Efficient Evaluation of Sobol’ Indices Utilizing Samples from an Auxiliary Probability Density Function

Abstract

A sample-based evaluation of Sobol’ sensitivity indices is discussed in this paper relying on Kernel Density Estimation (KDE) for achieving computational efficiency. The foundation of the approach is the definition of an auxiliary probability density function (PDF) for the vector of model parameters (i.e., random variables representing system input). The sensitivity index for each input can be then expressed through the marginal density related to this auxiliary joint PDF. The efficient estimation of the indices for all model parameters is ultimately facilitated by simulating first a single sample-set from the joint PDF and then utilizing this information to approximate all marginal distributions of interest through KDE. The same sample set is used for all approximations whereas this set is further exploited to improve the accuracy of the estimation of the integrals defining the sensitivity indices. An extension to facilitate calculation of higher order indices and total sensitivity indices is also examined. Once the sample set is generated, the computational burden of the approach is small and practically independent of the dimension of the model parameters. This makes the proposed approach particularly attractive for applications for which such samples are readily available (or can be efficiently obtained), as it can provide sensitivity information with small overall computational cost.