# A Design of CGK-Based Granular Model Using Hierarchical Structure
## 2 Data Clustering
### Fuzzy Clustering That Considers the Output Space  
The aforementioned fuzzy clustering is a clustering that considers the features of the data in the input space. A fuzzy clustering that considers the output space generates clusters by considering both the features of the data in the input space and also the similarity and features of the data in the output space. This clustering type includes context-based fuzzy C-means (CFCM) clustering and interval-based fuzzy C-means (IFCM) clustering [36], which differ according to how the output space is divided. Figure 1. illustrates the fuzzy clustering that considers the output space. In Figure 1a. triangle-shaped contexts (fuzzy sets), which are IGs, are created in the output space, while clusters that correspond to each context are created in the input space. In Figure 1b interval-shaped IGs are created in the output space, and clusters that correspond to each interval are created in the input space. In normal fuzzy clustering, clusters are created using only the Euclidean distance between the cluster centers and the data in the input space, without considering the features of the data in the output space. However, in context-based fuzzy clustering, triangle-shaped contexts (fuzzy sets) are created in the output space using the method proposed by [PERSON] [37,38], while clusters are created via fuzzy clustering in each context; hence, clusters can be created in a more sophisticated manner than in conventional fuzzy clustering. Figure 2a presents normal fuzzy clustering, and Figure 2b shows clusters that were created in context-based fuzzy clustering by considering the features of the output space. As illustrated in Figure 2, fuzzy clustering creates clusters using the distance between the cluster centers and the data in the input space without considering the properties of  
Figure 1: Context-based fuzzy clustering and interval-based fuzzy clustering concept: (**a**) context-based fuzzy clustering; (**b**) interval-based fuzzy clustering. Figure 2: Comparison of clusters created in fuzzy clustering and context-based fuzzy clustering: (**a**) presents normal fuzzy clustering; (**b**) context-based fuzzy clustering. the data in the output space. In contrast, context-based fuzzy clustering creates clusters by considering the properties of the data in the output space; hence, it can create clusters more efficiently than conventional fuzzy clustering. The context of the data in the output space can be expressed as \(D:T\ \rightarrow\ [0,\ 1]\). D represents all of the data in the output space. Here, it is assumed that the context for the given data can be adopted. \(f_{k}=T(d_{k})\) represents the extent to which the kth data belongs in the context created in the output space. \(f_{k}\) can be a value between zero and one, and the requirements for the membership matrix are as expressed in Equation (6) owing to the aforementioned properties. \[U(f)\ =\Big{\{}eqalignu_{ik}\in\ [0,1]\ |\ \sum_{i=1}^{c}u_{ik}=f_{k}\ \forall\ k\ \text{and}\ 0<\sum_{k=1}^{N}u_{ik}<N\Big{\}} \tag{6}\]  
\[u_{ik}=\frac{f_{k}}{\sum_{j=1}^{c}\big{(}\frac{\|x_{k}-c_{j}\|}{\|x_{k}-c_{j}\| }\big{)}^{\frac{2}{m-1}}} \tag{7}\]  
The membership matrix \(U\) updated by Equation (6) can be expressed as Equation (7). Here, m is the fuzzification coefficient, and \(m=2\) is generally used. For the contexts, the output space is uniformly divided into fuzzy set forms, while the degree of membership \(f_{k}\) is obtained. Usually, the output space is divided uniformly; however, it can be divided flexibly according to a Gaussian probability distribution according to the features of the data. The sequence in which the context-based fuzzy clustering is performed is presented below. **[Step 1]** Select the number of contexts that can be expressed linguistically and the number of clusters that can be created in each context, and then initialize the membership matrix U with arbitrary values between zero and one.