# Combining Fuzzy C-Means Clustering with Fuzzy Rough Feature Selection
## 3 Theoretical Background
### Fuzzy Rough Feature Selection  
The RSAR we mentioned only applies on the discrete datasets, but in real life, datasets usually contain real values and noises. Using fuzzy set theory, we can deal with this complex situation. Fuzzy mathematics is a mathematical theory in which studies utilize fuzzy phenomena. The core of fuzzy mathematics is the fuzzy set, which is different from the classical set. It has no definite elements and can only be mastered by membership function. The intersection, union, and complement operations of fuzzy sets are similar to those of classical sets, but in some cases, the general and operator may fail. The selection of the and operator should be analyzed in detail. T norm and s norm can be thought of as generalized operations, but they are not actually. This question remains uncertain. **Definition 6 (t-norm). **_A triangular norm or shortly t-norm better reflects the nature of the logic operator and. T-norm is a binary function \(T\) on \([0,1]\), which satisfies the exchange law, associative law, monotonicity, and the boundary condition. That is to say \(T:[0,1]\times[0,1]\rightarrow[0,1],\forall x,y,z\in[0,1]\) :_  
\[T(x,y) =T(y,x)\] \[T(x,T(y,z)) =T(T(x,y),z)\] \[y\leq z,T(x,y) \leq T(x,z)\] \[T(x,1)=x.\] \[There are some frequently used t-norms\otimes\] : _The standard min operator \[x\otimes y=min\{x,y\}\] The algebraic product \[x\otimes y=xy\] \[Lukasiewicz t-norm\ x\otimes y=max\{x+y-1,0\}.\]_  
**Definition 7 (s-norm). **_S-norm is also called a triangular conorm and shortly t-conorm. S-norm is a binary function S on \([0,1]\), which satisfies the exchange law, associative law, monotonicity, and the boundary condition. That is to say \(S:\ [0,1]\times[0,1]\rightarrow[0,1],\forall x,y,z\in[0,1]\) :_  
\[S(x,y) =S(y,x)\] \[S(x,S(y,z)) =S(S(x,y),z)\] \[T(x,T(y,z)) =T(T(x,y),z)\]0 \[T(x,T(y,z)) =T(T(x,y),z)\]1 _Three well known s-norms \(\oplus\) are:__The standard max operator_ \(x\oplus y=max\{x,y\}\)__  
_The probabilistic sum_ \(x\oplus y=x+y-xy\)__  
_The bounded sum_ \(x\oplus y=min\{x+y,1\}\)_._  
The fuzzy rule can be expressed as \"if \(A\) then \(B\)\", shortly \(A\to B\). \(A\) and \(B\) are fuzzy sets, the true degrees are expressed as \(A(x)\), \(B(y)\). The true degree of \(A\to B\) is expressed as \(A(x)\to B(y)\); the degree of the proposition depends on the true degree of the former and the latter. **Definition 8** (Fuzzy Implicator). : _Find a binary function \(R:[0,1]\times[0,1]\rightarrow[0,1]\) to represent the fuzzy rules properly. Meet left monotone (or right monotone)_  
\(R(1,0)=0,R(1,1)=R(0,1)=R(0,0)=1\)_._  
_The frequently used indicators are:_  
_Mandani implicator:_ \(x\to y=x\wedge y\)__  
_Zadeh implicator:_ \(x\to y=(1-x)\vee(x\wedge y)\)__  
_Kleene-Dienes implicator:_ \(x\to y=(1-x)\vee(x\wedge y)\)__  
_Lukasiewicz implicator:_ \(x\to y=min\{1,1-x+y\}\)_._  
Fuzzy upper approximation and lower approximation are defined with the fuzzy division of input [39] or with the following definition:  
\[T(x,T(y,z)) =T(T(x,y),z)\]2  
where \(I\) is the fuzzy implicator and \(R_{P}\) represents the fuzzy similar relation in the subset of feature set \(P\). \[T(x,T(y,z)) =T(T(x,y),z)\]3  
where \(\mu_{R_{a}}(x,y)\) represents the degree of similarity of \(x\) and \(y\) about feature \(a\). \[T(x,T(y,z)) =T(T(x,y),z)\]4  
The fuzzy positive region and dependency are defined as in rough set [40]. \[T(x,T(y,z)) =T(T(x,y),z)\]5  
\[T(x,T(y,z)) =T(T(x,y),z)\]6  
An example with the dataset in Table 1 follows. There are six objects, features (condition attribute) \(a,b\) and \(c\), label (decision attribute) \(q\). Assuming \(C=\{a,b,c\},D=\{q\},IND(q)=\{\{2,4,5\},\{1,3,6\}\}\), using Equation (6), we can obtain the fuzzy similarity matrixes as follows:  
\[U_{R_{a}}(x,y)=\left( \begin{array}{cccccc}1.0&1.0&0.987&0.013&0.006&0.006\\ 1.0&1.0&0.987&0.013&0.006&0.006\\ 0.987&0.987&1.0&0.026&0.019&0.019\\ 0.013&0.013&0.026&1.0&0.993&0.993\\ 0.006&0.006&0.019&0.993&1.0&1.0\\ 0.006&0.006&0.019&0.993&1.0&1.0\\ \end{array} \right)\]\[R_{b}(x,y)=\left(