Macsum: A new interval-valued linear operator

Abstract

Many applications in fields as diverse as chemistry, mechanics, medicine, economics, robotics, environment, ecology, meteorology, etc. are based on the notion of system modeling. A system is a real process associating in a deterministic way an output value to one or more input values. A model is a mathematical object that allows the analysis of real phenomena and the prediction of results at a given level of approximation. One of the difficulties of modeling is the choice of the model and how to measure, predict or control the level of approximation. The linear model, where the output is obtained by a weighted sum of the inputs, is a simple model, based on a reduced number of parameters, but describing the functioning of a system in a very approximate way, without the level of approximation being known. Non-linear models are much more specific but much more difficult to use, the level of approximation being even more difficult to measure. What we propose in this article is an imprecise linear model, so the simplicity of representation and use is quite comparable to that of a linear model. This model is imprecise in the sense that the output is imprecise, although the inputs are precise, thus potentially reflecting the inadequacy of a linear model to represent the behavior of the system: the more imprecise the output, the less likely a single linear model is to correctly describe the system. This imprecise linear model can be seen as a convex set of conventional linear models, the imprecise output of this model being the convex set of outputs that would have been obtained by each linear model individually. This modeling is based on non-monotonic real-valued concave set measures.