Particle Filters in high dimensions
description
Transcript of Particle Filters in high dimensions
![Page 1: Particle Filters in high dimensions](https://reader035.fdocuments.net/reader035/viewer/2022062309/568150d2550346895dbef5c9/html5/thumbnails/1.jpg)
Particle Filters in high dimensions
Peter Jan van Leeuwen and Mel AdesData-Assimilation Research Centre DARC
University of Reading
Lorentz Center 2011
![Page 2: Particle Filters in high dimensions](https://reader035.fdocuments.net/reader035/viewer/2022062309/568150d2550346895dbef5c9/html5/thumbnails/2.jpg)
Data assimilation: general formulation
Solution is pdf!
NO INVERSION !!!
Bayes theorem:
![Page 3: Particle Filters in high dimensions](https://reader035.fdocuments.net/reader035/viewer/2022062309/568150d2550346895dbef5c9/html5/thumbnails/3.jpg)
Parameter estimation:
with
Again, no inversion but a direct point-wise multiplication.
![Page 4: Particle Filters in high dimensions](https://reader035.fdocuments.net/reader035/viewer/2022062309/568150d2550346895dbef5c9/html5/thumbnails/4.jpg)
Nonlinear filtering: Particle filter
Use ensemble
with the weights.
![Page 5: Particle Filters in high dimensions](https://reader035.fdocuments.net/reader035/viewer/2022062309/568150d2550346895dbef5c9/html5/thumbnails/5.jpg)
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 !!!
![Page 6: Particle Filters in high dimensions](https://reader035.fdocuments.net/reader035/viewer/2022062309/568150d2550346895dbef5c9/html5/thumbnails/6.jpg)
Standard Particle filter
Not very efficient !
![Page 7: Particle Filters in high dimensions](https://reader035.fdocuments.net/reader035/viewer/2022062309/568150d2550346895dbef5c9/html5/thumbnails/7.jpg)
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)
![Page 8: Particle Filters in high dimensions](https://reader035.fdocuments.net/reader035/viewer/2022062309/568150d2550346895dbef5c9/html5/thumbnails/8.jpg)
A closer look at the weights IIAssume 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
![Page 9: Particle Filters in high dimensions](https://reader035.fdocuments.net/reader035/viewer/2022062309/568150d2550346895dbef5c9/html5/thumbnails/9.jpg)
A closer look at the weights III
The ratio of the weights is
Take M=1000 to find
Conclusion: the number of independent observations isresponsible for the degeneracy in particle filters.
![Page 10: Particle Filters in high dimensions](https://reader035.fdocuments.net/reader035/viewer/2022062309/568150d2550346895dbef5c9/html5/thumbnails/10.jpg)
Increased efficiency: proposal density
Instead of drawing samples from p(x) we draw samples froma proposal pdf q(x).
Use ensemble
with weights
![Page 11: Particle Filters in high dimensions](https://reader035.fdocuments.net/reader035/viewer/2022062309/568150d2550346895dbef5c9/html5/thumbnails/11.jpg)
Particle filter with proposal transition
density
![Page 12: Particle Filters in high dimensions](https://reader035.fdocuments.net/reader035/viewer/2022062309/568150d2550346895dbef5c9/html5/thumbnails/12.jpg)
Barotropic vorticity equation
256 X 256 grid points600 time stepsTypically q=1-3Decorrelation time scale= = 25 time steps
Observations Every 4th gridpointEvery 50th time step
24 particlessigma_model=0.03 sigma_obs=0.01
![Page 13: Particle Filters in high dimensions](https://reader035.fdocuments.net/reader035/viewer/2022062309/568150d2550346895dbef5c9/html5/thumbnails/13.jpg)
Vorticity field standard particle filter
Ensemble mean PFTruth
![Page 14: Particle Filters in high dimensions](https://reader035.fdocuments.net/reader035/viewer/2022062309/568150d2550346895dbef5c9/html5/thumbnails/14.jpg)
Vorticity field new particle filter
Ensemble meanTruth
![Page 15: Particle Filters in high dimensions](https://reader035.fdocuments.net/reader035/viewer/2022062309/568150d2550346895dbef5c9/html5/thumbnails/15.jpg)
Posterior weights
![Page 16: Particle Filters in high dimensions](https://reader035.fdocuments.net/reader035/viewer/2022062309/568150d2550346895dbef5c9/html5/thumbnails/16.jpg)
Rank histogram:How the truth ranks in the
ensemble
![Page 17: Particle Filters in high dimensions](https://reader035.fdocuments.net/reader035/viewer/2022062309/568150d2550346895dbef5c9/html5/thumbnails/17.jpg)
Equivalent Weights Particle Filter
Recall:
Assume
Find the minimum for each particle by perturbing each observation, gives
![Page 18: Particle Filters in high dimensions](https://reader035.fdocuments.net/reader035/viewer/2022062309/568150d2550346895dbef5c9/html5/thumbnails/18.jpg)
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.
![Page 19: Particle Filters in high dimensions](https://reader035.fdocuments.net/reader035/viewer/2022062309/568150d2550346895dbef5c9/html5/thumbnails/19.jpg)
Equivalent Weights Particle Filter
• This leads to 80% of the particle having an almost equal weight, so no degeneracy by construction!• Example …
![Page 20: Particle Filters in high dimensions](https://reader035.fdocuments.net/reader035/viewer/2022062309/568150d2550346895dbef5c9/html5/thumbnails/20.jpg)
Gaussian pdf in high dimensions
Assume variables are identically independently distributed:
Along each if the axes it looks like a standard Gaussian:
![Page 21: Particle Filters in high dimensions](https://reader035.fdocuments.net/reader035/viewer/2022062309/568150d2550346895dbef5c9/html5/thumbnails/21.jpg)
However, the probability mass as function of the distance to thecentre is given by:
d=100 d=400 d=900
r in standard deviation
The so-calledImportant Ring
![Page 22: Particle Filters in high dimensions](https://reader035.fdocuments.net/reader035/viewer/2022062309/568150d2550346895dbef5c9/html5/thumbnails/22.jpg)
Why?
In distribution
Fisher has shown
So we find
![Page 23: Particle Filters in high dimensions](https://reader035.fdocuments.net/reader035/viewer/2022062309/568150d2550346895dbef5c9/html5/thumbnails/23.jpg)
Importance Ring
r
d
Experimental evidence, sums of d squared random numbers:
![Page 24: Particle Filters in high dimensions](https://reader035.fdocuments.net/reader035/viewer/2022062309/568150d2550346895dbef5c9/html5/thumbnails/24.jpg)
Given this what do theseefficient particles represent???
![Page 25: Particle Filters in high dimensions](https://reader035.fdocuments.net/reader035/viewer/2022062309/568150d2550346895dbef5c9/html5/thumbnails/25.jpg)
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?
![Page 26: Particle Filters in high dimensions](https://reader035.fdocuments.net/reader035/viewer/2022062309/568150d2550346895dbef5c9/html5/thumbnails/26.jpg)
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