## 2 Materials and Methods
### Methods
#### 2.2.6 Fuzzy Rules Optimization  
In our experiments, the basic assumption was to analyse the influence of the shape and the FOU of the type-2 fuzzy sets used in the rule premises, derived from the corresponding fuzzy information system, on the classification accuracy. For this purpose, we have used the type-2 fuzzy sets shown in Table 4. Each of these type-2 membership functions was derived for every attribute of all benchmark data analyzed. Below, we explain the descriptions used in our research:  
1. The number of Gaussian functions for fuzzification: [3; 5; 7; 9; 11]. The fuzzification procedure is as follows: generate the _medium_ membership function and, next, the rest of the membership functions based on it. All Gaussians are transformed into type-2 fuzzy sets by changing the standard deviation value (called _sigma_offset_ parameter). The _sigma_offset_ defines the FOU of the corresponding IT2 fuzzy sets. The same value was applied for each Gaussian function.
2. If the standard deviations applied to the Gaussians are the same or not: {_equal_, _progressive_}_--_progressive_ assumes the concentration of membership functions around the mean or central values. For 3 Gaussians, _equal_ and _progressive_, give the same fuzzification result.
3. If the mean of the _medium_ basic function is derived directly from the corresponding data set or if a fixed value is used: {_mean_, _center_}_--_center_ assumes Gaussian distribution with fixed mean 0.5.  
For example, <3 _Gausses_, _Equal_, _Mean_> is considered as 3 Gaussian membership functions with equal standard deviation values used to define 3 type-2 fuzzy sets, and the _medium_ function is defined by its corresponding mean value derived from the considered data. The <5 _Gausses_, _Equal_, _Center_> is considered as 5 Gaussian membership functions and the _medium_ function is defined by a fixed mean value 0.5. The <7 _Gausses_, _Progressive_, _Mean_> is considered as 7 Gaussian membership functions with different standard deviation values, assuming concentration of membership functions around the mean value of the _medium_ function.  
Figure 5: Type-1 and type-2 fuzzification proposal.  
For the type-2 sets used in the conclusions, we define s-shaped functions representing each class in the classification problem considered. We optimize the FOU of each function by changing the cross-points. We called the corresponding parameter under optimization _center_offset_--see Figure 6. The final classification decision value was fixed = 0.5, assuming normalized domain [0, 1]. Changes of the classification threshold did not show a positive effect on the experiment.  
For each benchmark data, we optimized the _sigma_offset_ and _center_offset_ parameters for the type-2 fuzzy sets considered, using the grid search optimization algorithm applied in _k_-folded cross-validation process with evaluation on a held-out validation set. The _sigma_offset_ takes values in the range [0.01, 0.03] with step 0.005 and the _center_offset_ takes values in the range [0.01, 0.21] with step 0.02. The \(k\) value was chosen for any data set assuming a fair number of samples in the corresponding validation sets, presented in Table 5. Additionally, for any data partition, we have used stratified sampling to ensure balanced validation and test sets. For better clarity, we show the experiment information flow in Figure 7.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Data Set** & _k_**-Value** \\ \hline Blood Transfusion & 10 \\ \hline Breast Cancer Data & 10 \\ \hline Breast Cancer Wisconsin & 10 \\ \hline Data Banknote Authentication & 10 \\ \hline Haberman & 6 \\ \hline Heart & 5 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Number of folds applied in the cross-validation process.  
Figure 6: IT2 fuzzy sets used in the rule conclusions.