## 2 Background and Related Work
### Interrelationship Mining Theory  
Interrelationship mining, an extension of rough set-based data mining, was first proposed by Kudo et al [25]. It extends the domain of comparison of attributes from a value set, \(AV_{a}\) of attribute \(a\) of the Cartesian product \(AV_{a}\times AV_{b}\) with another attribute \(b\), and makes it possible to extract features with respect to a comparison of values of two different attributes. Herein, we provide some brief definitions and refer interested reader to [26; 27; 28] for more information about interrelationship mining.  
Let \(a,b\in AT\) be any attributes of a given information table, and \(R\subseteq AV_{a}\times AV_{b}\) be any binary relations. We recall that attributes \(a\) and \(b\) are interrelated by \(R\) if and only if there exists an item \(i_{x}\in I\) such that \((\rho(x,a),\rho(x,b))\in R\) holds, where \(\rho(x,a)\) denotes the value of item \(i_{x}\) at the attribute \(a\). We denote the set of objects that those values of attributes \(a\) and \(b\) that satisfy the relation \(R\) as follows:  
\[R(a,b)\triangleq\{i_{x}\in I\mid\rho(x,a),\rho(x,b))\in R\}, \tag{3}\]  
and we call the set \(R(a,b)\) the support set of the interrelation between \(a\) and \(b\) by \(R\).  
To evaluate interrelationships between different attributes, Kudo et al. [23] proposed an approach for constructing new binary relation by using given binary relations between values of different  
attributes is not considered. Thus, to extract the characteristic between different attributes using the given binary relations \(\mathcal{R}_{AT}\), the extended binary relations \(\mathcal{R}_{int}\) can be defined as:  
\[\mathcal{R}_{int}=\mathcal{R}_{AT}\cup\Big{\{}\{=\}\|\text{For each }a_{i}Ra_{j}\}\cup\Big{\{}\{ \big{\{}R_{a_{i}\times a_{j}}\big{\}}\Big{|}\exists a_{i},a_{j}\in AT\Big{\}}, \tag{6}\]  
where \(\Big{\{}R_{a_{i}\times a_{j}}\big{\}}=\Big{\{}R_{a_{i}\times a_{j}}^{1},\dots, R_{a_{i}\times a_{j}}^{n_{i}}\Big{\}}\) refers to the new binary relation set for attribute values of \(a_{i}\) and \(a_{j}\). Theoretically, the number of new binary relation can be unlimited. Based on extended binary relations, the interrelated attribute set \(AT_{int}\) can be expressed as:  
\[AT_{int}=\Big{\{}a_{i}Ra_{j}\Big{\}}\exists R\in\Big{\{}R_{a_{i}\times a_{j}} \Big{\}},R\big{(}a_{i},a_{j}\big{)}\
eq\varnothing\Big{\}}, \tag{7}\]  
Further, we explore three types of the binary relation \(\Big{\{}<_{le},=_{eq},>_{mo}\Big{\}}\) for each pair of attributes \(a_{i},a_{j}\in AT\), and the values of three newly defined interrelated attributes \(a_{i}<_{le}a_{j}\), \(a_{i}=_{eq}a_{j}\), and \(a_{i}>_{mo}a_{j}\) are defined as follows:  
\[\rho_{int}\Big{(}x,a_{i}<_{le}a_{j}\Big{)}=\left\{\begin{array}{c}1,\ \rho(x,a_{i})<\rho \Big{(}x,a_{j}\Big{)},\\ 0,\rho(x,a_{i})\geq\rho\Big{(}x,a_{j}\Big{)}.\end{array}\right. \tag{8}\]  
\[\rho_{int}\Big{(}x,a_{i}=_{eq}a_{j}\Big{)}=\left\{\begin{array}{c}1,\ \rho(x,a_{i})=\rho\Big{(}x,a_{j}\Big{)}=1,\\ 0,\quad\text{otherwise}.\end{array}\right. \tag{9}\]  
\[\rho_{int}\Big{(}x,a_{i}>_{mo}a_{j}\Big{)}=\left\{\begin{array}{c}1,\ \rho(x,a_{i})>\rho \Big{(}x,a_{j}\Big{)},\\ 0,\rho(x,a_{i})\leq\rho\Big{(}x,a_{j}\Big{)}.\end{array}\right. \tag{10}\]  
In this paper, we consider the interrelated attributes \(a_{i}Ra_{j}\) of \(a_{i}\) and \(a_{j}\) if and only if \(i<j\) holds. From Equations (8)-(10), we discover that \(\rho_{int}\Big{(}x,a_{i}<_{le}a_{j}\Big{)}=1\) implies that item \(i_{x}\) has attribute \(a_{j}\), but does not have attribute \(a_{i}\); \(\rho_{int}\Big{(}x,a_{i}=_{eq}a_{j}\Big{)}=1\) implies that item \(i_{x}\) owns both attribute \(a_{i}\) and attribute \(a_{j}\); and \(\rho_{int}\Big{(}x,a_{i}>_{mo}a_{j}\Big{)}=1\) implies that item \(i_{x}\) owns attribute \(a_{i}\), but does not own attribute \(a_{j}\).