Multivariate sensitivity analysis based on the direction of eigen space through principal component analysis

Abstract

In this paper, a new kind of sensitivity indices based on the principal component analysis (PCA) is proposed to measure the effects of input variables on multivariate outputs. Through PCA, the outputs are projected onto a new coordinate system (eigen space), which is constructed by the eigenvector (principal components). The existent sensitivity indices based on PCA focus on the variance of principal components, which can be considered as a magnitude of the uncertainty in the corresponding coordinate axes. In addition, the direction of the coordinate axes in the eigen space also contains another part of uncertainty of outputs (the direction of the uncertainty). The new sensitivity indices measure the effect of input variables on the direction of the coordinates axes through the angles between the unconditional and conditional eigenvectors. Thus, the new sensitivity indices can reflect different effect of input variables on the output compared to the existent sensitivity indices. The results of three numerical examples and an environmental model show the difference between the new sensitivity indices and the existent sensitivity indices. Since the new sensitivity indices measure the effects of input variables on the multivariate outputs from a different perspective compared to the existent sensitivity indices, they should be mutually complementary to each other.