  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.
