Particle Filters in high dimensions

1. Particle Filters in high dimensions Peter Jan van Leeuwen and Mel Ades Data-Assimilation Research Centre DARC University of Reading Lorentz Center 2011

2. Data assimilation: general formulation Bayes theorem: Solution is pdf! NO INVERSION !!!

3. Parameter estimation: with Again, no inversion but a direct point-wise multiplication.

4. Nonlinear filtering: Particle filter Use ensemble with the weights.

5. What are these weights? • The weight is the normalised value of the pdf of the observations given model state . • For Gaussian distributed variables is is given by: • One can just calculate this value • That is all !!!

6. Standard Particle filter Not very efficient !

7. A closer look at the weights I Probability space in large-dimensional systems is ‘empty’: the curse of dimensionality u(x1) u(x2) T(x3)

8. A closer look at the weights II • Assume particle 1 is at 0.1 standard deviations s of M • independent observations. • Assume particle 2 is at 0.2 s of the M observations. • The weight of particle 1 will be • and particle 2 gives

9. A closer look at the weights III The ratio of the weights is Take M=1000 to find Conclusion: the number of independent observations is responsible for the degeneracy in particle filters.

10. Increased efficiency: proposal density Instead of drawing samples from p(x) we draw samples from a proposal pdf q(x). Use ensemble with weights

11. Particle filter with proposal transitiondensity

12. Barotropicvorticity equation 256 X 256 grid points 600 time steps Typically q=1-3 Decorrelation time scale= = 25 time steps Observations Every 4th gridpoint Every 50th time step 24 particles sigma_model=0.03 sigma_obs=0.01

13. Vorticity field standard particle filter Truth Ensemble mean PF

14. Vorticity field new particle filter Truth Ensemble mean

15. Posterior weights

16. Rank histogram:How the truth ranks in the ensemble

17. Equivalent Weights Particle Filter Recall: Assume Find the minimum for each particle by perturbing each observation, gives

18. Equivalent Weights Particle filter Calculate the full weight for each of these Determine a target weight C that 80% of the particles can reach and determine in such that This is a line search, so doable.

19. Equivalent Weights Particle Filter • This leads to 80% of the particle having an almost equal weight, so no degeneracy by construction! • Example …

20. Gaussian pdf in high dimensions Assume variables are identically independently distributed: Along each if the axes it looks like a standard Gaussian:

21. However, the probability mass as function of the distance to the centre is given by: The so-called Important Ring d=400 d=900 d=100 r in standard deviation

22. Why? In distribution Fisher has shown So we find

23. Importance Ring Experimental evidence, sums of d squared random numbers: r d

24. Given this what do theseefficient particles represent???

25. Conclusions • Particle filters with proposal transition density: • solve for fully nonlinear posterior pdf • very flexible, much freedom • scalable => high-dimensional problems • extremely efficient • But what do they represent?

26. We need more people ! • In Reading only we expect to have 7 new PDRA positions available in the this year • We also have PhD vacancies • And we still have room in the Data Assimilation and Inverse Methods in Geosciences MSc program