Dominance-based rough approximation and knowledge reduction: a class-based approach

Abstract

The dominance-based rough set approach (DRSA) extends Pawlak’s rough set theory and has become prevailing for multicriteria decision-making. The literature has reported multiple versions of DRSA models, in which rough approximations are preserving the union of decision classes. In this paper, we propose a new type of rough approach based on the classes rather than the conventional class unions. We extend the class-based rough approximation to a series of DRSA models, including classical DRSA, variable consistency DRSA model, variable precision DRSA model, and believable rough set approach model. Besides, we explore the methods of criteria reduction under the framework of class-based rough approximation. We clarify the relations among the proposed and previous reducts in DRSA.

Appendix

Proposition 1

For the given decision table, five kinds of reducts in DRSA, including L, B, U, $L^{\diamondsuit }$, and Q reducts defined in Kusunoki and Inuiguchi (2010) are equivalent to each other.

Proof

Suppose subset $P \subseteq C$ is the $L^{\diamondsuit }$-reduct in a given decision table. Then subset $P \subseteq C$ is “the minimal condition attribute set preserving boundary regions of upward unions, as well as the minimal condition attribute set preserving boundary regions of downward unions” (Kusunoki and Inuiguchi 2010), represented as $Bn_{P} \left( {Cl_{t}^{ \ge } } \right) = Bn_{C} \left( {Cl_{t}^{ \ge } } \right)$ (for $t = 2, \ldots ,l$) and $Bn_{P} \left( {Cl_{t}^{ \le } } \right) = Bn_{C} \left( {Cl_{t}^{ \le } } \right)$ (for $t = 1, \ldots ,l - 1$). Since we also have:

$$ \gamma _{C} (CL) = \frac{{\left| {U - \left( { \cup _{{t = 2, \ldots ,l}} } \right)Bn_{c} \left( {Cl_{t}^{ \ge } } \right)} \right|}}{{|U|}} = \frac{{\left| {U - \left( { \cup _{{t = 2, \ldots ,l}} } \right)Bn_{P} \left( {Cl_{t}^{ \ge } } \right)} \right|}}{{|U|}} = \gamma _{P} (CL); $$

$$ \gamma _{C} (CL) = \frac{{\left| {U - \left( { \cup _{{t = 1, \ldots ,l - 1}} } \right)Bn_{C} \left( {Cl_{t}^{ \le } } \right)} \right|}}{{|U|}} = \frac{{\left| {U - \left( { \cup _{{t = 1, \ldots ,l - 1}} } \right)Bn_{P} \left( {Cl_{t}^{ \le } } \right)} \right|}}{{|U|}} = \gamma _{P} (CL), $$

we can say subset $P \subseteq C$ is also the minimal set affirming $\gamma _{C} (CL) = \gamma _{P} (CL)$. Then it affirms that subset $P \subseteq C$ is the $Q$-reduct as well, denoted by $L^{\diamondsuit } \Leftrightarrow Q$. According to the proof in Kusunoki and Inuiguchi (2010), we conclude: $L \Leftrightarrow B \Leftrightarrow U \Leftrightarrow L^{\diamondsuit } \Leftrightarrow Q$ as a proposition. $\square$

Proposition 2

For the given decision table, L $\beta$-reduct is the so-called ${L}^{\ge }$-reduct, as well as, H $\beta$-reduct is the so-called ${L}^{\le }$-reduct.

Proof

According to Kusunoki and Inuiguchi (2010), the $L^{ \ge }$-reduct is defined as a minimum criteria subset that fulfills $\underline{P} \left( {Cl_{t}^{ \ge } } \right) = \underline{C} \left( {Cl_{t}^{ \ge } } \right)$ for $t = 2, \ldots ,l$. For a given decision table, we have $Cl_{t}^{ \ge } - \underline{P} \left( {Cl_{t}^{ \ge } } \right) = Cl_{t}^{ \ge } - \underline{C} \left( {Cl_{t}^{ \ge } } \right)$. Based on the definition of Low boundary region, we conclude the definition of L $\beta$-reduct. Therefore, the L $\beta$-reduct is exactly the $L^{ \ge }$-reduct. In the same manner, we can prove that H $\beta$-reduct is the $L^{ \le }$-reduct. $\square$