- Lorentz Center

advertisement
Particle Filters in high dimensions
Peter Jan van Leeuwen and Mel Ades
Data-Assimilation Research Centre DARC
University of Reading
Lorentz Center 2011
Data assimilation: general formulation
Bayes theorem:
Solution is pdf!
NO INVERSION !!!
Parameter estimation:
with
Again, no inversion but a direct point-wise multiplication.
Nonlinear filtering: Particle filter
Use ensemble
with
the weights.
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 !!!
Standard Particle filter
Not very efficient !
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)
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
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.
Increased efficiency: proposal
density
Instead of drawing samples from p(x) we draw samples from
a proposal pdf q(x).
Use ensemble
with weights
Particle filter with
proposal transition
density
Barotropic vorticity 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
Vorticity field standard particle filter
Truth
Ensemble mean PF
Vorticity field new particle filter
Truth
Ensemble mean
Posterior weights
Rank histogram:
How the truth ranks in the ensemble
Equivalent Weights Particle Filter
Recall:
Assume
Find the minimum for each particle by
perturbing each observation, gives
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.
Equivalent Weights Particle Filter
• This leads to 80% of the particle having
an almost equal weight, so no degeneracy
by construction!
• Example …
Gaussian pdf in high dimensions
Assume variables are identically independently distributed:
Along each if the axes it looks
like a standard Gaussian:
However, the probability mass as function of the distance to the
centre is given by:
d=100
d=400
r in standard deviation
d=900
The so-called
Important Ring
Why?
In distribution
Fisher has shown
So we find
Importance Ring
Experimental evidence, sums of d squared random numbers:
r
d
Given this what do these
efficient particles represent???
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?
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
Download
Related flashcards

Quantum field theory

40 cards

Standard Model

11 cards

Particle accelerators

13 cards

Elementary particles

12 cards

Create Flashcards