## 2 Design of the CGK-Based Fuzzy Granular Model
### Optimal Assignment of IGs  
IGs are generated using clustering of original numerical data, and the GM is designed using the generated IGs. The following conditions should be satisfied when designing the GM. First, the data in the output space are granulated. Second, the data in the input space corresponding to each IG generated in the output space are granulated. The cluster's centroid generated through clustering in the input space is granulated. The considerations when designing the GM are as follows. The IG generation method is provided so that the model is granulated, and the IG is limited to the form specified by the user. The higher the value of the IG, the greater the design flexibility becomes. If the IG is 0, it is not granulated, and the model has the original numerical value. There are two general methods to generate IGs through information granulation described above: uniform and flexible generation methods. Figure 4 shows the original data conversion to IG through the information granulation method. The uniform generation method generates the data in the output space symmetrically and uniformly under the level of information granulation, and \(N\) IGs show the same form. In Equations (15) and (16), \(\varepsilon_{0}\) can represent \(\frac{\varepsilon}{N}\), which denotes the upper and lower levels of the IGs that have been generated uniformly. \(\mathrm{a}_{ij}^{-}\) and \(\mathrm{a}_{ij}^{+}\) represent the lower limit value and upper limit value of the information particle. For uniformly generated information particles, the distance between the lower limit value and the upper limit value is the same from the center. Each flexibly generated information particle is assigned a value based on the Gaussian probability distribution. \[a_{ij}^{-}=\left\{ \begin{matrix}\min\bigl{(}a_{ij}(1-\varepsilon_{0}),\ a_{ij}(1+ \varepsilon_{o})\bigr{)}\ if\ a_{ij}\
eq 0\\ -\varepsilon_{0}\ if\ a_{ij}=0\end{matrix} \right. \tag{15}\]  
\[a_{ij}^{+}=\left\{ \begin{matrix}\max\bigl{(}a_{ij}(1-\varepsilon_{0}),\ a_{ij}(1+ \varepsilon_{o})\bigr{)}\ if\ a_{ij}\
eq 0\\ \varepsilon_{0}\ if\ a_{ij}=0\end{matrix} \right. \tag{16}\]  
The flexible generation method [42] generates IGs based on data distribution in the output space using the Gaussian probability distribution. The size of each IG is not uniform. \(-\)epsilon and \(+\)epsilon are parameters that set the lower and upper bounds in the uniformly generated information particles, and their values are set based on the Gaussian probability distribution of the data. \[a_{ij}^{-}=\left\{ \begin{matrix}\min\bigl{(}a_{ij}\Bigl{(}1-\varepsilon_{ij}^ {-}\Bigr{)},\ a_{ij}\Bigl{(}1+\varepsilon_{ij}^{+}\Bigr{)}\Bigr{)}\ if\ a_{ij}\
eq 0\\ -\varepsilon_{ij}^{-}\ if\ a_{ij}=0\end{matrix} \right. \tag{17}\]  
\[a_{ij}^{+}=\left\{ \begin{matrix}\max\bigl{(}a_{ij}\Bigl{(}1-\varepsilon_{ij}^ {-}\Bigr{)},\ a_{ij}\Bigl{(}1+\varepsilon_{ij}^{+}\Bigr{)}\Bigr{)}\ if\ a_{ij}\
eq 0\\ \varepsilon_{ij}^{+}\ if\ a_{ij}=0\end{matrix} \right. \tag{18}\]  
Figure 4: Shape of IGs generated through the information granulation method. The method of generating IGs in the input space corresponding to each IG generated in the output space is to granulate the information around the cluster's centroid generated through clustering. For information granulation in the input space, \(x_{k}\in D(G(x_{k},\ V_{1},\ V_{2},\ \ldots,\ V_{c},\ U))\) must be satisfied for as many data of \(X\) as possible. When the centroid of the cluster is granulated, the granulation of a specific level that assumes the value of \([0,1]\) for \(V_{i}\) is allowed. \[V_{ij}=\left[v_{ij}-\varepsilon*range_{j},\ v_{ij}+\varepsilon*range_{j}\right] \tag{19}\]  
Here, \(i\) and \(j\) denote the number of clusters and the number of data, respectively. The cluster centroid is granulated at the same level for all data. Every coordinate of the cluster centroid is distributed symmetrically around \(v_{ij}\) and receives an equal effect by the given level of granulation. The higher the coverage, the higher the performance index (PI) [43].