1. Nonlinear Data Assimilation and Particle Filters
Peter Jan van Leeuwen

2. Data Assimilation Ingredients
Prior knowledge, the Stochastic model:
Observations:
Relation between the two:

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

4. Parameter estimation:
with
in which H(..) contains the numerical model.
Again, no inversion but a direct point-wise multiplication.

5. Propagation of pdf in time: Kolmogorov’s equation
Model equation:
Pdf evolution: Kolmogorov’s equation
(Fokker-Planck equation)
Too expensive !!!
advection
diffusion

6. Too expensive? But computers grow larger and larger…
How big is the nonlinear data-assimilation problem?
Assume we need 10 frequency bins for each variable to build the joint pdf of all variables.
Let’s assume we have a modest model with a million variables.
Then we need to store ,000 numbers.
The total number of atoms in the universe is estimated to be about 1080.
So the data-assimilation problem is larger than the universe…

7. Motivation ensemble methods: ‘Efficient’ propagation of pdf in time

8. How is DA used today in geosciences?
Present-day data-assimilation systems are based on linearizations and state covariances are essential.
4DVar, Representer method (PSAS):
- Gaussian pdf’s , solves for posterior mode, needs error covariance of initial state (B matrix), ‘no’ posterior error covariances
(Ensemble) Kalman filter:
- assumes Gaussian pdf’s for the state, approximates posterior mean and covariance, doesn’t minimize anything in nonlinear systems, needs inflation and localisation
Combinations of these: hybrid methods (!!!)

9. Non-linear Data Assimilation
Metropolis-Hastings Start from one sample, generate a new one and decide on acceptance (better, or by chance), etc. Slow convergence, but new smarter algorithms are being devised.
Langevin sampling Idem, but always accept, each sample expensive. Slow convergence, but smarter algorithms are being devised.
Hybrid Monte-Carlo idem, but almost always accept, each sample expensive, faster convergence

10. Non-linear Data Assimilation
Particle Filters/Smoothers Generate samples in parallel sequential over time and weight them according how good they are. Importance sampling. Can be made very efficient.
Combinations of MH and PF Expensive but good for e.g. parameter estimation.

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

12. 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 !!!

13. No explicit need for state covariances
3DVar and 4DVar need a good error covariance of the prior state estimate: complicated
The performance of Ensemble Kalman filters relies on the quality of the sample covariance, forcing artificial inflation and localisation.
Particle filter doesn’t have this problem, but…

14. Standard Particle filter
The standard particle filter is degenerate for moderate
ensemble size in moderate-dimensional systems.

15. Particle Filter degeneracy: resampling
With each new set of observations the old weights are multiplied with the new weights.
Very soon only one particle has all the weight…
Solution:
Resampling: duplicate high-weight particles and abandon low-weight particles

16. Standard Particle filter

17. A simple resampling scheme
1. Put all weights after each other on the unit interval: 1 w1 w2 w3 w4 w5 w6 w7 w8 w9
w10
2. Draw a random number from the uniform distribution over [0,1/N],
in this case with 10 members over [0,1/10].
3. Put that number on the unit interval: its end point is the first
member drawn.
4. Add 1/N to the end point: the new end point is our second member.
Repeat this until N new members are obtained. 1 1 w1 w2 w3 w4 w5 w6 w7 w8 w9
w10
5. In our example we choose m1 2 times, m2 2 times, m3, m4,
m5 2 times, m6 and m7.

18. 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)

19. 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

20. 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.

21. A closer look at the weights IV
The volume of a hypersphere of radius r in an M dimensional space is
Taking for the radius we find, using Stirling:
So very small indeed.

22. The volume in hyperspace occupied by observations
Log10 of Volume of hypersphere
of radius 1.
Number of observations

23. How can we make particle filters useful?
The joint-in-time prior pdf can be written as:
So the marginal prior pdf at time n becomes:
We introduced the transition densities

24. Meaning of the transition densities
Stochastic model:
Transition density:
So, draw a sample from the model error pdf, and use that in the stochastic model equations.
For a deterministic model this pdf is a delta function centered around the the deterministic forward step.
For a Gaussian model error we find:

25. 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:

26. The magic: the proposal density
Performing the integral over the sum of delta functions gives:
The posterior is now given as a sum of transition densities.
In the standard particle filter we use these to draw particles
at time n, which, remember, is running the stochastic model
from time n-1 to time n. We know that is degenerate.
So we introduce another transition density, the proposal.

27. 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.
Ensures that the particles end up close to the observations
because they know where the observations are.
2) Allows for an equal-weight filter, as the performance bounds
suggested by Snyder, Bickel, and Bengtsson do not apply.

28. What does this all mean?
The standard Particle Filter propagates the original model by drawing from p(xn|xn-1).
Now we draw from , so we propagate the state using a different model.
This model can be anything, e.g.

29. Examples of proposal transition densities
The proposal transition density is related to a proposed model.
For instance, add a relaxation term and change random forcing:
Or, run a 4D-Var on each particle (implicit particle filter).
This is a special 4D-Var:
initial condition is fixed
model error essential
needs extra random forcing
Or use the EnKF as proposal density.

30. How are the weights affected?
Draw samples from the proposal transition density q, to find:
Which can be rewritten as:
with weights
Likelihood weight
Proposal weight

31. How to calculate p/q in the weights?
Let’s assume that the original model has Gaussian distributed
model errors:
To calculate the value of this term realise it is the probability of
moving from xin-1 to xin. Since xin and xin-1 are known from the
proposed model we can calculate directly:

32. Example calculation of p
Assume the proposed model is
Then we find
We know all the terms, so this can be calculated

33. And q … The deterministic part of the proposed model is:
So the probability becomes
We did draw the stochastic terms, so we know what they are, so this term can be calculated too.

34. The weights
We can calculate p/q and we can calculate the likelihood so we can calculate the weights:

35. Example: EnKF as proposal
EnKF update:
Use model equation:
Regroup terms:
Leading to:

36. Algorithm Generate initial set of particles
Run proposed model conditioned on next observation
Accumulate proposal density weights p/q
Calculate likelihood weights
Calculate full weights and resample
Note, the original model is never used directly.

37. Particle filter with proposal transition density
Still degenerate …

38. Equal-weight Particle filtering
Define an implicit map as follows:
is the mode of the optimal proposal density,
which is given by
is a random draw from the density N(0,P), with the P covariance of the optimal proposal density,
is chosen such that all particles have equal weight (using the expression for the weights)

39. How to find Remember the new particle is given by:
in which and are known. Use this expression
In the weights and set all weights equal to a target weight:
and solve for Now all weights are equal by construction!

40. Experiments, model error and observation errors Gaussian, H linear
Linear model of Snyder et al
1000 dimensional independent Gaussian linear model
20 particles
Observations every time step
Lorenz 1996
1000 dimensional
Observations every 5 time steps, half of state

41. Linear model: Rank histogram 1000 time steps, 20 particles

42. Normalised pdf 1000 time steps 20 particles

43. Normalised pdf 1000 time steps 1000 particles -> Convergence!

44. Lorenz 1996 1000 dimensional 20 particles

45. Rank histograms 10000 time steps 20 particles, land-sea configuration
Observed gridpoint
Unobserved gridpoint

46. Conclusions
Large number of ‘nonlinear’ filters and smoothers available
Best method will be system dependent
Fully nonlinear equal-weight particle filters
for systems with arbitrary dimensions that
converge to the truth posterior pdf do exist.
Proposal-density freedom needs further exploration
Example shown for 1000 dimensional systems, but
methods have been applied to 2.3 million dimensional
systems too.