Generalized sensitivity indices based on vector projection for multivariate output

Analyzing the sensitivity of model outputs to inputs is important to assess risk and make decisions in engineering application. However, for model with multiple outputs, it is difficult to interpret the sensitivity index since the effect of the dimension and the correlation between multiple outputs are often ignored in the existing methods. In this paper, a new kind of sensitivity analysis method is proposed by use of vector projection and dimension normalization for multiple outputs. Through the dimension normalization, the space of multiple outputs can be unified into a dimensionless one to eliminate the effect of the dimension of the different output. After an affine coordinate system is constructed by considering the correlation of the multiple normalized outputs, a total variance vector for the multiple outputs can be composed by the individual variance of each output. Then, by projecting the variance contribution vector composed by the individual variance contribution of the input to each output on the total variance vector, the new sensitivity indices are proposed for measuring the comprehensive effect of the input on the total variance vector of multiple outputs, it is defined as the ratio of the projection of the variance contribution vector to the norm of the total variance vector. We derive that the Sobol’ indices for a scalar output and the covariance decomposition based indices for multiple outputs are special cases of the proposed vector projection based indices. Then, the mathematical properties and geometric interpretation of the proposed method are discussed. Three numerical examples and a rotating shaft model of an aircraft wing are used to validate the proposed method and show their potential benefits.