# Neighborhood Granule Classifiers
## 2 Neighborhood Granulation of a Classification System  
The rough set theory (RST) [51] proposed by [PERSON], a mathematician of Poland, is a widely used model in classification systems. In a classification system, the data with real values need to be discretized, but the process of discretization easily leads to a loss of classification information. Aiming at the limitation of Pawlak RST, a neighborhood RST is introduced to granulate these real-value data. In RST, an equivalence class is regarded as a granule. As for neighborhood RST, neighborhood granules are constructed. Suppose \(CS=(U,A,D,\delta)\) is a classification system, where \(U=\{x_{1},x_{2},...,x_{n}\}\) is a set of samples or objects; \(A=\{a_{1},a_{2},...,a_{m}\}\) is a set of condition features or attributes; \(D=\{d\}\) represents a decision feature or attribute; \(\delta\in[0,1]\) is a neighborhood parameter [44]. The data of samples in the condition feature set \(A\) are real values, and the data of samples in the decision feature \(D\) are symbolic or discrete values. Suppose \(CS=(U,A,D,\delta)\) is a classification system. For any sample \(x,y\in U\) and a feature subset \(P\subseteq A\), where \(P=\{a_{1},a_{2},...,a_{m}\}\), a distance function on the feature subset \(P\) is defined as [44]:  
\[\Delta_{P}(x,y)=(\sum_{i=1}^{m}(v(x,a_{i})-v(y,a_{i}))^{s})^{1/s}, \tag{1}\]  
where \(v(x,a_{i})\) represents the value of the sample \(x\) on a feature \(a_{i}\). If \(s=1\), it is named the Manhattan distance, and if \(s=2\), it is named the Euclidean distance. Suppose \(CS=(U,A,D,\delta)\) is a classification system. For any sample \(x\in U\) and a condition feature subset \(P\subseteq A\), the \(\delta\) neighborhood granule of \(x\) on \(P\) is formed by:  
\[g_{P}^{\delta}(x)=\{y|x,y\in U,\Delta_{P}(x,y)\leq\delta\}, \tag{2}\]  
which is called a process of neighborhood granulation. The data on decision features in classification systems are symbolic values. The samples are divided into equivalent classes on the decision features and the neighborhood parameter \(\delta=0\). Therefore, these samples are granulated into equivalent granules, which are called decision equivalent granules, and they are expressed as:  
\[g_{D}^{0}(x)=\{y|x,y\in U,\Delta_{D}(x,y)=0\}. \tag{3}\]  
It is obvious that a neighborhood granule is degenerated into an equivalent granule when the neighborhood parameter \(\delta\) is zero. Therefore, the equivalent granule is a special neighborhood granule. In a classification system, for any sample \(x\in U\) and a condition feature subset \(P\subseteq A\), it determines a neighborhood relation, which is defined as:  
\[NR_{\delta}(P)=\{(x,y)\in U\times U|\Delta_{P}(x,y)\leq\delta\}. \tag{4}\]  
The \(U/NR_{\delta}(P)\) is called a covering of the domain \(U\), which is induced by a neighborhood relation, and the covering is a collection of neighborhood granules. The neighborhood relation is a kind of similarity relation that satisfies reflexive and symmetric properties, rather than the equivalence relation. When a neighborhood parameter \(\delta\) is equal to zero, the neighborhood relation is degenerated into an equivalence relation. Therefore, the equivalence relation is a special neighborhood relation. The rough sets based on neighborhood relations can handle the real-value data, while the rough sets based on equivalence relations can only deal with the symbolic data. A classification system is \(CS=(U,A,D,\delta)\). For any sample \(x\in U\) and a feature subset \(P\subseteq A\), let \(g_{P}^{\delta}(x)\) be a neighborhood granule, then the size of granule \(g_{P}^{\delta}(x)\) is defined as:  
\[Size(g_{P}^{\delta}(x))=\frac{|g_{P}^{\delta}(x)|}{|U|}, \tag{5}\]  
where \(|.|\) represents the cardinality of a set. It is easy to know that the size of the neighborhood granule holds the property: \(\frac{1}{|U|}\leq Size(g_{P}^{\delta}(x))\leq 1\). **Example 1**.