Measuring the interactions among variables of functions over the unit hypercube

TL;DR

This paper introduces a new interaction index for functions over the unit hypercube, extending concepts from cooperative game theory and multilinear regression, with properties and interpretations that highlight variable interactions.

Contribution

It defines a novel interaction index based on least squares approximation, generalizing the Banzhaf index and linking it to derivatives and difference quotients of functions.

Findings

The index generalizes the Banzhaf interaction index.

It can be interpreted as an expected value of derivatives or difference quotients.

Applications demonstrate the usefulness of the interaction index.

Abstract

By considering a least squares approximation of a given square integrable function $f\colon[0,1]^n\to\R$ 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.