Gaussian process approximations of stochastic differential equations

Cédric Archambeau, Dan Cornford, Manfred Opper, John Shawe-Taylor

Research output: Contribution to journalArticle

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.
Original languageEnglish
Pages (from-to)1-16
Number of pages16
JournalJournal of Machine Learning Research
Volume1
Publication statusPublished - 11 Mar 2007

Fingerprint

Gaussian Process
Stochastic Equations
Differential equations
Environmental Modelling
Differential equation
Variational Approximation
Ornstein-Uhlenbeck Process
Approximation
Markov Process
Markov processes
Approximate Solution
Ensemble
Exact Solution
Regression
Path
Range of data
Experiment
Experiments
Standards
Context

Bibliographical note

JMLR Workshop and Conference Proceedings Volume 1: GPIP, 12-13 June 2006, Bletchley (UK). © 2007 C. Archambeau, D. Cornford, M. Opper and J. Shawe-Taylor.

Keywords

  • dynamical systems
  • stochastic processes
  • Bayesian inference
  • Gaussian processes

Cite this

Archambeau, C., Cornford, D., Opper, M., & Shawe-Taylor, J. (2007). Gaussian process approximations of stochastic differential equations. Journal of Machine Learning Research, 1, 1-16.
Archambeau, Cédric ; Cornford, Dan ; Opper, Manfred ; Shawe-Taylor, John. / Gaussian process approximations of stochastic differential equations. In: Journal of Machine Learning Research. 2007 ; Vol. 1. pp. 1-16.
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Archambeau, C, Cornford, D, Opper, M & Shawe-Taylor, J 2007, 'Gaussian process approximations of stochastic differential equations', Journal of Machine Learning Research, vol. 1, pp. 1-16.

Gaussian process approximations of stochastic differential equations. / Archambeau, Cédric; Cornford, Dan; Opper, Manfred; Shawe-Taylor, John.

In: Journal of Machine Learning Research, Vol. 1, 11.03.2007, p. 1-16.

Research output: Contribution to journalArticle

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AU - Archambeau, Cédric

AU - Cornford, Dan

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Archambeau C, Cornford D, Opper M, Shawe-Taylor J. Gaussian process approximations of stochastic differential equations. Journal of Machine Learning Research. 2007 Mar 11;1:1-16.