## 5 Discussion  
This research is a continuation of a previous one, published in [22]. Both papers present our newly introduced definition of a fuzzy information system, which has the following advantages:  
* The information function values are interpreted as fuzzy sets, labeled with corresponding linguistic variables. This gives the possibility to generalize information--we do not consider numerical values for pairs (object, attribute), but general descriptions such as'small','medium', and 'high'. * The decision table used is generated in an automatic manner for a considered data set, as the value'medium' is assumed as the Gaussian distribution of the data for each attribute. Then, the sets 'low' and 'high' are easy to be defined using the'medium' membership function. Next, a corresponding label (identifying the corresponding fuzzy set) is given for a pair (object, attribute), by using the maximum membership value. Additionally, we propose an easy transformation in order to change the applied type-1 fuzzy sets into type-2 fuzzy sets. Therefore, by defining such a decision table, the rule induction procedure described in [38; 39] is applied and next, type-2 fuzzy rules are derived. What is more, different fuzzification strategies are possible (see Table 4). \begin{table}
\begin{tabular}{p{85.4 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline  
In this research, the goal was to incorporate our fuzzy information system concept with the Takagi-Sugeno model, as it does not require the model outputs to be defined as fuzzy sets. Such an achievement would create new possibilities, using the advantages of the Takagi-Sugeno model. To incorporate the model, we had to optimize the model parameters with respect to the corresponding benchmark data. We have applied the CMA-ES (Covariance Matrix Adaptation Evolutionary Strategy) for this purpose. We conducted experiments with the same binary classification problems and the same fuzzification experiments, as presented in [22], in order to compare the results. While performing experiments, we have discovered further promising features and potential disadvantages of the method proposed. The classification process is able to fit training data, but sometimes has issues with generalization. Applying PCA allows us to minimize this issue, making the method proposed more robust. Additionally, performing optimization with an evolutionary strategy allows the model to find very well-fitting decision boundaries. Therefore, one might achieve even better performance by applying additional regularization methods. To solve the problem of imbalanced data, Random Oversampling was used, which was not beneficial in a few cases. If the imbalance is very high, ROS may produce too much redundancy; therefore, the information system may produce biased rules, ignoring important information contained in the data. In such a scenario, other sampling techniques should be applied instead. Regarding the attribute fuzzification procedure, only Gaussians were assumed as membership functions. We suppose, there is room for improvement, adjusting membership functions to the attributes' characteristics. Additionally, not only the first-order Takagi-Sugeno model could be applied. Despite such constraints, good classification results for several datasets were achieved. We believe that the application of the Takagi-Sugeno model in our approach will support the solving of multi-class classification problems. This is because, in the model used, there is no need to define decision classes, contrary to the Mamdani fuzzy model. Summarizing, a new classification concept was proposed in this research with the following main advantages:  
* Defining a decision table with fuzzy values. The fuzzification is provided in an automatic manner directly from a data set. * Using the rule induction method based on the information system concept, which has a solid mathematical background. Each rule is related to a corresponding class, regarding the classification problem considered.