Modelling data by the Choquet integral

It is well known that the main tool for finding dependencies among data is the linear regression model, which expresses one or several variables in term of a linear combination of the others. Linear regression is based on the least square principle, and has been studied at length, its statistical performance are well known. In real situations however, the linear assumption happens to be often rather far from the reality, and models with low accuracy are produced. For this reason, more general models that offer more flexibility are looked for, the price to pay being that they are often much more complex to use. In this paper, we are interested in using the Choquet integral [4] as a general non linear regression model. The Choquet integral is a generalization of the Lebesgue integral, defined with respect to a non classical measure, often called fuzzy measure, or non-additive measure or also capacity. When the underlying universe is finite, the Lebesgue integral reduces to a (convex) linear combination, hence can be assimilated to a particular class of regression models, where the coefficients are all positive and sum up to one. Hence, the Choquet integral offers a more general model, more precisely, as it will be seen below, it offers a set of (convex) linear models, each of them being defined in a simplex. The Choquet integral has been successfully applied many times in classification [37,19,24], decision making under uncertainty [31,2,32], multicriteria decision making [8,35,25] and also data modelling [36]. The main difficulty is to determine efficiently the 2− 2 coefficients of the model. This exponential complexity limits the applicability, although solutions to reduce this number to a polynomial size exist. In this paper, we make a quick overview of the main works along this line, and describe recent works we performed. First sections are devoted to introduce the necessary material. In all the paper we consider a single regression problem, i.e. we want to model one variable y by some other variables x1, . . . , xn.