Importance analysis for model with mixed uncertainties

In engineering, sparse or imprecise data often leads to epistemic uncertainty about the distribution parameters of the aleatory input variables. Separating and estimating the individual contributions of the aleatory variability and epistemic parameter uncertainties to the uncertainty in the model output can assist in resource allocation for data collection, as natural variability is irreducible whereas parameter uncertainty is reducible. For structural systems involving aleatory inputs with epistemic distribution parameters, a new kind of importance measure is proposed to distinguish and quantify the individual contributions of these two kinds of uncertainties, in which probability distribution is used to describe the aleatory uncertainty, and fuzzy membership function is employed to represent the epistemic distribution parameter. The mathematical properties of the proposed importance measures are discussed and proved. The defined importance measures are easy to apprehend, and can evaluate the contributions of the aleatory variability and epistemic parameter uncertainties even when the information of the epistemic parameters is very sparse. Thus, they can measure the importance of the two kinds of uncertainties more reasonably. For efficiently estimating the proposed importance measures, a monotonicity analysis is conducted for model function and the probability transformation process. The results indicate that the extreme values of model function at each membership level can generally be obtained by combining the bounds of membership intervals of the fuzzy distribution parameters. Based on the monotonicity analysis, an efficient algorithm is then formulated to compute the proposed importance measures. Several examples demonstrate the rationality and effectiveness of the proposed importance measures and the efficiency of the presented algorithm.