Learning indistinguishability from data

 In this paper we revisit the idea of interpreting fuzzy sets as representations of vague values. In this context a fuzzy set is induced by a crisp value and the membership degree of an element is understood as the similarity degree between this element and the crisp value that determines the fuzzy set. Similarity is assumed to be a notion of distance. This means that fuzzy sets are induced by crisp values and an appropriate distance function. This distance function can be described in terms of scaling the ordinary distance between real numbers. With this interpretation in mind, the task of designing a fuzzy system corresponds to determining suitable crisp values and appropriate scaling functions for the distance. When we want to generate a fuzzy model from data, the parameters have to be fitted to the data. This leads to an optimisation problem that is very similar to the optimisation task to be solved in objective function based clustering. We borrow ideas from the alternating optimisation schemes applied in fuzzy clustering in order to develop a new technique to determine our set of parameters from data, supporting the interpretability of the fuzzy system.