Sampling The Posterior: An Approach to Non-Gaussian Data Assimilation
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Amit Apte, Martin Hairer, Andrew M. Stuart and Jochen Voss:
Sampling The Posterior: An Approach to Non-Gaussian Data Assimilation.
Physica D: Nonlinear Phenomena, vol. 230, no. 1–2,
pp. 50–64, 2007.
The viewpoint taken in this paper is that data assimilation is
fundamentally a statistical problem and that this problem should be cast
in a Bayesian framework. In the absence of model error, the correct
solution to the data assimilation problem is to find the posterior
distribution implied by this Bayesian setting. Methods for dealing with
data assimilation should then be judged by their ability to probe this
distribution. In this paper we propose a range of techniques for probing
the posterior distribution, based around the Langevin equation; and we
compare these new techniques with existing methods.
When the underlying dynamics is deterministic, the posterior
distribution is on the space of initial conditions leading to a sampling
problem over this space. When the underlying dynamics is stochastic the
posterior distribution is on the space of continuous time paths. By
writing down a density, and conditioning on observations, it is possible
to define a range of Markov Chain Monte Carlo (MCMC) methods which sample
from the desired posterior distribution, and thereby solve the data
assimilation problem. The basic building-blocks for the MCMC methods that
we concentrate on in this paper are Langevin equations which are ergodic
and whose invariant measures give the desired distribution; in the case of
path space sampling these are stochastic partial differential equations
(SPDEs).
Two examples are given to show how data assimilation can be formulated
in a Bayesian fashion. The first is weather prediction, and the second is
Lagrangian data assimilation for oceanic velocity fields. Furthermore the
relationship between the Bayesian approach outlined here and the commonly
used Kalman filter-based techniques, prevalent in practice, is discussed.
Two simple pedagogical examples are studied to illustrate the application
of Bayesian sampling to data assimilation concretely. Finally a range of
open mathematical and computational issues, arising from the Bayesian
approach, are outlined.
Citations
I believe that this work is cited in the following texts.
If you know of any more citations, please let me know.
M. Hairer, A.M. Stuart and J. Voss:
Signal Processing Problems on Function Space: Bayesian Formulation, Stochastic PDEs and Effective MCMC Methods.
Pages 833–873 in The Oxford Handbook of Nonlinear Filtering,
Dan Crisan and Boris Rozovsky (editors), Oxford University Press, 2011.
link, preprint:pdf, more…
A. Beskos and A.M. Stuart:
Computational complexity of Metropolis-Hastings methods in high dimensions.
Pages 61–72 in Proceedings of MCQMC08,
Pierre L'Ecuyer and Art B. Owen (editors), 2010.
link
A.M. Stuart:
Inverse problems: A Bayesian perspective.
Acta Numerica, vol. 19, pp. 451–559, 2010.
DOI:10.1017/S0962492910000061
A. Amit, A. Didier and R. Mythily:
Variational data assimilation for discrete Burgers equation.
Pages 15–30 in Proceedings of the Eighth Mississippi State - UAB Conference on Differential Equations and Computational Simulations,
vol. 19 of Electronic Journal of Differential Equations: Conference,
2010.
P. Krause:
The Diffusion Kernel Filter.
Journal of Statistical Physics, vol. 134, no. 2,
pp. 365–380, 2009.
DOI:10.1007/s10955-008-9673-1
A. Duggento, D.G. Luchinsky, V.N. Smelyanskiy and P.V.E. McClintock:
Inferential framework for non-stationary dynamics: theory and applications.
Journal of Statistical Mechanics-Theory and Experiment, 2009.
DOI:10.1088/1742-5468/2009/01/P01025
M. Hairer, A.M. Stuart and J. Voss:
Sampling Conditioned Diffusions.
Pages 159–186 in Trends in Stochastic Analysis,
Cambridge University Press,
vol. 353 of London Mathematical Society Lecture Note Series, 2009.
link, preprint:pdf, more…
S.L. Cotter, M. Dashti, J.C. Robinson and A.M. Stuart:
Bayesian inverse problems for functions and applications to fluid mechanics.
Inverse Problems, vol. 25, 2009.
DOI:10.1088/0266-5611/25/11/115008
A. Beskos and A.M. Stuart:
MCMC methods for sampling function space.
Pages 337–364 in Proceedings of the 6th International Congress on Industrial and Applied Mathematicians (Zürich, 2007),
Rolf Jeltsch and Gerhard Wanner (editors), 2009.
J.M. Restrepo:
A path integral method for data assimilation.
Phys. D, vol. 237, no. 1, pp. 14–27, 2008.
DOI:10.1016/j.physd.2007.07.020
A. Beskos, G.O. Roberts, A.M. Stuart and J. Voss:
MCMC Methods for Diffusion Bridges.
Stochastics and Dynamics, vol. 8, no. 3, pp. 319–350,
2008.
DOI:10.1142/S0219493708002378, preprint:pdf, more…
D.G. Luchinsky, V.N. Smelyanskiy, A. Duggento and P.V.E. McClintock:
Inferential framework for nonstationary dynamics. I. Theory.
Physical Review E, vol. 77, no. 6, Part 1, 2008.
DOI:10.1103/PhysRevE.77.061105
A. Apte, C.K.R.T. Jones, A.M. Stuart and J. Voss:
Data Assimilation: Mathematical and Statistical Perspectives.
International Journal for Numerical Methods in Fluids, vol. 56,
no. 8, pp. 1033–1046, 2008.
DOI:10.1002/fld.1698, preprint:pdf, more…
A. Apte, C.K.R.T. Jones and A.M. Stuart:
A Bayesian approach to Lagrangian data assimilation.
Tellus Series A-Dynamic Meteorology and Oceanography, vol. 60,
no. 2, pp. 336–347, 2008.
DOI:10.1111/j.1600-0870.2007.00295.x
R. Barillec:
Bayesian Data Assimilation.
PhD thesis, Aston University, 2008.
link
M. Hairer, A.M. Stuart and J. Voss:
Analysis of SPDEs Arising in Path Sampling, Part II: The Nonlinear Case.
Annals of Applied Probability, vol. 17, no. 5,
pp. 1657–1706, 2007.
DOI:10.1214/07-AAP441, arXiv:math/0601092, more…
J.O. Ramsay, G. Hooker, D. Campbell and J. Cao:
Parameter estimation for differential equations: a generalized smoothing approach.
J. R. Stat. Soc. Ser. B Stat. Methodol., vol. 69, no. 5,
pp. 741–796, With discussions and a reply by the authors, 2007.
MR2368570
C. Archambeau, D. Cornford, M. Opper and J. Shawe-Taylor:
Gaussian process approximations of stochastic differential equations.
Pages 1–16 in Gaussian Processes in Practice,
Neil Lawrence, Anton Schwaighofer and Joaquin Quiñonero Candela (editors),
vol. 1 of JMLR: Workshop and Conference Proceedings, 2007.
link
R. van Handel:
Filtering, Stability, and Robustness.
PhD thesis, California Institute of Technology, Passadena, California, 2007.