Estimating parameters in stochastic systems:a variational Bayesian approach


This work is concerned with approximate inference in dynamical systems, from a variational Bayesian perspective. When modelling real world dynamical systems, stochastic differential equations appear as a natural choice, mainly because of their ability to model the noise of the system by adding a variation of some stochastic process to the deterministic dynamics. Hence, inference in such processes has drawn much attention. Here a new extended framework is derived that is based on a local polynomial approximation of a recently proposed variational Bayesian algorithm. The paper begins by showing that the new extension of this variational algorithm can be used for state estimation (smoothing) and converges to the original algorithm. However, the main focus is on estimating the (hyper-) parameters of these systems (i.e. drift parameters and diffusion coefficients). The new approach is validated on a range of different systems which vary in dimensionality and non-linearity. These are the Ornstein–Uhlenbeck process, the exact likelihood of which can be computed analytically, the univariate and highly non-linear, stochastic double well and the multivariate chaotic stochastic Lorenz ’63 (3D model). As a special case the algorithm is also applied to the 40 dimensional stochastic Lorenz ’96 system. In our investigation we compare this new approach with a variety of other well known methods, such as the hybrid Monte Carlo, dual unscented Kalman filter, full weak-constraint 4D-Var algorithm and analyse empirically their asymptotic behaviour as a function of observation density or length of time window increases. In particular we show that we are able to estimate parameters in both the drift (deterministic) and the diffusion (stochastic) part of the model evolution equations using our new methods.

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Divisions: ?? 50811700Jl ??
College of Engineering & Physical Sciences > Systems analytics research institute (SARI)
Additional Information: NOTICE: this is the author’s version of a work that was accepted for publication in Physica D: Nonlinear Phenomena. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Vrettas, M, Cornford, D & Opper, M, 'Estimating parameters in stochastic systems: a variational Bayesian approach', PHYSICA D-NONLINEAR PHENOMENA, vol 240, no. 23, pp. 1877–1900., (2011) DOI
Uncontrolled Keywords: Bayesian inference,variational techniques,dynamical systems,stochastic differential equations,parameter estimation,Condensed Matter Physics,Statistical and Nonlinear Physics
Publication ISSN: 1872-8022
Last Modified: 05 Jun 2024 07:09
Date Deposited: 19 Apr 2012 11:39
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Related URLs: http://www.scop ... tnerID=8YFLogxK (Scopus URL)
PURE Output Type: Article
Published Date: 2011-11-15
Published Online Date: 2011-09-18
Authors: Vrettas, Michail
Cornford, Dan (ORCID Profile 0000-0001-8787-6758)
Opper, Manfred



Version: Accepted Version

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