A simple fuzzy rule-based system through vector membership and kernel-based granulation

Abstract

It is widely recognized that the human reasoning can be approximated by fuzzy rule-based (FRB) systems which can be seen as one of the basic frameworks for representation of intelligent systems. During the last quarter of a century two particular types of FRB systems, namely Zadeh-Mamdani (ZM) and Takagi-Sugeno (TS) dominated the field. In this paper we propose an alternative type which is simpler and more intuitive while preserving the advantages of its predecessors, such as flexibility, modularity, human-intelligibility. The newly proposed concept of vector membership (VM) and kernel-based granulation (KG) of complex systems (respectively their mathematical descriptions) we see as the next, more efficient form of system modelling that is widely applicable to a plethora of applications ranging from time-series prediction, clustering, classification, control, decision support systems to other problems where conventional fuzzy rule-based systems are used. The proposed simple FRB based on VM and KG are non-parametric and fully represent the real data. Contrast this to the mere approximation of the real data distributions that is provided by Gaussian (scalar), triangular, trapezoidal etc. parametric types of membership functions that are used in currently existing types of FRB (ZM and TS). Note that even probabilistic models that are usually based on Gaussian distributions or a mixture of Gaussians or other parametric representations provide only an approximation of the real data distribution (it should be noted that particle filters are perhaps the only form of non-parametric representation that is similar in this sense to the newly proposed simple FRB with VM and KG, but they are computationally cumbersome with exponentially growing complexity). The main contribution of the proposed simple FRB with VM and KG is that while preserving all the advantages of `traditional' FRB systems they avoid the well known problems related to (multiple scalar) membership functions definition, identification and update. They fully take into account and exactly represent the spatial distribution and similarity of all the real data by proposing an innovative and much simplified form of the antecedent part. At the same time, transformations to the “traditional” (ZM and TS) fuzzy sets expressed by parametric membership functions per variable are also possible. In papers that will follow we will demonstrate on practical examples (including classification, prediction, decision support and other classes of problems) the benefits of this scheme.