Bayesian active learning approach for estimation of empirical copula-based moment-independent sensitivity indices

The moment-independent global sensitivity method is an important branch among the prosperous developments of global sensitivity analysis. It can quantify the influence of input variables on the uncertainty of model output by taking the entire distribution ranges into account. However, the fast and accurate estimation still remains a challenging task in engineering practices. This article aims at developing a robust and efficient sensitivity analysis approach by leveraging the superiority of Bayesian active learning technology. An algorithm called active learning of cumulative distribution function (AL-CDF) is proposed to efficiently derive an accurate CDF of model output with a small group of training data. In AL-CDF algorithm, a modified U-learning function is defined to determine the best point to guide the learning process of CDF. Moreover, an innovative stopping criterion is specially designed based on functional samples of posterior Gaussian process, aided by an advanced Gaussian process generator. Once the AL-CDF is completed, the Bayesian inference of moment-independent indices by empirical-Copula method can be directly applied in a pure statistic manner, with no more evaluations of the complex performance function. From this perspective, the main computational cost is consumed in the AL-CDF procedure. In addition, benefiting from the sampling strategy from posterior GPR model, the posterior variations of moment-independent sensitivity indices can be derived as by-products. Finally, the effectiveness of the proposed work is demonstrated by a nonlinear numerical example, a wing flutter model as well as the NASA Langley multidisciplinary challenge.