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

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

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

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

6. 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 !!!

7. Standard Particle filter Not very efficient !

8. 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)

9. 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

10. 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.

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

12. Particle filter with proposal transition density

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

14. Vorticity field standard particle filter Ensemble mean PFTruth

15. Vorticity field new particle filter Ensemble meanTruth

16. Posterior weights

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

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

19. 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.

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

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

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

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

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

25. Given this what do these efficient particles represent???

26. 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?

27. 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