Equivalent Representations of Set Functions
Mathematics of Operations Research
Discrete Applied Mathematics - Special issue on Boolean functions and related problems
How to improve ACTS: an alternative representation of the importance of criteria in MCDM
International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems - Special issue on aggregation operators
Axiomatic characterizations of generalized values
Discrete Applied Mathematics
Approximations of Lovász extensions and their induced interaction index
Discrete Applied Mathematics
Collective coin flipping, robust voting schemes and minima of Banzhaf values
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
The influence of variables on Boolean functions
SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
Aggregation Functions (Encyclopedia of Mathematics and its Applications)
Aggregation Functions (Encyclopedia of Mathematics and its Applications)
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By considering a least squares approximation of a given square integrable function f : [0, 1]n → R by a multilinear polynomial of a specified degree, we define an index which measures the overall interaction among variables of f. This definition extends the concept of Banzhaf interaction index introduced in cooperative game theory. Our approach is partly inspired from multilinear regression analysis, where interactions among the independent variables are taken into consideration. We show that this interaction index has appealing properties which naturally generalize the properties of the Banzhaf interaction index. In particular, we interpret this index as an expected value of the difference quotients of f or, under certain natural conditions on f, as an expected value of the derivatives of f. These interpretations show a strong analogy between the introduced interaction index and the overall importance index defined by Grabisch and Labreuche [7]. Finally, we discuss a few applications of the interaction index.