Importance analysis for models with dependent input variables by sparse grids

To get a better understanding on the output uncertainty contributed by an individual variable as well as the correlated variables of models with dependent inputs, a method for decomposing Sobol's first-order effect indices into uncorrelated variations and correlated variations is investigated. Instead of using Monte Carlo simulation or full tensor product-based numerical integration approaches, a new sparse grid numerical integration method is proposed for estimating Sobol's main effect indices as well as the two decomposed sensitivity measures. Before conducting the sparse grid numerical integration-based algorithm, an orthogonal transformation is used to transform the dependent input variables and model performance function into independent space as the joint probability density function of the correlated variables cannot be written as the product of univariate density functions. An obvious advantage of the sparse grid numerical integration-based method is that it can decrease the computational cost of the conventional methods significantly while keeping the accuracy level controllable, particularly for high-dimensional problems. The proposed approach is compared with other alternative approaches through theoretical and applied numerical experiments to demonstrate its efficiency, accuracy and high-dimensional adaptivity.