Approximate Bayesian Computation by Subset Simulation using hierarchical state-space models. (1st February 2017)

• Record Type: Journal Article
• Authors: Vakilzadeh, Majid K.; Huang, Yong; Beck, James L.; Abrahamsson, Thomas
• Is Part Of: Mechanical systems and signal processing. Volume 84:Part B(2017)
• Page Start: 2
• Page End: 20
• Publication Date: 2017-02-01
• Subjects: Approximate Bayesian Computation -- Subset Simulation -- Self-regulating ABC-SubSim algorithm -- Optimal proposal variance scaling; Structural dynamics -- Periodicals; Vibration -- Periodicals; Constructions -- Dynamique -- Périodiques; Vibration -- Périodiques; Structural dynamics; Vibration; Periodicals; 621
• DOI: 10.1016/j.ymssp.2016.02.024
• Languages: English
• ISSNs: 0888-3270

Abstract

Abstract: A new multi-level Markov Chain Monte Carlo algorithm for Approximate Bayesian Computation, ABC-SubSim, has recently appeared that exploits the Subset Simulation method for efficient rare-event simulation. ABC-SubSim adaptively creates a nested decreasing sequence of data-approximating regions in the output space that correspond to increasingly closer approximations of the observed output vector in this output space. At each level, multiple samples of the model parameter vector are generated by a component-wise Metropolis algorithm so that the predicted output corresponding to each parameter value falls in the current data-approximating region. Theoretically, if continued to the limit, the sequence of data-approximating regions would converge on to the observed output vector and the approximate posterior distributions, which are conditional on the data-approximation region, would become exact, but this is not practically feasible. In this paper we study the performance of the ABC-SubSim algorithm for Bayesian updating of the parameters of dynamical systems using a general hierarchical state-space model. We note that the ABC methodology gives an approximate posterior distribution that actually corresponds to an exact posterior where a uniformly distributed combined measurement and modeling error is added. We also note that ABC algorithms have a problem with learning the uncertain error variances in a stochastic state-space model and so we treat them as nuisance parameters and analytically integrate them out of the posterior distribution. In addition, the statistical efficiency of the original ABC-SubSim algorithm is improved by developing a novel strategy to regulate the proposal variance for the component-wise Metropolis algorithm at each level. We demonstrate that Self-regulated ABC-SubSim is well suited for Bayesian system identification by first applying it successfully to model updating of a two degree-of-freedom linear structure for three cases: globally, locally and un-identifiable model classes, and then to model updating of a two degree-of-freedom nonlinear structure with Duffing nonlinearities in its interstory force-deflection relationship. Highlights: This paper exploits the potential of a recently published multi-level MCMC algorithm, ABC-SubSim, for Bayesian inference of the model parameters of a general stochastic state-space model. It is shown that the approximate posterior actually corresponds to an exact posterior under the assumption of the existence of uniformly-distributed measurement errors. The original ABC-SubSim algorithm is improved by using an adaptive method to choose the proposal variance for the component-wise Metropolis algorithm. Illustrative examples include the application of ABC-SubSim to Bayesian model updating of both linear and nonlinear dynamical systems.