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A Deterministic Filter for non-Gaussian State Estimation
Oliver Pajonk, Bojana Rosic, Alexander Litvinenko, Hermann G. Matthies
ISUME 2011, Prag, 2011-05-03
Picture: smokeonit (via Flickr.com)
Outline
 Motivation / Problem Statement
 State inference for dynamic system from measurements
 Proposed Solution
 Hilbert space of random variables (RVs) + representation of RVs by
PCE  a recursive, PCE-based, minimum variance estimator
 Examples
 Method applied to: a bi-modal truth; the Lorenz-96 model
 Conclusions
3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation
Motivation
 Estimate state of a dynamic system from measurements
 Lots of uncertainties and errors
 Bayesian approach: Model “state of knowledge” by probabilities
 New data should change/improve “state of knowledge”
[Tarantola, 2004]
 Methods:
 Bayes’ formula (expensive) or simplifications (approximations)
 Common: Gaussianity, linearity  Kalman-filter-like methods
[Evensen, 2009]
 KF, EKF, UKF, Gaussian-Mixture, … popular: EnKF
 All: Minimum variance estimates in Hilbert space
 Question: What if we “go back there”?
3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation
Outline
 Motivation / Problem Statement
 State inference for dynamic system from measurements
 Proposed Solution
 Hilbert space of random variables (RVs) + representation of RVs by
PCE  a recursive, PCE-based, minimum variance estimator
 Examples
 Method applied to: a bi-modal truth; the Lorenz-96 model
 Conclusions
3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation
Tool 1: Hilbert Space of Random Variables
[Luenberger, 1969]
*Under usual assumptions of uncorrelated errors!
3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation
Tool 2: Representation of RVs by
Polynomial Chaos Expansion (1/2)
[e.g. Holden, 1996]
* Of course, there are still more representations – we skip them for brevity.
3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation
Tool 2: Representation of RVs by
Polynomial Chaos Expansion (2/2)
“min-var-update”:
[Pajonk et al, 2011]
3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation
Outline
 Motivation / Problem Statement
 State inference for dynamic system from measurements
 Proposed Solution
 Hilbert space of random variables (RVs) + representation of RVs by
PCE  a recursive, PCE-based, minimum variance estimator
 Examples
 Method applied to: a bi-modal truth; the Lorenz-96 model
 Conclusions
3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation
Example 1: Bi-modal Identification
1
2
…
10
3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation
Example 2: Lorenz-84 Model
3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation
[Lorenz, 1984]
Example 2: Lorenz-84 – Application of PCE-based updating
 PCE  “Proper”
uncertainty
quantification
 Updates 
Variance reduction
and shift of mean
at update points
 Skewed structure
clearly visible,
preserved by
updates
3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation
Example 2: Lorenz-84 – Comparison with EnKF
3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation
Example 2: Lorenz-84 – Variance Estimates – PCE-based upd.
3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation
Example 2: Lorenz-84 – Variance Estimates – EnKF
3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation
Example 2: Lorenz-84 – Non-Gaussian Identification
(a) PCE-based
(b) EnKF
3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation
Conclusions & Outlook
 Recursive, deterministic, non-Gaussian minimum variance estimation
method
 Skewed & bi-modal identification possible
 Appealing mathematical properties: Rich mathematical structure of
Hilbert spaces available
 No closure assumptions besides truncation of PCE
 Direct computation of update from PCE efficient
 Fully deterministic: Possible applications with security & real time
requirements
 Future: Scale it to more complex systems, e.g. geophysical applications
 “Curse of dimensionality” (adaptivity, model reduction,…)
 Development of algebra (numerical & mathematical)
3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation
References & Acknowledgements
 Pajonk, O.; Rosic, B. V.; Litvinenko, A. & Matthies, H. G., A Deterministic
Filter for Non-Gaussian Bayesian Estimation, Physica D: Nonlinear
Phenomena, 2011, Submitted for publication
 Preprint: http://www.digibib.tu-bs.de/?docid=00038994
 The authors acknowledge the financial support from SPT Group for a
research position at the Institute of Scientific Computing at the TU
Braunschweig.
 Lorenz, E. N., Irregularity: a fundamental property of the atmosphere, Tellus A, Blackwell Publishing Ltd,
1984, 36, 98-110
 Evensen, G., The ensemble Kalman filter for combined state and parameter estimation, IEEE Control
Systems Magazine, 2009, 29, 82-104
 Tarantola, A., Inverse Problem Theory and Methods for Model Parameter Estimation, SIAM,
Philadelphia, 2004
 Luenberger, D. G., Optimization by Vector Space Methods, John Wiley & Sons, 1969
 Holden, H.; Øksendal, B.; Ubøe, J. & Zhang, T.-S., Stochastic Partial Differential Equations, Birkhäuser
Verlag, 1996
3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation