## 2 Materials and Methods
### Methods
#### 2.2.1 Fuzzy Sets and Interval Type-2 Fuzzy Sets  
A fuzzy set or type-1 fuzzy set F consists of a domain X of real numbers together with a function \(\mu_{\text{F}}\): X \(\rightarrow\) [0, 1], [3] i.e.,:  
\[\text{F}\ =\ \text{df}\int_{\text{X}}\mu_{\text{F}}(\text{x})/\ \text{x}\,\ \text{x}\ \in\ \text{X} \tag{1}\]  
Here, the integral denotes the collection of all points x \(\in\) X with associated membership grade \(\mu_{\text{F}}\) (x) \(\in\) [0, 1]. The function \(\mu_{\text{F}}\) is also known as the membership function of the fuzzy set F, as its value represents the grade of membership of the elements of X to the fuzzy set F. The idea is to use membership functions to describe imprecise or vague information.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Benchmark**} & **Number of Attributes** & **Number of Samples** & **Class** \\ \hline Blood Transfusion [35] & 7 & 748 & 76\%/24\% \\ \hline Breast Cancer Data & 4 & 569 & 63\%/37\% \\ \hline Breast Cancer Wisconsin [36] & 9 & 683 & 65\%/35\% \\ \hline Data Banknote Authentication & 4 & 1372 & 56\%/44\% \\ \hline Haberman & 3 & 305 & 73\%/27\% \\ \hline Heart & 13 & 303 & 46\%/54\% \\ \hline HTRU 2 [37] & 8 & 17898 & 9\%/91\% \\ \hline Immunotherapy [38] & 7 & 90 & 79\%/21\% \\ \hline Indian Liver Patient & 10 & 583 & 71\%/29\% \\ \hline Ionosphere & 34 & 351 & 64\%/36\% \\ \hline Parkinson [39] & 22 & 187 & 74\%/26\% \\ \hline Pima Indians Diabetes & 8 & 768 & 65\%/35\% \\ \hline Vertebral & 6 & 310 & 68\%/32\% \\ \hline \hline \end{tabular}
\end{table}
Table 1: The data sets used.  
By expanding this concept with the assumption that membership function values can be fuzzified themselves, the idea of type-2 fuzzy sets was introduced. A type-2 fuzzy set, denoted as \(\widetilde{\text{F}}\), is defined as follows [4]:  
\[\widetilde{\text{F}}\ =\text{{}_{df}}\int_{x\in X}\int_{u\in X_{h}}\mu_{\widetilde{ \text{F}}}(x,\ u)/(x,\ u),\ \text{J}_{x}\ \subseteq\ [0,\ 1] \tag{2}\]  
where \(\iint\) denotes union over all admissible x and u. Interval type-2 (IT2) fuzzy sets [4], a special case of type-2 fuzzy sets, are the most widely used because of their acceptable computational complexity and easy interpretation. Uncertainty about \(\widetilde{\text{F}}\) is conveyed by the so-called _footprint of uncertainty_ (_FOU_) of \(\widetilde{\text{F}}\):  
\[\text{FOU}(\widetilde{\text{F}})\ =\text{{}_{df}}\bigcup_{x\in X}\text{J}_{x} \ \subseteq\ [0,\ 1] \tag{3}\]  
The size of an FOU (the corresponding surface) is directly related to the uncertainty that is conveyed by an interval type-2 fuzzy set, and what follows, an FOU with more area is more uncertain than the one with less area. The _upper membership function_ and _lower membership function_ of \(\widetilde{\text{F}}\) are two type-1 membership functions \(\underline{\text{E}}\) and \(\widetilde{\text{F}}\) that bound the FOU, which might be used to describe \(\text{J}_{x}\), i.e.,:  
\[\text{J}_{x}=\text{{}_{df}}\left[\mu_{\underline{\text{F}}}(x),\ \mu_{ \overline{\text{F}}}(x)\right] \tag{4}\]  
which leads to the following:  
\[\text{FOU}(\widetilde{\text{F}})\ =\text{{}_{df}}\bigcup_{x\in X}\left[\mu_{ \underline{\text{F}}}(x),\ \mu_{\overline{\text{F}}}(x)\right] \tag{5}\]  
Figure 1 clearly illustrates the above definitions.