Loia Vincenzo - Fuzzy Logic and Soft Computing Applications: 11th International Workshop, WILF 2016, Naples, Italy, December 19-21, 2016, Revised Selected Papers

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Springer International Publishing AG 2017

Alfredo Petrosino , Vincenzo Loia and Witold Pedrycz (eds.) Fuzzy Logic and Soft Computing Applications Lecture Notes in Computer Science 10147 10.1007/978-3-319-52962-2_1

Towards a Framework for Singleton General Forms of Interval Type-2 Fuzzy Systems

Gonzalo Ruiz-Garca 1, Hani Hagras 2 , Ignacio Rojas 1 and Hector Pomares 1

(1)

Department of Computer Architecture and Computer Technologies, Universidad de Granada, Granada, Spain

(2)

The Computational Intelligence Centre, School of Computer Science and Electronic Engineering, University of Essex, Colchester, UK

Hani Hagras

Email:

Abstract

Recently, it has been shown that interval type-2 fuzzy sets (IT2FSs) are more general than interval-valued fuzzy sets (IVFSs), and some of these IT2FSs can actually be non-convex. Although these IT2FSs could be considered within the general type-2 fuzzy sets (GT2FSs) scope, this latter have always been studied and developed under certain conditions considering the convexity and normality of their secondary grades. In recent works the operations of intersection and union for GT2FSs have been extended to include non-convex secondary grades. Hence, there is a need to develop the theory for those general forms of interval type-2 fuzzy logic systems (gfIT2FLSs) which use IT2FSs that are not equivalent to IVFSs and can have non-convex secondary grades. In this chapter, we will present the mathematical tools to define the inference engine for singleton gfIT2FLSs. This work aims to introduce the basic structure of such singleton gfIT2FLSs, paying special attention to those blocks presenting significant differences with the already well known type-2 FLSs which employ IT2FSs which are equivalent to IVFSs (we will term IVFLSs).

Introduction

It is a well-known fact that both Type-1 (T1) and Type-2 (T2) fuzzy logic systems (FLSs) have received significant attention from the research community, and both have been successfully used in many real world applications, such as robotics [] and many others.

Type-2 fuzzy logic and systems are usually divided in the literature between Interval Type-2 (IT2) and General Type-2 (GT2) fuzzy logic and systems. Some authors argue that the term General Type-2 Fuzzy Logic should not be used, as there is no formal definition for it, so Type-2 Fuzzy Logic should be the proper name to use; nonetheless, the term is widely used to make explicit difference with the IT2 case, and we will do so in this Chapter. Although GT2FLSs were defined as soon as 1999, their practical application has been limited due to their higher computational complexity, favouring the simpler version of IT2FLSs. More recently, some authors have proposed some approximations of GT2FLSs based on several IT2FLSs working in parallel ( -planes representation in []); however, these systems require the secondary grades to be convex type-1 fuzzy sets.

Traditionally IT2FSs have been considered to be equivalent to IVFSs [] that IT2FSs are more general than IVFSs. In addition, some of these IT2FSs have secondary grades which are non-convex T1FSs. Hence, although there is a big literature in both IT2 and GT2FLSs, most of the existing work focuses either on IVFSs or GT2FSs with convex and normal secondary grades. In this Chapter, we consider those IT2FSs whose secondary grades can be non-convex, and thus are not equivalent to IVFSs. Those sets will be referred to in this text as general forms of interval type - 2 fuzzy sets (gfIT2FSs).

These new gfIT2FSs can easily capture the faced uncertainty without introducing unneeded and unrealistic uncertainty to the IT2FS. For instance, a method frequently used in the literature to obtain IVFSs [(b).

Fig. 1.

(a) Type-1 MFs obtained from different people in a survey. (b) IVFS obtained aggregating the T1 MFs.

However, if the different T1 MFs provided are sparse (which might happen frequently, as shown in Fig. (b) might include huge number of emerging non triangular embedded sets which might not represent the surveyed population opinion.

This problem can be naturally solved by using a specific class of the gfIT2FSs, which is called multi - singleton IT2FSs , whose secondary membership is comprised of several singletons at each point within the X-domain (as depicted in Fig. ). These sets can easily represent all sets gathered in the survey to model the faced uncertainty without adding extra unneeded and/or unrealistic uncertainty to the final type-2 fuzzy set.

Fig. 2.

A multi-singleton IT2FS to model the uncertainty amongst the type-1 fuzzy sets in Fig. (a).

In [] the theory for the join and meet operations on GT2FSs with non-convex/arbitrary secondary grades was presented; in addition, special attention was drawn to the case of non-convex IT2FSs which are gfIT2FSs. Once these set theoretic operations are available, the fuzzy inference engine for the gfIT2FLSs can be defined. In this Chapter we aim to present the structure of a singleton gfIT2FLSs. We will specially focus on those parts having significant differences, as many of the elements are analogous to those in other kinds of FLS (i.e. IVFLSs and GT2FLSs).

This chapter is organised as follows: Sect. presents applications and future work.

Theoretical Background on General Forms of IT2FSs (gfIT2FSs)

In this section, we will review the theoretical background of the gfIT2FSs. In order to do so, we will revisit the initial notion of T2FSs, their initial formal definitions and, finally, their generalisation.

Based on Zadehs initial notion of a type-2 fuzzy set in []:

Definition 1:

A type-2 fuzzy set in a non-empty universe of discourse X is given by:

(1)

Definition 2:

Using Definition ] IT2FSs are defined as follows:

If all then is an IT2FSs.

It is important to note that, from Definition ), which is summarised subsequently:

Fig. 3.

Examples of all types of IT2FSs. (a) T1FS. (b) Multi-singleton IT2FS. (c) IVFS. (d) Multi-interval IT2FS. (e) gfIT2FS containing both singletons and intervals [].

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