## 2 Materials and Methods
### Function Approximation Using IT2 Fuzzy Logic Systems  
The type-2 fuzzy set of system is a general type of the type-1 fuzzy system which can include uncertainties. A schematic diagram of an IT2-FLS is illustrated in Figure 1a. The structure of IT2-FLS is very similar to the conventional type-1 fuzzy logic system (T1-FLS) with an additional unit of 'type reducer'. There are five units in the IT2-FLS, i.e., fuzzifier, fuzzy rules, an inference engine, a type-reducer, and defuzzifier. Figure 1b shows a Gaussian membership function (MF). By considering the rule base of IT2 fuzzy, the IT2 fuzzy rule is expressed as follows [23]:  
\[Rule^{i}:IF\ x_{1}\ is\ \widetilde{F}_{1}^{i},\ \ldots,\ and\ x_{n}\ is\ \widetilde{F}_{n}^{i},\ \ then\ y_{1}\ is\ \widetilde{G}_{1}^{i},\ \ldots,\ y_{m}\ is\ \widetilde{G}_{m}^{i},i=1,\ldots,\ M \tag{5}\]  
where \(n\) and \(m\) are the number of inputs and the outputs, respectively. \(M\) indicates the number of fuzzy rules to be considered for the IT2 fuzzy system. Then, \(x=[x_{1},\ \ldots,\ x_{n}]\) are singleton inputs; \(y_{1},\ \ldots,\ y_{m}\) are outputs of IT2 fuzzy; \(\widetilde{F}^{i}\) and \(\widetilde{G}^{i}\) are the membership functions of the antecedent and the consequent parts of the IT2 fuzzy rules, respectively. Using the firing strength \(F^{i}\) for the \(i\)th rule, the IT2 function set is given as  
\[F^{i}=\left[f_{l}^{i}\ f_{h}^{i}\right] \tag{6}\]  
where \(f_{l}^{i}\) and \(f_{h}^{i}\) are defined as  
\[f_{l}^{i}=\mu_{l\widetilde{F}_{1}^{i}}(x_{1})\times\cdots\times\mu_{l \widetilde{F}_{n}^{i}}(x_{n})=\prod_{j=1}^{n}\mu_{l\widetilde{F}_{j}^{i}}(x_{ j}), \tag{7}\]  
\[f_{h}^{i}=\mu_{h\widetilde{F}_{1}^{i}}(x_{1})\times\cdots\times\mu_{h \widetilde{F}_{n}^{i}}(x_{n})=\prod_{j=1}^{n}\mu_{h\widetilde{F}_{j}^{i}}(x_{ j}). \tag{8}\]  
The center-of-sets (COS) type-reducer method combines firing intervals and the corresponding rule consequences as  
\[Y_{\cos j}(x)=\mathop{\cup}\limits_{f^{i}\in F^{i}(x),y_{j}^{i}\in Y^{i}} \frac{\sum_{i=1}^{M}f^{i}y_{j}^{i}}{\sum_{i=1}^{M}f^{i}}=\left[y_{l_{j}},\ y_{r_{j}} \right],\ j=1,\ldots,m, \tag{9}\]  
where the defuzzified output is given as  
\[y_{l_{j}}=\frac{\sum_{i=1}^{M}f_{l}^{i}y_{l_{j}}^{i}}{\sum_{i=1}^{M}f_{l}^{i} }=\frac{\sum_{i=1}^{L}f_{h}^{i}y_{l_{j}}^{i}+\sum_{i=L+1}^{M}f_{l}^{i}y_{l_{j} }^{i}}{\sum_{i=1}^{L}f_{h}^{i}+\sum_{i=L+1}^{M}f_{l}^{i}}, \tag{10}\]  
\[y_{r_{j}}=\frac{\sum_{i=1}^{M}f_{r}^{i}y_{r_{j}}^{i}}{\sum_{i=1}^{M}f_{r}^{i} }=\frac{\sum_{i=1}^{R}f_{l}^{i}y_{r_{j}}^{i}+\sum_{i=R+1}^{M}f_{h}^{i}y_{r_{j} }^{i}}{\sum_{i=1}^{R}f_{l}^{i}+\sum_{i=R+1}^{M}f_{h}^{i}}. \tag{11}\]  
\(y_{l_{j}}\) and \(y_{r_{j}}\) in the above equations are obtained using the Karnik-Mendel algorithm [39]. Finally, the crisp defuzzified output of the IT2-FLS is calculated as  
\[y_{j}=\frac{y_{l_{j}}+y_{r_{j}}}{2},\ j=1,\ldots,m. \tag{12}\]  
The design of the IT2-FLS approximator \(\hat{f}(X|\Theta)\) is carried out to approximate the unknown \(f(x)\). The IT2-FLS approximator based on (10) and (11) is defined as \[ \begin{array}{rl}\hat{f}(X|\Theta)&=\dfrac{1}{2}\left(\dfrac{\sum_{i=1}^{T} \overline{\rho}_{i}^{i}+\sum_{l=1}^{M}\ell_{l+1}^{l}\ell_{l}^{i}}{\sum_{l=1}^{L }\overline{\ell}+\sum_{l=1}^{M}\ell^{i}}+\dfrac{\sum_{i=1}^{R}\ell_{l}^{i} \ell_{l}^{i}+\sum_{l=1}^{M}\overline{\ell}_{l}^{i}\ell_{l}^{i}}{\sum_{l=1}^{L }\overline{\ell}+\sum_{l=1}^{M}\overline{\ell}}\right)\\ &=\dfrac{1}{2}\left(\sum_{i=1}^{M}\Theta_{l}^{i}\Xi_{l}^{i}+\sum_{l=1}^{M} \Theta_{l}^{i}\Xi_{r}^{i}\right)=\dfrac{1}{2}\big{(}\[PERSON]_{l}^{T}\[PERSON]_{l}+ \[PERSON]_{r}^{T}\[PERSON]_{r}\big{)}\\ &=\[PERSON]^{T}\[PERSON],\end{array}