Gaussian Process Approximations of Stochastic Differential Equations

Cedric Archambeau, Dan Cornford, Manfred Opper, John Shawe-Taylor
Gaussian Processes in Practice, PMLR 1:1-16, 2007.

Abstract

Stochastic differential equations arise naturally in a range of contexts, from financial to environmental modeling. Current solution methods are limited in their representation of the posterior process in the presence of data. In this work, we present a novel Gaussian process approximation to the posterior measure over paths for a general class of stochastic differential equations in the presence of observations. The method is applied to two simple problems: the Ornstein-Uhlenbeck process, of which the exact solution is known and can be compared to, and the double-well system, for which standard approaches such as the ensemble Kalman smoother fail to provide a satisfactory result. Experiments show that our variational approximation is viable and that the results are very promising as the variational approximate solution outperforms standard Gaussian process regression for non-Gaussian Markov processes.

Cite this Paper

BibTeX
@InProceedings{pmlr-v1-archambeau07a, title = {Gaussian Process Approximations of Stochastic Differential Equations}, author = {Archambeau, Cedric and Cornford, Dan and Opper, Manfred and Shawe-Taylor, John}, booktitle = {Gaussian Processes in Practice}, pages = {1--16}, year = {2007}, editor = {Lawrence, Neil D. and Schwaighofer, Anton and Quiñonero Candela, Joaquin}, volume = {1}, series = {Proceedings of Machine Learning Research}, address = {Bletchley Park, UK}, month = {12--13 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v1/archambeau07a/archambeau07a.pdf}, url = {https://proceedings.mlr.press/v1/archambeau07a.html}, abstract = {Stochastic differential equations arise naturally in a range of contexts, from financial to environmental modeling. Current solution methods are limited in their representation of the posterior process in the presence of data. In this work, we present a novel Gaussian process approximation to the posterior measure over paths for a general class of stochastic differential equations in the presence of observations. The method is applied to two simple problems: the Ornstein-Uhlenbeck process, of which the exact solution is known and can be compared to, and the double-well system, for which standard approaches such as the ensemble Kalman smoother fail to provide a satisfactory result. Experiments show that our variational approximation is viable and that the results are very promising as the variational approximate solution outperforms standard Gaussian process regression for non-Gaussian Markov processes.} }
Endnote
%0 Conference Paper %T Gaussian Process Approximations of Stochastic Differential Equations %A Cedric Archambeau %A Dan Cornford %A Manfred Opper %A John Shawe-Taylor %B Gaussian Processes in Practice %C Proceedings of Machine Learning Research %D 2007 %E Neil D. Lawrence %E Anton Schwaighofer %E Joaquin Quiñonero Candela %F pmlr-v1-archambeau07a %I PMLR %P 1--16 %U https://proceedings.mlr.press/v1/archambeau07a.html %V 1 %X Stochastic differential equations arise naturally in a range of contexts, from financial to environmental modeling. Current solution methods are limited in their representation of the posterior process in the presence of data. In this work, we present a novel Gaussian process approximation to the posterior measure over paths for a general class of stochastic differential equations in the presence of observations. The method is applied to two simple problems: the Ornstein-Uhlenbeck process, of which the exact solution is known and can be compared to, and the double-well system, for which standard approaches such as the ensemble Kalman smoother fail to provide a satisfactory result. Experiments show that our variational approximation is viable and that the results are very promising as the variational approximate solution outperforms standard Gaussian process regression for non-Gaussian Markov processes.
RIS
TY - CPAPER TI - Gaussian Process Approximations of Stochastic Differential Equations AU - Cedric Archambeau AU - Dan Cornford AU - Manfred Opper AU - John Shawe-Taylor BT - Gaussian Processes in Practice DA - 2007/03/11 ED - Neil D. Lawrence ED - Anton Schwaighofer ED - Joaquin Quiñonero Candela ID - pmlr-v1-archambeau07a PB - PMLR DP - Proceedings of Machine Learning Research VL - 1 SP - 1 EP - 16 L1 - http://proceedings.mlr.press/v1/archambeau07a/archambeau07a.pdf UR - https://proceedings.mlr.press/v1/archambeau07a.html AB - Stochastic differential equations arise naturally in a range of contexts, from financial to environmental modeling. Current solution methods are limited in their representation of the posterior process in the presence of data. In this work, we present a novel Gaussian process approximation to the posterior measure over paths for a general class of stochastic differential equations in the presence of observations. The method is applied to two simple problems: the Ornstein-Uhlenbeck process, of which the exact solution is known and can be compared to, and the double-well system, for which standard approaches such as the ensemble Kalman smoother fail to provide a satisfactory result. Experiments show that our variational approximation is viable and that the results are very promising as the variational approximate solution outperforms standard Gaussian process regression for non-Gaussian Markov processes. ER -
APA
Archambeau, C., Cornford, D., Opper, M. & Shawe-Taylor, J.. (2007). Gaussian Process Approximations of Stochastic Differential Equations. Gaussian Processes in Practice, in Proceedings of Machine Learning Research 1:1-16 Available from https://proceedings.mlr.press/v1/archambeau07a.html.

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