## 2 Basic Theory  
### Basic Concepts of Neighborhood Rough Sets  
Classic rough set theory, which is based on the equivalence relation, can only handle discrete feature data. However, in practical applications where continuous feature data is commonly encountered, it usually requires preprocessing through discretization. There are various existing discretization methods, and different algorithms have varying impacts on the accuracy of rough set models. Nevertheless, regardless of the algorithm used, it inevitably leads to the problem of information loss [9; 10]. To address this issue, [PERSON] and [PERSON] et al. introduce a neighborhood-based rough set model called neighborhood rough sets [11], which directly analyzes and handles continuous data. Suppose \((U,C\cup\{d\})\) is a decision information system that contains continuous features, where \(U\) is a non-empty finite set consisting of objects \(x_{i}(i=1,2,\cdots,n)\), called the universe; \(C\) is a collection of continuous conditional features \(a_{k}(k=1,2,\cdots,m)\) describing the objects, known as the conditional feature set; and \(d\) represents discrete decision features about the objects. Let \(B\subseteq C\) be a conditional characteristic subset in the decision information system \((U,C\cup\{d\})\),\(\delta>0\). For any \(x_{i}\in U\), the neighborhood can be represented as:  
\[\delta_{B}(x_{i})=\big{\{}\,x_{j}\in U\big{|}D_{B}(x_{i},x_{j})\leq\delta\big{\}} \tag{1}\]  
where \(D_{B}(x_{i},x_{j})\) is the distance between object \(x_{i}\) and \(x_{j}\), with respect to \(B\) in \(U\). There are multiple forms of distance function \(D_{B}\), among which the most representative is the Minkowski distance series [12], which includes Manhattan distance, Euclidean distance, Chebyshev distance, etc. This article adopts the following Chebyshev distance function to construct \(D_{B}\), namely  
\[D_{B}(x_{i},x_{j})=\max_{a_{k}\in B}\big{|}a_{k}(x_{i})-a_{k}(x_{j})\big{|} \tag{2}\]  
where \(a_{k}(x_{i})\) and \(a_{k}(x_{j})\) are the eigenvalues of \(x_{i}\) and \(x_{j}\), respectively, with respect to \(a_{k}\). If \(x_{j}\in\delta_{B}(x_{i})\), then for any \(a_{k}\in B\), there is \(x_{j}\in\delta_{a_{k}}(x_{i})\), i.e.,  
\[\delta_{B}(x_{i})=\cap_{a_{k}\in B}\delta_{a_{k}}(x_{i}) \tag{3}\]  
By the neighborhood \(\delta_{B}(x)\), a neighborhood relation on the domain \(U\) of can be obtained, satisfying:  
\[\left\{ \begin{array}{l}N_{B}^{\delta}=\big{\{}\,(x_{i},x_{j})\in U^{2}\big{|} x_{j}\in\delta_{B}(x_{i})\big{\}}\\ N_{B}^{\delta}=\cap_{a_{k}\in B}N_{a_{k}}^{\delta}\end{array} \right. \tag{4}\]  
The set of all neighborhood relations guided by the feature set \(C\) is denoted as \(N=\big{\{}\,N_{B}^{\delta}\big{|}B\subseteq C\big{\}}\), and the binary tuple \((U,N)\) is referred to as the neighborhood approximation space. Given a decision system \((U,C\cup\{d\})\), \(B\subseteq C\), if \(Y\in U/\{d\}\) is divided by the decision attribute \(d\) into decision classes, then in the neighborhood approximation space \((U,N)\), the lower and upper approximation sets of \(Y\) concerning \(B\) and \(\delta\) are, respectively. \[\left\{ \begin{array}{l}\underline{N}_{B}^{\delta}(Y)=\big{\{}x\in U|\delta _{B}(x)\subseteq Y\big{\}}\\ \overline{N}_{B}^{\delta}(Y)=\big{\{}x\in U|\delta_{B}(x)\cap Y\
eq\varnothing \big{\}}\end{array} \right. \tag{5}\]  
The relative positive domains of \(d\) in \((U,N)\) with respect to \(B\) and \(\delta\) are  
\[POS_{B}^{\delta}(d)=\underset{Y\in U/\{d\}}{\cup}\underline{N}_{B}^{\delta}(Y) \tag{6}\]  
Feature selection (attribute reduction) is a core aspect of rough set theory and methodology research. Its objective is to select and construct a set containing the minimum or fewer features while maintaining the classification ability of decision information systems. Thisset is referred to as the minimal feature set or minimal reduction, to eliminate redundant features and improve classifier performance.