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

1. Comparative survey on non linear filtering methods : thequantization and the particle filtering approachesAfef SELLAMI Chang Young Kim

2. 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

3. 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

4. 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

5. Bayes Filter • Bayesian approach: We attempt to construct the πnfof the state given all measurements. • Prediction • Correction

6. Bayes Filter • One step transition bayes filter equation • By introducint the operaters , sequential definition of the unnormalized filter πn • Forward Expression

7. 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

8. Quantization based filters • Zero order scheme • First order schemes • One step recursive first order scheme • Two step recursive first order scheme

9. Zero order scheme • Quantization • Sequential definition of the unnormalized filter πn • Forward Expression

10. Zero order scheme

11. 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:  

12. 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.

13. One step recursive scheme • The recursive definition of the differential term estimator • Forward Expression

15. Two step recursive scheme • An integration by part formula where where

17. Comparisons of convergence rate • Zero order scheme • First order schemes • One step recursive first order scheme • Two step recursive first order scheme

18. 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

19. Particle filtering • Consists of two basic elements: • Monte Carlo integration • Importance sampling

20. p ( x ) ` wl = q ( x ) ` Importance sampling Proposal distribution: easy to sample from Original distribution: hard to sample from, easy to evaluate Importance weights

21. 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

22. draw xit-1from Bel(xt-1) draw xitfrom p(xt | xit-1) Importance factor for xit: SIS Filter Algorithm

23. 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

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

25. 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

26. Elements for a comparison • Complexity • Numerical performances in three state models: • Kalman filter (KF) • Canonical stochastic volatility model (SVM) • Explicit non linear filter

27. Complexity comparison

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

29. 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)

30. Canonical stochastic volatility model (SVM) • The time discretization of a continuous diffusion model. • State Model

31. Explicit non linear filter • A non linear non Gaussian state equation • Serial Gaussian distributions SG() • State Model

32. Numerical performance Results • Convergence tests • three test functions: • Kalman filter: d=1

33. Numerical performance Results : Convergence rate improvement • Kalman filter: d=3 <Regression slopes on the log-log scale representation (d=3)>

34. 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>

35. Numerical performance Results • Non linear explicit filter • <Explicit filter estimators as function of grid sizes >

36. 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