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 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 π n f 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 π n  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 π n Forward Expression

Zero order scheme

Recalling Taylor Series Let's call our point x 0 and let's define a new variable that simply measures how far we are from x 0 ; call the variable h = x –x 0. 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

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

Importance sampling Proposal distribution: easy to sample from Original distribution: hard to sample from, easy to evaluate Importance weights wlwl = 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

draw x i t  1 from Bel (x t  1 ) draw x i t from p ( x t | x i t  1 ) Importance factor for x i t : 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

Sampling-Importance Resampling(SIR) x 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 C0N2C0N2 One step recursive first order scheme C1N2d3C1N2d3 Two step recursive first order scheme C2N2dC2N2d SIS particle filter C3NC3N SIR particle filter C4NC4N

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  Kalman filter: d=3

Numerical performance Results Stochastic volatility model

Numerical performance Results Non linear explicit filter

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