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Mathematical and computational models with correlated multivariate output are commonly used for risk assessment and decision support in engineering. Traditional methods for sensitivity analysis of the model with scalar output fail to provide satisfactory results for this multivariate case. In this work, we introduce a new sensitivity index which looks at the influence of input uncertainty on the entire distribution of the multivariate output without reference to a specific moment of the output. The definition of the new index is based on the multivariate probability integral transformation (PIT), which can take into account both of the uncertainties and the correlations among multivariate output. The mathematical properties of the proposed sensitivity index are discussed and its differences with the sensitivity indices previously introduced in the literature are highlighted. Two numerical examples and a rotating shaft model of an aircraft wing are employed to illustrate the validity and potential benefits of the new sensitivity index.

The target in this work is to validate the component functions of model output between physical observation and computational model with the area metric. Based on the theory of high dimensional model representations (HDMR) of independent input variables, conditional expectations are component functions of model output, and the conditional expectations reflect partial information of model output. Therefore, the model validation of conditional expectations tells the discrepancy between the partial information of the computational model output and that of the observations. Then a calibration of the conditional expectations is carried out to reduce the value of model validation metric. After that, a recalculation of the model validation metric of model output is taken with the calibrated model parameters, and the result shows that a reduction of the discrepancy in the conditional expectations can help decrease the difference in model output. At last, several examples are employed to demonstrate the rationality and necessity of the methodology in case of both single validation site and multiple validation sites.