# Neighborhood Rough Fuzzy Penetration Control Method with Variable Precision Based on GMAW
## 2 Materials and Methods  
_Theory and Method of Variable Precision Neighborhood Rough Sets_  
The classical rough set is suitable for dealing with discrete datasets, but the discretization procedure alters the original properties of the attributes, which can have an impact on knowledge reduction. The neighborhood rough set (Neighborhood-RS) is proposed to solve this problem. By combining variable precision rough set (VPRS) with neighborhood rough set (Neighborhood-RS), a variable precision neighborhood rough set (VPNRS) knowledge modeling method is proposed. Here are the concepts of \(\delta\)-neighborhood, upper and lower approximations, dependence degree, attribute reduction, and attribute importance in the variable precision neighborhood rough set. **Definition 1** ([15]). : _Define an \(N\)-dimensional space \(\Omega\) where \(\Delta\): \(R^{N}\times R^{N}\to R\), then \(\Delta\) is said to belong to a metric on \(R^{N}\), if \(\Delta\) satisfies:_  
1. \(\Delta(x_{1},x_{2})\geq 0,\Delta(x_{1},x_{2})=0\)_, if and only if_ \(x_{1}=x_{2},\forall x_{1},x_{2}\in R^{N}\)_;_
2. \(\Delta(x_{1},x_{2})=\Delta(x_{2},x_{1}),x_{1},x_{2}\in R^{N}\)_;_
3. \(\Delta(x_{1},x_{3})\leq\Delta(x_{1},x_{2})+\Delta(x_{2},x_{3}),\forall x_{1}, x_{2},x_{3}\in R^{N}\)_._  
We call \(<\Omega,\Delta>\) the metric space, and the distance on the metric space can be expressed as follows: \(\Delta(x_{i},x_{j})=\left(\sum_{k=1}^{N}\left|x_{ik}-x_{jk}\right|_{p}\right)^ {1/p}\). If \(p=1\), \(\Delta\) is the Hamiltonian distance. For \(p=2\), \(\Delta\) is the Euclidean distance. If \(p=\infty\), \(\Delta\) is expressed as Chebyshev distance. **Definition 2** ([15]). : _A nonempty finite set \(U=\{x_{1},x_{2},\ldots,x_{n}\}\) on a metric space \(\Omega\), whose \(\delta\)-neighborhood is denoted by: \(\delta(x_{i})=\{x|x\in U,\Delta(x,x_{i})\leq\delta\}\), where \(x\in U\), \(\delta\geq 0\)._  
**Definition 3** ([16]). : _Let \(A\) and \(B\) be two nonempty finite sets, and if \(I(A,B)=\text{card}(A\cap B)/\text{card}(A)\) is denoted, then \(I(A,B)\) is the degree to which \(A\) is contained in \(B\)._  
**Definition 4** ([16]). : _Given a decision system \(<\)\(U,A,D\)\(>\), where \(U\) is the sample set, \(A\) is the condition attribute, and decision attribute \(D\) divides \(U\) into \(n\) equivalence classes: \(X_{1},X_{2},\ldots,X_{n}\), \(\forall B\subseteq A\) and generates the neighborhood relation \(n_{B}\) on \(U\) with \(\beta\in(0.5,1]\), then the upper and lower approximations of decision \(D\) with respect to attribute subset \(B\) are respectively:_  
\[ \begin{array}{l}\overline{\text{N}_{B}^{\beta}}(D)=\cup_{i=1}^{n}\overline{ n_{B}}(X_{i})=\{x_{i}|I(\delta_{B}(x_{i}),X_{i})>1-\beta,x_{i}\in U\},\\ \underline{\text{N}_{B}^{\beta}}(D)=\cup_{i=1}^{n}\underline{n_{B}}(X_{i})=\{ x_{i}|I(\delta_{B}(x_{i}),X_{i})\geq\beta,x_{i}\in U\}.\end{array}  \tag{1}\]  
_Similar to the definition of classical rough set, the lower approximation \(N_{B}^{\beta}(D)\) of attribute subset \(B\) is the positive region \(POS_{B}^{\beta}(D)\), and the irrelevant region of attribute subset \(B\) is the negative region._  
**Definition 5** ([2]). : _Define the decision system \(<\)\(U,A,D\)\(>\), where \(U\) is the set of samples, and the dependency degree of decision attribute \(D\) with respect to condition attribute \(B=A\) is defined as follows:_  
\[\gamma_{B}(D)=\frac{\text{card}(POS_{B}(D))}{\text{card}(U)} \tag{2}\]  
_It follows that \(\gamma_{B}(D)\) is monotone, and if \(B_{1}\subseteq B_{2}\subseteq\cdots\subseteq A\), then \(\gamma_{B_{1}}(D)\leq\gamma_{B_{2}}(D)\leq\cdots\leq\gamma_{A}(D)\)._  
**Definition 6** ([2]). : _Given a decision system \(<\)\(U,A,D\)\(>\), \(B\subseteq A\) and \(a\in B\), we call \(B\) an attribute reduction of \(A\) that satisfies the following two conditions:_  
1. \(\gamma_{B}(D)=\gamma_{A}(D)\)_._
2. \(\forall a\in B,\gamma_{B-a}(D)<\gamma_{B}(D)\)_._  
**Definition 7** ([2]).