  Variance-based global sensitivity analysis, in particular Sobol' analysis, is
widely used for determining the importance of input variables to a
computational model. Sobol' indices can be computed cheaply based on spectral
methods like polynomial chaos expansions (PCE). Another choice are the recently
developed Poincar\'e chaos expansions (PoinCE), whose orthonormal
tensor-product basis is generated from the eigenfunctions of one-dimensional
Poincar\'e differential operators. In this paper, we show that the Poincar\'e
basis is the unique orthonormal basis with the property that partial
derivatives of the basis form again an orthogonal basis with respect to the
same measure as the original basis. This special property makes PoinCE ideally
suited for incorporating derivative information into the surrogate modelling
process. Assuming that partial derivative evaluations of the computational
model are available, we compute spectral expansions in terms of Poincar\'e
basis functions or basis partial derivatives, respectively, by sparse
regression. We show on two numerical examples that the derivative-based
expansions provide accurate estimates for Sobol' indices, even outperforming
PCE in terms of bias and variance. In addition, we derive an analytical
expression based on the PoinCE coefficients for a second popular sensitivity
index, the derivative-based sensitivity measure (DGSM), and explore its
performance as upper bound to the corresponding total Sobol' indices.
