## 2 Material and Methods
### Methods  
#### 2.2.1 Type-1 and Type-2 Fuzzy Sets  
A Type-1 fuzzy set F consists of a non-empty domain X and a function \(\mu_{\text{F}}:\text{X}\rightarrow[0,1]\), called a membership function [32]. Considering a continuous membership function, a fuzzy set of type-1 as defined in Equation (1):  
\[\text{F}=_{\text{df}}\int_{\text{X}}\mu_{\text{F}}(\text{x})/\text{ x},\text{ x}\ \in\ \text{X} \tag{1}\]  
The above integral denotes the collection of all points \(\text{x}\in\text{X}\) with associated membership grade \(\mu_{\text{F}}(\text{x})\in[0,1]\). The function values define the grade of membership of the elements of X to the fuzzy set F. Membership functions are assumed to describe imprecise or vague information.  
The type-1 fuzzy sets concept was expanded by fuzzifying the membership function values themselves and type-2 fuzzy sets were introduced. A type-2 fuzzy set, denoted as \(\widetilde{\text{F}}\), is defined in Equation (2) [5]:  
\[\widetilde{\text{F}}=_{\text{df}}\int_{\text{x}\in\text{X}}\int_{\text{u}\in \text{I}_{\text{X}}}\mu_{\widetilde{\text{F}}}(\text{x},\,\text{u})/(\text{ x},\,\text{u}),\text{J}_{\text{x}}\subseteq\ [0,1] \tag{2}\]  
where \(\iint\) denotes the union over all admissible x and u. The most widely used special cases of type-2 fuzzy sets, mostly because of their easy interpretation, are the interval type-2  
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Benchmark** &
\begin{tabular}{c} **Number of** \\ **Attributes** \\ \end{tabular} & **Number of Samples** & **Class Proportions** \\ \hline Blood Transfusion & 7 & 748 & 76\%/24\% \\ \hline Breast Cancer Data & 4 & 569 & 63\%/37\% \\ \hline Breast Cancer & 9 & 683 & 65\%/35\% \\ Wisconsin & & \\ \hline Data Banknote & 4 & 1372 & 56\%/44\% \\ Authentication & & \\ \hline Haberman & 3 & 305 & 73\%/27\% \\ \hline Heart & 13 & 303 & 46\%/54\% \\ \hline HTRU 2 & 8 & 17,898 & 9\%/91\% \\ \hline Immunotherapy & 7 & 90 & 79\%/21\% \\ \hline Indian Liver Patient & 10 & 583 & 71\%/29\% \\ \hline Ionosphere & 34 & 351 & 64\%/36\% \\ \hline Parkinson & 22 & 187 & 74\%/26\% \\ \hline Pima Indians & 8 & 768 & 65\%/35\% \\ Diabetes & & \\ \hline Vertebral & 6 & 310 & 68\%/32\% \\ \hline \hline \end{tabular}
\end{table}
Table 1: The benchmark data used.  
(IT2) fuzzy sets [5]. Uncertainty about \(\widetilde{\text{F}}\) is handled by the so-called footprint of uncertainty (FOU) of \(\widetilde{\text{F}}\), as shown in Equation (3):  
\[\text{FOU}(\widetilde{\text{F}})=_{\text{df}}\bigcup_{\text{x}\in\text{X}}J_{ \text{x}},\ J_{\text{x}}\subseteq\ [0,1] \tag{3}\]  
The area of a FOU is directly related to the uncertainty that is conveyed by an interval type-2 fuzzy set, and what follows, a FOU with more area is more uncertain than the one with less area. The lower membership function (LMF) and upper membership function (UMF) of \(\widetilde{\text{F}}\) are two type-1 membership functions \(\widetilde{\text{E}}\) and \(\widetilde{\text{F}}\) that bound the FOU, which are used to describe J\({}_{\text{x}}\), see Equation (4):  
\[J_{\text{x}}=_{\text{df}}\ [\mu_{\text{E}}(\text{x}),\ \mu_{\text{F}}(\text{x})] \tag{4}\]  
which leads to Equation (5):  
\[\text{FOU}(\widetilde{\text{F}})=_{\text{df}}\bigcup_{\text{x}\in\text{X}} \biggl{[}\mu_{\text{E}}(\text{x}),\ \mu_{\text{F}}(\text{x})\biggr{]} \tag{5}\]  
Figure 1 illustrates the graphical interpretation of the above definitions.