Learning in the framework of fuzzy lattices

Abstract

A basis for rigorous versatile learning is introduced theoretically, that is the framework of fuzzy lattices or FL-framework for short, which proposes a synergetic combination of fuzzy set theory and lattice theory. A fuzzy lattice emanates from a conventional mathematical lattice by fuzzifying the inclusion order relation. Learning in the FL-framework can be effected by handling families of intervals, where an interval is treated as a single entity/block the way explained here. Illustrations are provided in a lattice defined on the unit-hypercube where a lattice interval corresponds to a conventional hyperbox. A specific scheme for learning by clustering is presented, namely /spl sigma/-fuzzy lattice learning scheme or /spl sigma/-FLL (scheme) for short, inspired from adaptive resonance theory (ART). Learning by the /spl sigma/-FLL is driven by an inclusion measure /spl sigma/ of the corresponding Cartesian product to be introduced here. We delineate a comparison of the /spl sigma/-FLL scheme with various neural-fuzzy and other models. Applications are shown to one medical data set and two benchmark data sets, where /spl sigma/-FLL's capacity for treating efficiently real numbers as well as lattice-ordered symbols separately or jointly is demonstrated. Due to its efficiency and wide scope of applicability the /spl sigma/-FLL scheme emerges as a promising learning scheme.