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. Comparisons are usually evaluated through a measure of comparison. The used measure is often a distance. But, more and more, a similarity or a dissimilarity measure is chosen. But the choice of an appropriate measure among all avalaible measures in literature is not an easy task. It is linked to the problem of the characterization of relevant properties for the considered task. We have proposed [2] a framework in order to deal with measures of comparison. This framework displays the main families of measures of comparison according to the properties they satisfy. Hence, the existing measures of comparison can be classified. Fyrther, it is known that a classification simplifies a problem. However, the problem of the choice of a measure of comparison within a same family is still present. This paper gives an explicit method to choose a measure of comparison. The proposed solution lies in a new representation of measures of comparison obtained by normalization of arguments. This normalization leads to a desirable property: measures of comparison do not depend on the scale of the system. Another consequence of this normalization is the fact that measures of satisfiability can be described by a unique argument and measures of resemblance by two arguments. The analysis of behaviours of measures of comparison is easy thanks to a geometrical interpretation and to a natural definition of the power of discrimination of a given measure of comparison. 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 we point out and we propose new measures satisfying particular properties. For measures of satisfiability, the parameter controls the severity towards difference between sets whereas the parameter penalizes the differences between two sets for a measure of resemblance.