Fuzzy set covering of a set of ordinal attributes without parameter sharing

Ordinal attributes with values such as “poor”, “very poor”, “good”, and “very good” are neither entirely numerical nor entirely qualitative. This leads to difficulties in clustering. The ordinal attributes have the numerical property of order but yet it is meaningless to take differences of values of these attributes. However, underlying the ordinal values may be values on an interval scale for which it is meaningful to take differences. The ordinal values may then be interpreted as linguistic values of linguistic variables that correspond to fuzzy sets.This paper discusses a method to represent ordinal values by fuzzy sets on a fixed interval of real values. The method consists of determining centroids from a frequency distribution on the ordinal values and then finding triangular and trapezoidal fuzzy sets that have these centroids using the method of gradient descent. The fuzzy sets that are obtained do not share parameters as would be the case if the end point of one fuzzy set was the vertex of an adjacent fuzzy set. Simulations show that the method is quite effective.