Regional importance measure of the basic variable and its sparse grid solution

To improve the stability and convergence of the existing contribution to sample mean (CSM) regional importance measure (RIM), a new RIM is proposed to estimate the contribution to the mean of the model output by the different regions of basic variable, and it is named as improved contribution to sample mean (ICSM). An extended version of the ICSM, which is named as contribution to first order variance (CFOV), is developed to analyze the effect of the different regions of the basic variable on its corresponding first order variance in the variance decomposition. The properties of the two proposed RIMs are analyzed and their relationships with the existing CSM and contribution to sample variance (CSV) RIM are derived. Furthermore, based on the characteristics of the proposed RIMs, their highly efficient sparse grid integration (SGI) solutions are also established. Several numerical and engineering examples show that the newly defined ICSM can act as effectively as the CSM, but the convergence and stability of ICSM is better than those of CSM. The proposed CFOV can provide more detailed information than the existing CSV, which can effectively instruct the engineer on how to achieve a targeted reduction of the main effect of each basic variable. The established SGI-based method can improve the efficiency of the regional importance analysis considerably in case of acceptable accuracy.