Bayesian approach to order selection of arx models with applications to structural health monitoring

Abstract eng:
A Bayesian framework for model class selection of ARX models is developed and applied to actual earthquake response data obtained by the structural health monitoring system of a high-rise building. System identification using ARX models is a powerful method for estimating the modal parameters of structures. In order to conduct system identification using ARX models, however, the model order has to be selected first, which is always a problem. Although one can utilize Akaike’s Information Criterion (AIC) for the model selection, it does not always give an appropriate model order in practice. Therefore a fully probabilistic framework using Bayes’ theorem for model order selection of ARX models is developed, where the posterior probabilities of alternative model classes that have different ARX orders are compared. The evidence, which determines the posterior probability of each model class, is evaluated by using Laplace’s asymptotic approximation. The derivation of an analytical form of the Hessian matrix in the evidence contributes to a significant reduction of computing time. The model selection method is applied to seismic response data from a high-rise building. The model order of ARX models is selected appropriately by the Bayesian framework by maximizing the evidence, which has a distinct peak at the optimal model order, even in the case where the commonly-used AIC does not work. In the overall analysis for the building response records during 43 earthquakes over nine years, the modal parameters up to the 4th mode in each horizontal direction are estimated properly in all cases, showing that the model selection method developed here is very effective and robust. According to the results, the estimates of natural frequency depend significantly on the response amplitude, while the estimates of damping ratio and participation factor do not seem to have an apparent correlation with the response amplitude. The natural frequency is then compensated by an empirical correction so that the influence of the response amplitude is removed. The compensated natural frequency is much more stable along the timeline, indicating that the building has no significant change in its global dynamic characteristics during the nine-year period.

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