Macsum aggregation learning

Abstract

Identification of a multi-input/single-output system enables the creation of a mathematical model that as accurately as possible represents the input-output relation induced by a physical system. When there is little information about the physical laws related to the system or when the system is too complex, methods such as parametric identification are used to define the system model. In that case, preliminary assumptions can be made about the system leading to a parametric aggregation function based model. This parametric model may be learned by estimating the parameters of this aggregation function from a representative set of inputs/outputs. A major difficulty is to design a model that is relatively simple yet precise enough to meet the user's needs. Linear models are commonly used because they meet these two contradictory constraints. However, use of a linear model is often at the expense of the accuracy of input-output relationship description. In a recent paper, a new modeling approach was proposed, under the name of macsum modeling [31], which aims at replacing the linear model concept by a set of linear models. The obtained aggregation function leads to an interval-valued output. This represents the lack of accuracy of the model to predict the system output when the inputs are known. An interesting feature of this model is that it is ruled by a single precise parametric vector whereas the output is imprecise. Moreover, the vector dimension is equal to that of the input space. In this paper, we propose a method to learn such a model so that it best reflects the input-output relationship of the considered system. This method is based, as in the case of linear modeling, on a regression method. This is particularly new approach because the macsum aggregation is based on a Choquet integral and very few authors have proposed learning of an aggregation function based on the Choquet integral.