## 2 Granulation Techniques
### Standard Granulation  
Let us consider the decision system \((U,A,d)\), where \(U\) is the universe of objects, \(A\) the set of conditional attributes, \(d\
ot\in A\) is the decision attribute, and \(r_{gran}\) granulation radius from the set \(\{0,\frac{1}{|A|},\frac{2}{|A|},...,1\}\). The standard rough inclusion \(\mu\), for \(u,v\in U\) and for selected \(r_{gran}\) is defined as  
\[\mu(v,u,r_{gran})\Leftrightarrow\frac{|IND(u,v)|}{|A|}\geq r_{gran},\text{ where }IND(u,v)=\{a\in A:a(u)=a(v)\}, \tag{1}\]  
For each object \(u\in U\), and selected \(r_{gran}\), we compute the _standard granule_\(g_{r_{gran}}(u)\) as follows:  
\[g_{r_{gran}}(u)\text{ is }\{v\in U:\mu(v,u,r_{gran})\}. \tag{2}\]  
In the next step, we use a selected strategy to cover the training decision system \(U\) by computed granules--the random choice is the simplest among the most effective studied in [30]). All studied methods are available in [30] (pages 105-220). Toy Example  
For a given training decision system from Table 1, the granulation radius \(r_{gran}\in\{0,.25,.5,.75,1\}\), the steps of the standard granulation are as follows. In case of \(r_{gran}=0\), each single granule is equal \(U\) because objects are treated as indiscernible even if they are completely different. In addition, we expected only one object as the granular reflection of the training data. The second boundary case is \(r_{gran}=1\); each granule contains only their central object or duplicates because the objects are indiscernible. Now, allow us to show how the standard granulation works for radius \(r_{gran}=0.5\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Day** & **Outlook** & **Temperature** & **Humidity** & **Wind** & **Play.golf** \\ \hline \(u_{1}\) & Sunny & Hot & High & Weak & No \\ \(u_{2}\) & Sunny & Hot & High & Strong & No \\ \(u_{3}\) & Overcast & Hot & High & Weak & Yes \\ \(u_{4}\) & Rainy & Mild & High & Weak & Yes \\ \(u_{5}\) & Rainy & Cool & Normal & Weak & Yes \\ \(u_{6}\) & Rainy & Cool & Normal & Strong & No \\ \(u_{7}\) & Overcast & Cool & Normal & Strong & Yes \\ \(u_{8}\) & Sunny & Mild & High & Weak & No \\ \(u_{9}\) & Sunny & Cool & Normal & Weak & Yes \\ \(u_{10}\) & Rainy & Mild & Normal & Weak & Yes \\ \(u_{11}\) & Sunny & Mild & Normal & Strong & Yes \\ \(u_{12}\) & Overcast & Mild & High & Strong & Yes \\ \(u_{13}\) & Overcast & Hot & Normal & Weak & Yes \\ \(u_{14}\) & Rainy & Mild & High & Strong & No \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Exemplary decision system \((U,A,d)\) by [PERSON] [36]. \[ \begin{array}{c}\text{Assuming that }g_{r_{gran}}(u_{i})=\{u_{j}\in U_{trn}:\frac{| IND(u_{i},u_{j})|}{|A|}\geq r_{gran}\,\}\\ IND(u_{i},u_{j})=\{a\in A;a(u_{i})=a(u_{j})\},\;U_{trn}\text{ is the universe of training objects, }\\ \text{and }|X|\text{ is the cardinality of set}\end{array} \]  
The sample standard granules with a 0.5 radius, derived from decision systems from Table 1 look as follows,  
\[ \begin{array}{l}g_{05}(u_{1})=\{u_{1},u_{2},u_{3},u_{4},u_{8},u_{9},u_{13}, \,\},\,g_{0.5}(u_{2})=\{u_{1},u_{2},u_{3},u_{8},u_{11},u_{12},u_{14},\,\}\\ g_{05}(u_{3})=\{u_{1},u_{2},u_{3},u_{4},u_{8},u_{12},u_{13},\,\},\,g_{0.5}(u_{4} )=\{u_{1},u_{3},u_{4},u_{5},u_{8},u_{10},u_{12},u_{14},\,\}\\ g_{05}(u_{5})=\{u_{4},u_{5},u_{6},u_{7},u_{9},u_{10},u_{13},\,\},\,g_{0.5}(u_{6} )=\{u_{5},u_{6},u_{7},u_{9},u_{10},u_{11},u_{14},\,\}\\ g_{0.5}(u_{7})=\{u_{5},u_{6},u_{7},u_{9},u_{11},u_{12},u_{13},\,\},\,g_{0.5}(u_ {8})=\{u_{1},u_{2},u_{3},u_{4},u_{8},u_{9},u_{10},u_{11},u_{12},u_{14},\,\}\\ g_{0.5}(u_{9})=\{u_{1},u_{5},u_{6},u_{7},u_{8},u_{9},u_{10},u_{11},u_{13},\,\}, \,g_{0.5}(u_{10})=\{u_{4},u_{5},u_{6},u_{8},u_{10},u_{10},u_{11},u_{13},u_{14}, \,\}\\ g_{0.5}(u_{11})=\{u_{2},u_{6},u_{7},u_{8},u_{9},u_{10},u_{11},u_{12},u_{14},\,\},\,g_{0.5}(u_{12})=\{u_{2},u_{3},u_{4},u_{7},u_{8},u_{11},u_{12},u_{14},\,\}\\ g_{0.5}(u_{13})=\{u_{1},u_{3},u_{5},u_{7},u_{9},u_{10},u_{13},\,\},\,g_{0.5}(u_ {14})=\{u_{2},u_{4},u_{6},u_{8},u_{10},u_{11},u_{12},u_{14},\,\}\end{array}