Some geometrical properties of trapezoidal functions for their association with linguistic labels leading to a reduced 2-D representation

Abstract

Linguistic labels are often associated with trapezoidal membership functions in the framework of fuzzy set theory. Those functions are determined by the abcissae of the vertex of the trapeze, and therefore belong to a 4 dimensions space. A particular attention is here devoted to the coordinates of the gravity center of the trapeze: the abcissa is connected to the semantic position of the label, and the vertical coordinate is related to the shape. In this paper is described a new approach that, by imposing a shape factor reduces theoretically to 3 the dimension of the parametric space of representation, and it is proved that under slight additionnal constraints a linguistic label can be well represented in a 2-dimension space by its total area (mass) and the abcissa of its gravity center. This dimensionnal reduction makes more easy to handle and to visualize linguistic informations. Further research is being done to partition the plane of all possible linguistic labels according to a given ordered set of qualitative descriptors.