The choice of a measure of comparison in granular learning methods

The comparison of two objects is a usual task for many classification and learning methods. We focus on two families of measures of similitude: measures of satisfiability and measures of resemblance. For each family, we discuss the role of the new parameter that we point out, and we propose new measures satisfying particular properties. For measures of satisfiability, the parameter φ controls the severity towards the difference between sets whereas the parameter ρ penalizes the differences between two sets for a measure of resemblance. This paper should enable a better analysis of the behaviour of a similitude measure, thanks its discrimination power. This analysis is very useful in learning methods in general, because a lot of them are based on comparisons of attribute values of objects in the learning database. Furthermore, the classification step is also often based on comparisons. Each time a method needs a similitude measure, the questions are: when does the measure have to be severe, and when does it have to be tolerant? The answers are given by the discrimination power described in this paper. We have given an example of a measure of satisfiability, the Fermi-Dirac function, with a parameter that controls the severity towards the difference with respect to the reference