Shapley Effects for Global Sensitivity Analysis: Theory and Computation

Variance-based global sensitivity analysis decomposes the variance of the output of a computer model, resulting from uncertainty about the model's inputs, into variance components associated with each input's contribution. The two most common variance-based sensitivity measures, the first-order effects and the total effects, may fail to sum to the total variance. They are often used together in sensitivity analysis, because neither of them adequately deals with interactions in the way the inputs affect the output. Therefore Owen proposed an alternative sensitivity measure, based on the concept of the Shapley value in game theory, and showed it always sums to the correct total variance if inputs are independent. We analyze Owen's measure, which we call the Shapley effect, in the case of dependent inputs. We show empirically how the first-order and total effects, even when used together, may fail to appropriately measure how sensitive the output is to uncertainty in the inputs when there is probabilistic dependence or structural interaction among the inputs. Because they involve all subsets of the inputs, Shapley effects could be expensive to compute if the number of inputs is large. We propose a Monte Carlo algorithm that makes accurate approximation of Shapley effects computationally affordable, and we discuss efficient allocation of the computation budget in this algorithm.