Measuring the Interactions among Variables of Functions over the Unit hypercube

English

Marichal, Jean-Luc[University of Luxembourg > FSTC > Mathematics Research Unit > >]

Mathonet, Pierre[University of Luxembourg > FSTC > Mathematics Research Unit > >]

2010

Modeling Decisions for Artificial Intelligence 7th International Conference, MDAI 2010

Springer

Lecture notes in artificial intelligence 6408

19-30

Yes

No

International

0302-9743

Berlin

Germany

Modeling Decisions for Artificial Intelligence, 7th International Conference, MDAI 2010

October 27-29, 2010

Perpignan

France

[en] Interaction index ; multilinear polynomial ; least squares approximation ; difference operator ; aggregation function ; cooperative game

[en] By considering a least squares approximation of a given square integrable function f:[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. Finally, we discuss a few applications of the interaction index.