Learning non-stationary model of prediction errors with hierarchical Bayesian modeling

The hierarchical Bayesian modeling (HBM) framework has proven its effectiveness in addressing the model updating problem. However, the assumption of Gaussian white noise for prediction errors in HBM overlooks their inherent non-stationary uncertainties, which are prevalent in engineering applications. Ignoring the non-stationaries of prediction errors can lead to significant errors in identifying model parameters, ultimately resulting in biased predictions and reduced reliability of the updated model. To comprehensively estimate the non-stationary uncertainties of prediction errors while simultaneously identifying unknown physical model parameters, a new HBM framework is proposed, wherein the prediction errors are modeled using a non-stationary Gaussian process (GP). In this framework, the hyper parameters consist of two sets: one representing the statistics of the GP model and the other representing the distribution parameters of the physical model parameters. Due to the complexity stemming from the large number of parameters required in the non-stationary GP model, directly inferring the joint posterior distribution of all the hyperparameters is computationally infeasible. To address this issue, a sequential process is designed to infer the marginal distribution of each set of parameters individually. The product of these two marginal distributions is then used as an approximation of the joint distribution. Furthermore, an iterative procedure is proposed to ensure the consistency between the two distributions, ultimately achieving the optimal approximation of the joint posterior distribution. The effectiveness of the proposed framework is validated by identifying the structural parameters and prediction errors of time-history responses in a structural dynamic example using simulated data. It is then successfully applied to identify the fatigue crack growth (FCG) model using experimental data, resulting in improved predictive accuracy, as evidenced by the significantly narrower predicted interval for FCG life compared to the prediction made by the HBM.