The new importance measures based on vector projection for multivariate output: application on hydrological model

Abstract. Analyzing the effects of the inputs on the correlated multivariate output is important to assess risk and make decisions in Hydrological processes. However, the existing methods, such as output decomposition approach and covariance decomposition approach, cannot provide sufficient information of the effects of the inputs on the multivariate output, since these methods only measure the influence of input variables on the magnitudes of variances of the dimensionalities in the multiple output space and ignore the effects on the dimensionality directions of output variances. In this paper, a new kind of sensitivity indices based on vector projection for the multivariate output is proposed. By the projection of the conditional vectors on the unconditional vector in the dimensionless multiple output space, the new sensitivity indices measure the influence of the input variables on the magnitudes of variances and directions of the dimensionalities simultaneously. The mathematical properties of the proposed index are discussed, and its link with the Sobol indices is derived. And Polynomial Chaos Expansion (PCE) is used to estimate the proposed sensitivity indices. The results for two numerical examples and a hydrological model indicate the validity and potential benefits of the vector projection index and the efficiency of estimation approach.