# Incorporating Grey Total Influence into Tolerance Rough Sets for Classification Problems
## 2 Tolerance Rough Sets
### Traditional Similarity Measure  
Let \(m\) and \(n\) denote the numbers of patterns and attributes, respectively. \(\textbf{x}_{i}\) and \(\textbf{x}_{j}\) (\(1\leq i,j\leq m\)) are some objects in \(U\). A simple distance function can be defined by a similarity measure \(S_{a}(\textbf{x}_{i},\textbf{x}_{j})\) to measure the closeness between \(a(\textbf{x}_{i})\) and \(a(\textbf{x}_{j})\) as in [14; 15]:  
\[S_{a}(\textbf{x}_{i},\textbf{x}_{j})\ =1-\frac{\left|a(\textbf{x}_{i})-a(\textbf{x}_ {j})\right|}{d_{max}} \tag{6}\]  
where \(a(\textbf{x}_{i})\) and \(a(\textbf{x}_{j})\) are attribute values of \(\textbf{x}_{i}\) and \(\textbf{x}_{j}\), respectively, in \(V_{a}\), and \(d_{max}\) is the maximum value among \(\left|\,a(\textbf{x}_{i})\ -\ a(\textbf{x}_{j})\,\right|.\)\(d_{max}\) can be replaced by \((\max_{a}-\min_{a})\)[17], where \(\max_{a}\) and \(\min_{a}\) are the maximum and minimum values, respectively, of the domain interval of \(a\)[12]. Then, \(\textbf{x}_{i}\) and \(\textbf{x}_{j}\) are similar with respect to \(a\) when \(S_{a}(\textbf{x}_{i},\textbf{x}_{j})\geq t_{a}\), where \(t_{a}\in[0,1]\) is the similarity threshold with respect to \(a\). The tolerance relation \(R_{a}\) has a relation with \(S_{a}(\textbf{x}_{i},\textbf{x}_{j})\):  
\[a(\textbf{x}_{i})\ R_{a}\ a(\textbf{x}_{j})\Leftrightarrow S_{a}(\textbf{x}_{ i},\textbf{x}_{j})\geq t_{a} \tag{7}\]  
This means that \(\textbf{x}_{i}\) and \(\textbf{x}_{j}\) are similar with respect to attribute \(a\) when \(a(\textbf{x}_{i})R_{a}\ a(\textbf{x}_{j})\). \(S_{A}(\textbf{x}_{i},\textbf{x}_{j})\), an overall similarity measure, can be further defined as:  
\[S_{A}(\textbf{x}_{i},\textbf{x}_{j})\ =\ \frac{\sum\limits_{a\in A}S_{a}(\textbf{x}_{ i},\textbf{x}_{j})}{\left|A\right|} \tag{8}\]  
where \(\left|\,A\,\right|\) is the number of attributes in \(A\), and \(\left|\,A\,\right|=n\) here. [PERSON] and [PERSON] [14] used \(\textbf{x}_{i}\ \tau_{A}\ \textbf{x}_{j}\) to denote the above similarity between objects \(\textbf{x}_{i}\) and \(\textbf{x}_{j}\) with respect to all attributes \(A\). As a result, the tolerance relation \(\tau_{A}\) can be related to \(S_{A}\) as  
\[\textbf{x}_{i}\ \tau_{A}\ \textbf{x}_{j}\Leftrightarrow S_{A}(\textbf{x}_{i}, \textbf{x}_{j})\geq t_{A} \tag{9}\]  
where \(t_{A}\)_in_ [0, 1] is a similarity threshold based on \(A\).  
Patterns that have a tolerance relation with \(\textbf{x}_{i}\) form a tolerance class \(TC(\textbf{x}_{i})\) of \(\textbf{x}_{i}\):  
\[TC(\textbf{x}_{i})=\{\textbf{x}_{j}\in U\,|\,\textbf{x}_{i}\ \tau_{A}\ \textbf{x}_{j}\} \tag{10}\]  
\(X\) can be approximated by the lower (\(\underline{\tau_{A}}X\)) and upper approximations (\(\overline{\tau_{A}}X\)). For subset approximations, \(\underline{\tau_{A}}X\) and \(\overline{\tau_{A}}X\) are defined as [26]:  
\[\underline{\tau_{A}}X=\cup\{TC(\textbf{x})|\textbf{x}\in U,TC(\textbf{x}) \subseteq X\} \tag{11}\]  
\[\overline{\tau_{A}}X=\cup\{TC(\textbf{x})|\textbf{x}\in U,TC(\textbf{x})\cap X \
eq\varphi\} \tag{12}\]  
For concept approximations, \(\underline{\tau_{A}}X\) and \(\overline{\tau_{A}}X\) are defined as:  
\[\underline{\tau_{A}}X=\cup\{TC(\textbf{x})|\textbf{x}\in X,TC(\textbf{x}) \subseteq X\} \tag{13}\]  
\[\overline{\tau_{A}}X=\cup\{TC(\textbf{x})|\textbf{x}\in X,TC(\textbf{x})\cap X \
eq\varphi\} \tag{14}\]  
The tuple \(\langle\underline{\tau_{A}}X\), \(\overline{\tau_{A}}X\rangle\) is the tolerance rough set. The main difference between these two approximations is associated with objects belonging to \(U\) or \(X\).