## 2 Materials and Methods
### Methods
#### 2.2.4 Type-1 Fuzzy Information System  
The major assumption of our research is the interpretation of a fuzzy information system. First, we define the values of the information function as type-1 fuzzy sets. This assumption solves the problem of crisp values by providing a generalisation. Therefore, we propose better data granulation and thus more robustness to vague or imprecise data. Next, by assuming Gaussian distribution for each attribute, we are able to define any information function value as one of the very basic fuzzy sets: _low, medium_, and _high._ Therefore, the value of any pair object-attribute is generalised by a fuzzy set. Below in Table 2, we give an illustration of a type-1 fuzzy information system with a decision attribute, according to our assumptions. %clearpage  
where:  
\[\mu_{medium}(\text{a})=\text{\text{df}}\,\text{e}^{\frac{-(\text{a}-\text{ ab})^{2}}{2\sigma^{2}}},\ \text{a}\ \in\ \text{V}_{\text{a}}\text{ with expected value }(\text{a}_{0})\text{ and standard deviation }(\sigma) \tag{11}\]  
The fuzzy sets _low_ and _high_ attribute values are easy to define as well (see Figure 5):  
\[\mu_{\text{low}}(\text{a})=\text{\text{df}}\left\{ \begin{array}{l}1-\ \text{e}^{\frac{-(\text{a}-\text{ ab})^{2}}{2\sigma^{2}}}:\ \text{a}\ <\ \text{a}_{0}\end{array} \right\}, \tag{12}\]  
If a pair object-attribute\({}_{\text{j}}\) has a value defined as _low_, for example, it means that:  
\[\mu_{\text{low}}(\text{f}(\text{object}_{i},\text{attribute}_{\text{j}}))\geq \max\{\mu_{\text{medium}}(\text{f}(\text{object}_{i},\text{attribute}_{\text{j}})) \},\mu_{\text{high}}(\text{f}(\text{object}_{i},\text{attribute}_{\text{j}}))\}.\]  
It is important to note that our assumption was also the direct application of the original Skowron and [PERSON] algorithm [41; 42] for rule induction. As a result, we did not have to modify the introduced indistinguishability relation (IND) to define fuzzy rough  
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline  & **Attribute\({}_{1}\)** & **Attribute\({}_{2}\)** & **Attribute\({}_{3}\)** & \(\ldots\) & **Decision Attribute** \\ \hline Object\({}_{1}\) & \(\left\{\textit{low, medium, high}\right\}\) & \(\left\{\textit{low, medium, high}\right\}\) & \(\left\{\textit{low, medium, high}\right\}\) & \(\ldots\) & D\({}_{1}\) \\ \hline Object\({}_{2}\) & \(\left\{\textit{low, medium, high}\right\}\) & \(\left\{\textit{low, medium, high}\right\}\) & \(\left\{\textit{low, medium, high}\right\}\) & \(\ldots\) & D\({}_{2}\) \\ \hline Object\({}_{3}\) & \(\left\{\textit{low, medium, high}\right\}\) & \(\left\{\textit{low, medium, high}\right\}\) & \(\left\{\textit{low, medium, high}\right\}\) & \(\ldots\) & D\({}_{3}\) \\ \hline \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \sets or modify the quality of the approximation introduced by Equation (8). We directly used the fuzzified values of the information function as if they were crisp values. This means that \(\mathit{object}_{i}\) is related to \(\mathit{object}_{j}\) with respect to IND in accordance with our model if the values of the corresponding attributes define the same fuzzy sets. For example, below we show the partition (\(P\)) of the set of objects {Object1,..., Object5} implied by IND for a sample information system given in Table 3.  
The corresponding partition with respect to IND has three equivalence classes:  
\[P/\mathit{IND}(\mathit{attribute1},\mathit{Attribute2})=\mathrm{df}.\ \{\{\mathit{Object1} _{1}\mathit{ Objects}\},\ \{\mathit{Object2}_{2}\mathit{ Object3}\},\ \{\mathit{Object4}\}\}.\]  
The above assumption made possible the use of classical algorithms introduced in information systems theory, while at the same time generalizing the information given by the attributes. The use of membership degrees inferred from corresponding membership functions is performed after the induction of rules and their transformation into fuzzy rules, in accordance with the assumptions of the fuzzy model used. Of course, such an assumption may be too strong a generalization.