Gaussian Mixture–Based Autoregressive Error Model with a Conditionally Heteroscedastic Hierarchical Framework for Bayesian Updating of Structures

Abstract

The heteroscedastic Bayesian model updating framework assigned different variances to the modal errors using heteroscedastic parameters modeled by gamma distributions. However, the error density shows significant asymmetry, not captured by the assumed Student’s t-distribution. Thereby, the effect of heteroscedasticity is not adequately reflected in the variances of the updated stiffness and the prediction error variances due to hindered error propagation along the Markov chain Monte Carlo (MCMC) chain. This is overcome in the present study by proposing a Gaussian mixture–based autoregressive model in a conditional heteroscedastic framework (which is termed GMARCH). The GMARCH model adjusts the error intermittently at different stages of the MCMC chain and models the unknown error and its variances at any stage with respect to the previous stages. The proposed heteroscedastic error model obtains a direct estimate of the most probable values of the heteroscedastic parameters for the modal observables at different modes. An existing experimental data set derived from a multi-degree-of-freedom spring-mass model is used to illustrate the effectiveness of the model in addition to simulated data from a multistory shear building. The accuracy and computational effectiveness of the proposed approach are compared to those of the existing methods.