An efficient computational method of a moment-independent importance measure using quantile regression

The moment-independent uncertainty importance measure technique for exploring how uncertainty allocates from the output to the inputs has been widely used to help engineers estimate the degree of confidence of decision results and assess risks. The moment-independent importance measure (also called delta index) can better reflect the effect of the input on the whole distribution of the output instead of any specific moment. However, because the conditional probability density function (PDF) of the output is difficult to obtain, the computation process of delta index becomes quite complex. Therefore, an efficient computational algorithm by using the quantile regression is developed to estimate the delta index in this paper. Firstly, the non-linear quantile regression is employed to approximate the relationships between each input and the conditional quantiles of the output where only a set of input-output samples is needed. Secondly, at a certain value of the input, the conditional quantile points can be computed according to the obtained quantile regression models, which can be considered as the samples of the conditional output. Thirdly, the unconditional and conditional PDF of the output are evaluated by using the univariate kernel density estimation according to the original output samples and these quantile points respectively. Finally, the delta index is computed by estimating the area difference between the unconditional PDF and conditional PDF of the output. The number of model evaluations of this proposed method is dramatically decreased and is free of the dimensionality of the model inputs. Test examples show the performance of the proposed method and its usefulness in practice.