## 2 Materials and Methods
### Methods
#### 2.2.2 Mamdani Type-2 Fuzzy System  
In Figure 2, we show the information flow within an IT2 fuzzy system. It is very similar to its type-1 analogue. The major difference is that because of the IT2 fuzzy sets used in the rulebase, the outputs of the inference engine are IT2 fuzzy sets too. Therefore, a type reducer [6; 43] must be applied to convert them into type-1 fuzzy sets to enable the defuzzification procedure.  
Figure 1: An interval type-2 fuzzy set \(\widetilde{\text{F}}\).  
Figure 2: Information flow in a type-2 fuzzy system.  
Below, we give a brief description of the basic steps of the computations in an IT2 fuzzy system [40]. Let consider the rule base of an IT2 fuzzy system consisting of \(N\) rules taking the following form:  
\[\widetilde{\mathrm{R}}^{n}\cdot_{\mathrm{df}}\mathrm{IF}\ (x_{1}\mathrm{ is}\ \widetilde{\mathrm{X}}_{1}^{n})\ \mathrm{o}\ \ldots\ \mathrm{o}\ (x_{i}\mathrm{ is}\ \widetilde{\mathrm{X}}_{i}^{n})\ \mathrm{THEN \ y\ is}\ \widetilde{\mathrm{Y}}_{j}^{n}\]  
where \(\widetilde{\mathrm{X}}_{1}^{n}\) (i = 1, \(\ldots\), I; n = 1, \(\ldots\), N) are IT2 fuzzy sets defined over corresponding domains, and \(\widetilde{\mathrm{Y}}_{j}^{n}\) (j = 1, \(\ldots\), M) is an IT2 fuzzy set, assuming a Mamdani type-2 system, which represents the corresponding rule conclusion. The logical operator '\(\mathrm{o}\)' is defined as a fuzzy conjunction or disjunction, i.e., \(\mathrm{o}\in\{\otimes,\oplus\}\), where \(\otimes\) and \(\oplus\) are binary operators (\(t\)-_norm_ and \(s\)-_norm_ respectively) defined over [0, 1] (\(\otimes\), \(\oplus\): [0, 1]\({}^{2}\rightarrow[0,\) 1]). The \(t\)-_norm_ operator provides the characterization of the AND fuzzy operator, while the \(s\)-_norm_ provides the characterization of the OR fuzzy operator [44]. In our research, we applied the [PERSON]'s \(t\)- and \(s\)-_norms_, which correspond to the \(min\) and \(max\) operators, respectively.  
Assuming an input vector \(\mathbf{\overline{x}}=\{x_{1}^{\prime},x_{2}^{\prime},x_{3}^{\prime},\ldots,x_{ i}^{\prime}\}\), typical computations of an IT2 fuzzy system consist of the following steps:  
1. Compute the membership intervals of \(x_{i}^{\prime}\) for each \(\widetilde{\mathrm{X}}_{i}^{n}\), \([\mu_{\widetilde{\mathrm{X}}_{i}^{n}}(x_{i}^{\prime}),\ \mu_{\widetilde{\mathrm{X}}_{i}^{n}}(x_{i}^{\prime})]\), i = 1, \(\ldots\), I; n = 1, \(\ldots\), N
2. Compute the firing interval of the \(n\)-th rule: \[\mathrm{F}^{n}(\mathbf{\overline{x}})=_{\mathrm{df}}[\mu_{\widetilde{\mathrm{ X}}_{1}^{n}}(x_{1}^{\prime})\ \mathrm{o}\ldots\mathrm{o}\ \mu_{\widetilde{\mathrm{X}}_{1}^{n}}(x_{i}^{\prime}),\ \mu_{\widetilde{\mathrm{X}}_{1}^{n}}(x_{1}^{\prime})\ \mathrm{o}\ldots\ \mathrm{o}\ \mu_{\widetilde{\mathrm{X}}_{1}^{n}}(x_{i}^{ \prime})]\equiv[\ell^{n},\widetilde{\ell}^{n}]\]
3. Apply type reduction to combine \(\mathrm{F}^{n}(\mathbf{\overline{x}})\) with corresponding rule consequents.  
There are some methods of type reduction [6; 45; 46], but the most commonly used one is the _center-of-set_ (COS) type reducer [6] using the Karnik-Mendel algorithms [6; 43] or their variants [47; 48]. As the whole procedure concerning Mamdani type-2 fuzzy system is well described in the literature, we will omit further details. For more clarity, in Figures 3 and 4, we show the idea of the introduced calculations.