Understanding how to explain predictions with "explanation vectors"

In a recent post I introduced three existing approaches to explain individual predictions of any machine learning model. After the posts focused on LIME and Shapley values, now it's the turn of Explanation vectors, a method presented by David Baehrens, Timon Schroeter and Stefan Harmeling in 2010. As we have seen in the mentioned posts, explaining a decision of a black box model implies understanding what input features made the model give its prediction for the observation being explained. Intuitively, a feature has a lot of influence on the model decision if small variations in its value cause large variations of the model's output, while a feature has little influence on the prediction if big changes in that variable barely affect the model's output. Since a model is a scalar function, its gradient points in the direction of the greatest rate of increase of the model's output, so it can be used as a measure of features' influence.