# Combining Fuzzy C-Means Clustering with Fuzzy Rough Feature Selection
## 4 New Method
### Reduction  
The fuzzy c-means clustering algorithm (FCM) has a wide range application and is more successful in numerous fuzzy clustering algorithms. It obtains the membership degree of every sample point to the class center through the optimization of objective function [42]. Objective function is represented by the Euclidean distance of clustering center and sample point. Solving every clustering center to the minimum of the value function of the non-similarity index. The vague generalization is:  
\[\mathrm{J}(\mathrm{U},c_{1},c_{2},\ldots,c_{c})=\sum_{i=1}^{c}l_{i}=\sum_{i=1} ^{c}\sum_{j}^{n}u_{ij}^{m}d_{ij}^{2} \tag{12}\]where \(u_{ij}\) is between 0 and 1, \(c_{i}\) is the clustering center of the fuzzy set, \(d_{ij}=\|c_{i}-x_{j}\|\) is the Euclidean distance of \(i\)th clustering center \(j\)th sample point. \(m\) is the weighted index number. To construct the Lagrangian multiplier of the constraint formula, derivate all the input parameters to make Equation (12) reach the minimum:  
\[c_{i}=\frac{\sum_{j=1}^{n}u_{ij}^{m}x_{j}}{\sum_{j=1}^{n}u_{ij}^{m}} \tag{13}\]  
\[\mu_{ij}=\frac{\sum_{k=1}^{c}{\frac{d_{ij}}{2d_{ij}}}^{2/(m-1)}}. \tag{14}\]  
The whole processors of the clustering algorithm are as in Algorithm 2. The output of FCM are centers \(c\) and membership matrix \(U\). \(U\) contains the degree of every object belonging to each center. ```
0: Matrix of membership degree \(\mathbf{U}\) initialized with random numbers between 0 and 1, and satisfied the normalization, accuracy of convergence \(\epsilon\), maximum number of iterations \(\mathbf{T}\) and weighted index \(\mathbf{m}\). 0: U
1: t = 1
2: do
3:\(\mathbf{c}_{i},\mathbf{i=1,2,\ldots,c}\) with Equation (13)
4:\(\mathbf{U}=[\mathbf{u}_{ij}]\) with Equation (14). 5: Update \(\mathbf{c}_{i}\) and \(\mathbf{U}\)
6: t\(\leftarrow\) t + 1
7:Until \(\mathbf{t=T\,or\,}\|\mathbf{U}_{t}-\mathbf{U}_{t-1}\|\leq\epsilon\)
8:Return U
```  
**Algorithm 2**Fuzzy C-Means (FCM)  
The definition of the lower approximation of fuzzy rough set is:  
\[\mu_{R_{p}X}(x)=inf_{y\in U}I\big{(}\mu_{R_{p}}(x,y),\mu_{X}(y)\big{)} \tag{15}\]  
where \(I\) is the fuzzy indicator. \(\mu_{R_{p}}(x,y)\) represents the similarity relation between \(x\) and \(y\) in the whole subsets of feature set \(P\). In order to contain only one similarity relation, we take the intersection of relations in all subsets of \(P\), where the intersection is calculated with the t-norm. \[\mu_{R_{p}}(x,y)=\bigcap\limits_{a\in P}\{\mu_{R_{a}}(x,y)\} \tag{16}\]  
where \(\mu_{R_{a}}(x,y)\) represents the similarity degree of \(x\) and \(y\) about feature \(a\). Fuzzy clustering on every feature with Algorithm 2 to get the membership degree of every object to the feature:  
\[\mu_{R_{a}}(x)=U_{FCM(a)}[x]. \tag{17}\]  
Because equivalence relations are used to model equality, fuzzy equivalence relations are commonly considered to represent approximate equality or similarity [41; 43]. We use the fuzzy equivalence relation R in the literature [44]:  
\[R(x,y)=\frac{a-a|x-y|+bmin(x,y)}{a-(a-1)|x-y|+bmin(x,y)}. \tag{18}\]  
Figure 1 shows that with different values of \(a\) and \(b\), the equation has a different form. The value of \(a\) defines the basic shape of the function. Additionally, the value of \(b\) defines the size of the opening on one side. As we use the function to describe the similarity of two objects, so we choose \(b\) equal to 0 to make sure that the function is in balance of the two objects even if they are great values or  
small values. When \(a\) goes to 0, the function performs a crisp relation that when two objects are equal the function output is 1, otherwise it is 0. Thus, we chose a small value on \(a\), because after the FCM algorithm, the difference between objects was small.