# Transformation-Based Fuzzy Rule Interpolation Using Interval Type-2 Fuzzy Sets
## 1 Introduction
As another example, suppose that the concept of interest is eye contact [22], denoted by \(x\) with \(x\) belonging to the intensity range of \([0,10]\), where zero indicates no eye contact and 10 represents maximum eye contact. One of the terms that may characterise the amount of perceived eye contact is \"some eye contact\". Suppose that 10 people are surveyed, and they are asked to locate the distribution for some eye contact on the scale 0-10. Surely, it would not be expected that the same, precisely-described results are obtained from 10 individuals. Instead, different membership functions to depict such a fuzzy concept will be provided. Figure 1 shows such an example. The question is how such diverse information about the underlying concept may be captured and expressed. One approach might be to construct a type-1 fuzzy set with a triangle-shaped membership function whose support or base endpoints (on the \(x\)-axis) are the two average endpoint values (over the 10 surveyed results) and whose apex is midway between these two endpoints. However, this approach  
Figure 1: Different membership functions for some eye contact perceived by different people [23]. completely ignores the uncertainty associated with the different results. A second approach is to make use of the average endpoint values and the standard deviation of each endpoint to establish an uncertainty interval about each average endpoint value. In so doing, for each \(x\), the membership of some eye contact is no longer a single value; instead, it is itself a function of the membership values. Such a concept has, therefore, to be represented as a type-2 fuzzy set [24], which is illustrated in Figure 2, where two endpoints \(l\) and \(r\) have two uncertainty intervals associated with them. The apex point \(n\) is assumed to have a full certainty value. However, in general, \(n\) can also have an uncertainty interval associated with it, just as with \(l\) and \(r\). The concept of type-2 fuzzy sets was proposed as an extension of the concept of type-1 fuzzy sets. A type-1 fuzzy set has a grade of membership that is crisp, whereas a type-2 fuzzy set has grades of membership that are themselves fuzzy [25; 26]. That is, in set representation, membership functions of type-1 fuzzy sets are two-dimensional, whereas membership functions of type-2 fuzzy sets are three-dimensional. It is the new third dimension that provides additional degrees of freedom for handling uncertainty [22]. Such sets are useful in circumstances where it is difficult to determine the exact and precise membership functions [27]. Note that apart from the use of type-2 fuzzy sets, other extensions have also been developed to represent various types of uncertainty that are associated with defining type-1 fuzzy sets, including interval-valued fuzzy sets [28; 29], R-fuzzy sets [30], fuzzy complex numbers [31] and rough-fuzzy sets [23]. Type-2 fuzzy sets also include a specific group of fuzzy sets that are referred to as interval type-2 fuzzy sets [32; 33], where all of the values of the secondary membership grades are the same. In this case, the third dimension is no longer needed because it conveys no new information. Although general type-2 fuzzy sets have one extra degree of freedom than interval type-2 fuzzy sets, it is not yet known how to best choose their secondary membership functions [34; 35]. The third dimension is therefore often ignored in order to reduce the computational complexity, although this may result in the loss of an intuitively desirable degree of freedom in concept representation. Of particular interest are the very recent developments, e.g., [36; 37; 38] that have been proposed to deal with FRI using interval type-2 fuzzy sets. However, none of these extend T-FRI. Motivated by this observation and based on the initial investigation of [39], this paper extends the existing T-FRI approach with the use of interval type-2 fuzzy sets.