1. Introduction to the Particle Filter Computer Practical
Peter Jan van Leeuwen and Phil Browne

2. The Particle filter
Use ensemble
with
the weights.

3. What are these weights?
For Gaussian distributed variables is is given by:
One can just calculate this value
That is all !!!

4. Standard Particle filter

5. How to make particle filters useful?
Introduce localisation to reduce the number of observations.
Use proposal-density freedom.
Several ad-hoc combinations of Particle Filters and Ensemble Kalman Filters

6. Bayes Theorem and the proposal density
Bayes Theorem now becomes:
We have a set of particles at time n-1 so we can write
and use this in the equation above to perform the integral:

7. The standard particle filter
Performing the integral over the sum of delta functions gives:
The posterior is now given as a sum of transition densities.
This is the standard particle filter, also called
Sequential Importance Resampling, or SIR.
This scheme is degenerate when the number of observations
Is large.

8. The proposal transition density
Multiply numerator and denominator with a proposal density q:
Note that
1) the proposal depends on the future observation, and
2) the proposal depends on all previous particles, not just one.

9. Between observations: relaxation
We add a relaxation term to the model equation:
How strong should the relaxation be?

10. How are the weights affected?
Draw samples from the proposal transition density q, to find:
with weights
Likelihood weight
Proposal weight

11. The equivalent-weights Particle Filter
At the last time step towards the observations make sure all weights are approximately equal:
Set a target weight
Move each particle such that it has the target weight, using:
Give each particle small random perturbation.

12. The weights as function of the position in state space1 4 3
Target weight 2 5
xin
What should the target weight be?

13. Practical: the barotropic vorticity equation
Stochastic barotropic vorticity equation:
256 by 256 grid - 65,536 variables
Double periodic boundary conditions
Semi-Langrangian time stepping scheme
Identical twin experiments over 1200 time steps
Observations every 50 time steps – decorrelation time of 42
24-48 particles

14. ¼ Observations over half of state
Truth
Mean of particle filter ensemble

15. Experiment
Study influence of relaxation strength, or nudging strength (nudgefac)
Study influence of target weight, so how many particles can reach that weight, so percentage of particles to keep (keep).
Looking at:
Trajectories, fields, RMSE
Histograms, pdfs

16. Rank histogram

17. Rank Histograms SST (observed) Meridional wind high up in
Atmosphere (unobserved)

18. Enjoy !!!