Building fuzzy graphs: Features and taxonomy of learning for non-grid-oriented fuzzy rule-based systems

The use of Mamdani-type fuzzy rule-based systems (FRBSs) allows us to deal with the modeling of systems building a linguistic model clearly interpretable by human beings. However, the accuracy obtained is not sometimes as good as desired. This fact relates to the restriction imposed when using linguistic variables, which forces the membership functions considered in each fuzzy linguistic rule to belong to a common set of them, i.e., to use a global grid. To solve this problem, in the last few years a new variant has been proposed working directly with fuzzy variables in the fuzzy rules instead of linguistic terms, thus ignoring the said restriction. Therefore, these systems, which are totally equivalent to fuzzy graphs (defined by Zadeh as granular representations of functional dependencies and relations), do not consider a global grid and could be named nongrid-oriented (NGO) FRBSs. Of course, the main objective of these models is the accuracy of the system instead its interpretability. Until now, NGO FRBSs have been little considered and developed in the literature. However, and due to their good accuracy, their use is increasing thus making necessary a wide analysis on the features and associated learning methods in the NGO domain. This contribution aims at analyzing the structure and framework of NGO FRBSs, as well as making a taxonomy of learning methods considering the constrains imposed on the fuzzy sets in the generation process. Some automatic learning techniques and methods proposed in the literature to build these fuzzy graphs will be also reviewed and analyzed when solving several applications of different nature.