Empirical Fuzzy Sets

Abstract

In this paper, we introduce a new form of describing fuzzy sets (FSs) and a new form of fuzzy rule-based (FRB) systems, namely, empirical fuzzy sets (eFSs) and empirical fuzzy rule-based (eFRB) systems. Traditionally, the membership functions (MFs), which are the key mathematical representation of FSs, are designed subjectively or extracted from the data by clustering projections or defined subjectively. eFSs, on the contrary, are described by the empirically derived membership functions (eMFs). The new proposal made in this paper is based on the recently introduced Empirical Data Analytics (EDA) computational framework and is closely linked with the density of the data. This allows to keep and improve the link between the objective data and the subjective labels, linguistic terms, and classes definition. Furthermore, eFSs can deal with heterogeneous data combining categorical with continuous and/or discrete data in a natural way. eFRB systems can be extracted from data including data streams and can have dynamically evolving structure. However, they can also be used as a tool to represent expert knowledge. The main difference from the traditional FSs and FRB systems is that the expert does not need to define the MF per variable; instead, possibly multimodal, densities will be extracted automatically from the data and used as eMFs in a vector form for all numerical variables. This is done in a seamless way whereby the human involvement is only required to label the classes and linguistic terms. Moreover, even this intervention is optional. Thus, the proposed new approach to define and design the FSs and FRB systems puts the human “in the driving seat.” Instead of asking experts to define features and MFs correspondingly, to parameterize them, to define algorithm parameters, to choose types of MFs, or to label each individual item, it only requires (optionally) to select prototypes from data and (again, optionally) to label them. Numerical examples as well as a naive empirical fuzzy (eF) classifier are presented with an illustrative purpose. Due to the very fundamental nature of the proposal, it can have a very wide area of applications resulting in a series of new algorithms such as eF classifiers, eF predictors, eF controllers, and so on. This is left for the future research.