Active subspace methods and derivative-based Shapley effects for functions with non-independent variables

Lower-dimensional subspaces that impact estimates of uncertainty are often described by Linear combinations of input variables, leading to active variables. This paper extends the derivative-based active subspace methods and derivative-based Shapley effects to cope with functions with non-independent variables, and it introduces sensitivity-based active subspaces. While derivative-based subspace methods focus on directions along which the function exhibits significant variation, sensitivity-based subspace methods seek a reduced set of active variables that enables a reduction in the function's variance. We propose both theoretical results using the recent development of gradients of functions with non-independent variables and practical settings by making use of optimal computations of gradients, which admit dimension-free upper-bounds of the biases and the parametric rate of convergence. Simulations show that the relative performance of derivative-based and sensitivity-based active subspaces methods varies across different functions.