  Following up on the success of the analysis of variance (ANOVA) decomposition
and the Sobol indices (SI) for global sensitivity analysis, various related
quantities of interest have been defined in the literature including the
effective and mean dimensions, the dimension distribution, and the Shapley
values. Such metrics combine up to exponential numbers of SI in different ways
and can be of great aid in uncertainty quantification and model interpretation
tasks, but are computationally challenging. We focus on surrogate based
sensitivity analysis for independently distributed variables, namely via the
tensor train (TT) decomposition. This format permits flexible and scalable
surrogate modeling and can efficiently extract all SI at once in a compressed
TT representation of their own. Based on this, we contribute a range of novel
algorithms that compute more advanced sensitivity metrics by selecting and
aggregating certain subsets of SI in the tensor compressed domain. Drawing on
an interpretation of the TT model in terms of deterministic finite automata, we
are able to construct explicit auxiliary TT tensors that encode exactly all
necessary index selection masks. Having both the SI and the masks in the TT
format allows efficient computation of all aforementioned metrics, as we
demonstrate in a number of example models.
