## 2 Design of the CGK-Based Fuzzy Granular Model
### Context-Based GK (CGK) Fuzzy Granular Model  
The CGK-based fuzzy granular model is designed by generating contexts using context-based GK clustering and then developing clusters of geometric shapes in each context. Figure 2 illustrates the structure of the CGK-based fuzzy granular model that generates three contexts and nine clusters. This figure shows there are the premise and conclusion variables. First, the conclusion variable is an IG of the context shape generated in the output space. Next, the premise variable is an IG of the cluster type generated in each context. The final output of the CGK-based fuzzy granular model is expressed as follows:  
\[Y=\sum_{\bigoplus}W_{t}\otimes z_{t} \tag{7}\]  
Figure 2: Structure of CGK-based fuzzy granular model.  
In Equation (7), \(\bigoplus\), \(\bigotimes\) are addition and multiplication signs that express the operations for IG. The fuzzy set is generated by processing the premise part of the CGK-based fuzzy granular model. The clusters generated by context-based GK clustering are expressed as an activation layer of the CGK-based fuzzy granular model. The part between the conditional and output layers is expressed as a linguistically descriptive context. The sum of final outputs of the CGK-based fuzzy granular model is expressed as follows using every context in the output space:  
\[\begin{array}{l}Y=(z_{11}\bigotimes A_{1}\bigoplus z_{12}\bigotimes A_{1} \ldots z_{1n1}\bigotimes A_{1})\bigoplus(z_{21}\bigotimes A_{2}\bigoplus z_{22 }\bigotimes A_{2}\ldots z_{2n2}\bigotimes A_{2})\\ \ldots(z_{c1}\bigotimes A_{c}\bigoplus z_{c2}\bigotimes A_{c}\ldots z_{cnc} \bigotimes A_{c})\end{array} \tag{8}\]  
The final output of the CGK-based fuzzy granular model is expressed as a fuzzy number with the context shape and as a fuzzy set as follows:  
\[Y_{i}=\big{(}y_{i}^{-},\ y_{i},\ y_{i}^{+}\big{)} \tag{9}\]  
where \(y_{i}^{-},\ y_{i},\ \text{and}\ y_{i}^{+}\) are the lower limit, model value, and upper limit of the CGK-based fuzzy granular model, respectively. These values are expressed as Equations (10)-(12):  
\[y_{i}^{-}=\big{(}z_{11}a_{1}+z_{12}a_{1}^{-}+\ldots+z_{1n1}a_{1}^{-}\big{)}+ \big{(}z_{c1}a_{c}^{-}+z_{c2}a_{c}^{-}+\ldots+z_{cnc}a_{c}^{-}\big{)} \tag{10}\]  
\[y_{i}=(z_{11}a_{1}+z_{12}a_{1}+\ldots+z_{1n1}a_{1})+(z_{c1}a_{c}+z_{c2}a_{c}+ \ldots+z_{cnc}a_{c}) \tag{11}\]  
\[y_{i}^{+}=\big{(}z_{11}a_{1}+z_{12}a_{1}^{+}+\ldots+z_{1n1}a_{1}^{+}\big{)}+ \big{(}z_{c1}a_{c}^{+}+z_{c2}a_{c}^{+}+\ldots+z_{cnc}a_{c}^{+}\big{)} \tag{12}\]  
When context-based GK clustering is performed, the membership matrix is expressed as a value between 0 and 1. The requirements of the membership matrix are expressed as follows:  
\[U(f)=\mu_{ik}\in[0,\ 1]\ \Bigg{|}\ \sum_{i=1}^{c}\mu_{ik}=f_{k}\forall k\ and\ 0<\sum_{k=1}^{N}\mu_{ik}<N \tag{13}\]  
The structural description of the CGK-based fuzzy granular model is as follows. The input layer receives data and passes them to the next layer. The activation layer is a process of generating clusters in the input space corresponding to contexts and a step for generating clusters of geometric shapes. For the premise layer, conditional clustering is performed in each context. The activation and premise layers are interconnected. Clusters are generated using context-based GK clustering when the contexts of the output space are given. In each context, as many clusters as selected are created, and the number of nodes summed in each output layer is similar to the number of contexts. The final output summed in the output layer is expressed as a context.