( | ) ( )

( | ) ( )

( | )

it

t t t t t

t t t

t t

p xw

q x

p y x p x x Bel x

p x x Bel x

p y x

1 1 1( ) ( | ) ( | ) ( )t t t t t t tBel x p y x p x x Bel x dx SIS Filter Algorithm

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

2

1

1

( )eff n

ik

i

Nw

Sampling-Importance Resampling(SIR)

( )1

1

1,

ni

ki

xn

x

( ) ( )

1,

ni ik k i

x w

( )1

1

1,

ni

ki

xn

( ) ( )1 1 1,

ni ik k i

x w

( )2

1

1,

ni

ki

xn

Sensor model

Update

Resampling

Prediction

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

Complexity Numerical performances in three state

models:Kalman filter (KF)Canonical stochastic volatility model (SVM)Explicit non linear filter

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

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)

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) The time discretization of a continuous diffusion model.

State Model

Explicit non linear filter

A non linear non Gaussian state equation Serial Gaussian distributions SG()

State Model

Numerical performance Results Convergence tests

three test functions:

Kalman filter: d=1

Numerical performance Results : Convergence rate improvement

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

Kalman filter: d=3

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 Results Non linear explicit filter

<Explicit filter estimators as function of grid sizes >

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