# A Design of CGK-Based Granular Model Using Hierarchical Structure
## 2 Data Clustering
### Fuzzy Clustering  
Fuzzy clustering is a method that was developed by [PERSON] and improved by [PERSON] [35], which exhibits the feature of allowing the given data to belong to two or more clusters. In non-fuzzy clustering, the given data can only belong to exactly one cluster; hence, it is divided into separate clusters. In fuzzy clustering, data can belong to two or more clusters according to the membership values. For example, a banana can be yellow or green (non-fuzzy clustering criteria, or it can be yellow and green (fuzzy clustering criteria). Here, certain parts of the entire banana can be yellow, and they can be green. The banana can belong to green (green = 1), and it can belong to yellow (yellow = 0.5) and green (green = 0.5), which is not yellow (yellow = 0). The membership values can be between zero and one, while the sum of the membership values is 1. Membership values are assigned to the given data. These membership values numerically indicate the extent to which the data belongs to each cluster. If the data has a low membership value, it can be known that it is on the edge of the cluster; conversely, if it has a high membership value, it can be deduced that it is in the center part of the cluster.  
Fuzzy clustering can be generalized by the following formulas.  
\[J_{m}=\sum_{i=1}^{N}\sum_{k=1}^{c}u_{ki}^{m}d^{2}(x_{i},\ v_{k}) \tag{1}\]  
\[\sum_{k=1}^{c}u_{ki}=1,\ \forall i\ \in\ \{1,\ 2,\ \ldots,\ N\} \tag{2}\]  
where \(X=\{x_{1},\ x_{2},\ldots,\ x_{N}\}\ \in\ R^{N\times D}\) and \(x_{i}\ \in\ R^{1\times D}\) represents the data and data items, respectively. \(N\) denotes the number of data items, and \(c\) is the number of clusters, which is \(2\ \leq c\ \leq N\). \(u_{ki}\ \in\ R\) represents the membership value of the \(k\)th \(x_{i}\), and \(m\ \in Z^{+}\) is the fuzzification coefficient for the fuzzy membership values.  
The cluster center and fuzzy membership function are obtained via an iterative process by minimizing Equation (1) according to the constraint conditions defined in Equation (2). Therefore, the objective function is modified using Lagrange multipliers and expressed as:  
\[J_{m}=\sum_{i=1}^{N}\Bigl{(}\sum_{k=1}^{c}u_{ki}^{m}d^{2}(x_{i},v_{k})+\lambda _{i}\Bigl{(}1-\sum_{k=1}^{c}u_{ki}\Bigr{)}\Bigr{)} \tag{3}\]  
where \(\lambda_{i}\) denotes the Lagrange multiplier. Therefore, the clustering problem involves identifying the cluster center set \(v^{*}=\ \{v_{k}{}^{*},\ \forall k\ \in\ \{1,\ 2,\ \ldots,\ c\}\}\) and the fuzzy membership function set \(U^{*}=\ \{u_{ki}{}^{*},\ \forall k\ \in\ \{1,\ 2,\ \ldots,\ c\},\ \forall i\ \in\ \{1,\ 2,\ \ldots,\ N\}\}\) by minimizing Equation (3). The minimization of cluster centers \(v_{k}{}^{*}\) can be obtained via Equation (4), and the minimization of the fuzzy membership functions can be obtained using Equation (5), which are expressed as:  
\[v_{k}{}^{*}=\frac{\sum_{i=1}^{N}u_{ki}^{m}x_{i}}{\sum_{i=1}^{N}u_{ki}^{m}} \tag{4}\]  
\[u_{ki}{}^{*}=\frac{1}{\sum_{j=1}^{c}\left(\frac{d^{2}(x_{i},\ v_{k})}{d^{2}(x_ {i},\ v_{j})}\right)^{\frac{1}{(m-1)}}} \tag{5}\]  
Equations (4) and (5) are computed repeatedly to obtain the final cluster centers and fuzzy membership functions.