Chapter27 Bayesian System Identification of Dynamical Systems Using Reversible Jump Markov Chain Monte Carlo D. Tiboaca, P.L. Green, R.J. Barthorpe, and K. Worden Abstract The purpose of this contribution is to illustrate the potential of Reversible Jump Markov Chain Monte Carlo (RJMCMC) methods for nonlinear system identification. Markov Chain Monte Carlo (MCMC) sampling methods have come to be viewed as a standard tool for tackling the issue of parameter estimation using Bayesian inference. A limitation of standard MCMC approaches is that they are not suited to tackling the issue of model selection. RJMCMC offers a powerful extension to standard MCMC approaches in that it allows parameter estimation and model selection to be addressed simultaneously. This is made possible by the fact that the RJMCMC algorithm is able to “jump" between parameter spaces of varying dimension. In this paper the background theory to the RJMCMC algorithm is introduced. Comparison is made to a standard MCMC approach. Keywords Nonlinear dynamics • System identification • Bayesian inference • MCMC • RJMCMC 27.1 Introduction Since their invention in 1953, Markov Chain Monte Carlo (MCMC) sampling methods have been used in many research areas where they have proved their capacity of sampling from probability density functions (PDFs) with complex geometries. In the domain of system identification (SID), MCMC has been extensively used as a tool for parameter estimation. MCMC sampling methods are part of a group of algorithms that, through the use of generated samples from geometrically complicated PDFs, can be implemented to estimate the parameters on which a system depends. Because they make use of PDFs, MCMC algorithms have proven to work extremely well within a Bayesian framework. By joining these two concepts, one can conduct SID for either linear or nonlinear models efficiently. This is of great interest in structural dynamics, at a time when nonlinear models still remain difficult to identify and understand. The aim of this contribution is to give a better understanding of the RJMCMC algorithm and its application in system identification. A comparison is made between the Metropolis-Hastings (one of the MCMC samplers) algorithm and the RJMCMC algorithm in order to demonstrate the advantages of the later in model selection. The detailed balance principle is explained and it is proven that it is respected by both Metropolis-Hastings and RJMCMC algorithms. The power of RJMCMC is demonstrated through its ability of dealing efficiently with model selection and parameter estimation (simultaneously) for both linear and nonlinear models. Work of particular relevance in SID, with the use of Bayesian inference and MCMC algorithms, was conducted by Beck and Au [1]. The authors proposed a MCMC approach to sample from PDFs with multiple modes. A Metropolis-Hasting algorithm with a version of the Simulated Annealing algorithm were used together to obtain the “regions of concentration” of the posterior PDF. In [1] there are two different models used to demonstrate the validity of their proposed method, one locally identifiable and one unidentifiable (the locally identifiable model had multiple optimum parameter vectors while the unidentifiable model had a continuum of optimum parameter vectors). Paper [1] tackles the problems of uncertainty and reliability as well. Rather than selecting the model considering the data as was done in previous work with SID, the paper proposes a predictive approach which puts together all possible models according to their probability of being the D. Tiboaca ( ) • P.L. Green • R.J. Barthorpe • K. Worden Department of Mechanical Engineering, University of Sheffield, Sir Frederick Mappin Building, Mappin Street, Sheffield S1 3JD, UK e-mail: daniela.tiboaca@sheffield.ac.uk R. Allemang (ed.), Topics in Modal Analysis II, Volume 8: Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-04774-4__27, © The Society for Experimental Mechanics, Inc. 2014 277
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