Processing fuzzy set membership functionals as vectors

Abstract

A framework is presented for processing fuzzy sets for which the universe of discourse X = {x} is a separable Hilbert Space, which, in particular, may be a Euclidian Space. In a given application, X would constitute a feature space. The membership functions of sets in such X are then "membership functionals", that is, mappings from a vector space to the real line. This paper considers the class ? of fuzzy sets A, the membership functionals μ A of which belong to a Reproducing Kernel Hilbert Space (RKHS) F(X) of bounded analytic functionals on X, and satisfy the constraint $$0\,\leqslant \mu (x) \leqslant 1,x \in X$$ . These functionals can be expanded in abstract power series in x, commonly known as Volterra functional series in x. Because of the one-to-one relationship between the fuzzy sets A and their respective μ A , one can process the sets A as objects using their μ A as intermediaries. Thus the structure of the uncertainty present in the fuzzy sets can be processed in a vector space without descending to the level of processing of vectors in the feature space as usually done in the literature in the field. Also, the framework allows one to integrate human and machine judgments in the definition of fuzzy sets; and to use concepts analogous to probabilistic concepts in assigning membership values to combinations of fuzzy sets. Some analytical and interpretive consequences of this approach are presented and discussed. A result of particular interest is the best approximation $$\hat{\mu}_{A}$$ of a membership functional μ A in F(X) based on interpolation $$\hat {\mu}_{A} (v^{i}) = u^{i}$$ on a training set {(v i , u i ),i = 1, . . . , q} and under the positivity constraint. The optimal analytical solution comes out in the form of an Optimal Interpolative Neural Network (OINN) proposed by the author in 1990 for best approximation of pattern classification systems in a F(X) space setting. An example is briefly described of an application of this approach to the diagnosis of Alzheimer's disease.