Computational Intelligence Using Type-2 Fuzzy Logic Framework

1. Introduction

The concept was first proposed by (Zadeh, 1975). Subsequently Mizumoto and Tanaka (1976) developed and discussed some properties of type-2 fuzzy sets but no work was done to further develop it into a useful and practical tool. Karnik et al. (1999) introduced the concept of “Footprint of Uncertainty” and the upper and lower membership functions to describe type-2 fuzzy sets. Using interval type-2 fuzzy sets they developed the singleton and non-singleton type-2 fuzzy architectures for practical applications in engineering. A simple and straightforward treatment of s was given by Mendel and John (2002). The three dimensional nature of type-2 fuzzy sets suggest that uncertainties could be better accommodated compared to the two dimensional type-1 fuzzy sets.

The overall work plan of the proposed chapter is as follows:

• A.

  An introduction to standard and interval (type-2) fuzzy sets.

• B.

  Basic operations of type-2 fuzzy sets.

• C.

  Introduction to type-2 fuzzy system.

• D.

  Introduction to various hybrid type-2 FL system:

• E.

  A case study on the applicability of Students’ Performance in Oral Presentation Using Interval Type-2 Fuzzy Approach.

• F.

  Future directions for type-2 FL system research and applications.

The objectives of this chapter is to:

• •

  Review the literature;

• •

  Perspective on the issues and problems related to computational intelligence using interval type-2 fuzzy logic based on the application to Students’ Performance in Oral Presentation research; and,

• •

  Identify future research directions.

This chapter is organized as follows. Section 2 introduces detail definitions and discussions of type-2 fuzzy sets and incorporate views of others. In section 3, concepts of type-2 fuzzy system are introduced. A brief introduction and logic of various hybrid type-2 fuzzy systems is stated in section 4. A case study on the applicability of Students’ Performance in Oral Presentation using Interval Type-2 Fuzzy Approach will be framed in Section 5. Section 6 puts emphasis on the issues of future directions for type-2 FL system. The conclusions to this chapter are given in Section 7.

2. Background

A type-2 fuzzy set is characterized by a fuzzy membership function, i.e., the membership grade for each element of this set is a fuzzy set in [0, 1], unlike a type-1 set where the membership grade is a crisp number in [0, 1]. Such sets can be used in situations where there is uncertainty about the membership grades themselves, e.g., an uncertainty in the shape of the membership function or in some of its parameters. Consider the transition from ordinary sets to fuzzy sets. When we cannot determine the membership of an element in a set as 0 or 1, we use fuzzy sets of type-1. Similarly, when the situation is so fuzzy that we have trouble determining the membership grade even as a crisp number in [0, 1], we use fuzzy sets of type-2. This does not mean that we need to have extremely fuzzy situations to use type-2 fuzzy sets. There are many real-world problems where we cannot determine the exact form of the membership functions, e.g., in time series prediction because of noise in the data. Another way of viewing this is to consider type-1 fuzzy sets as a first order approximation to the uncertainty in the real world. Then type-2 fuzzy sets can be considered as a second order approximation. Of course, it is possible to consider fuzzy sets of higher types but the complexity of the fuzzy system increases very rapidly. For this reason, we will only consider very briefly type-2 fuzzy sets.