The dominance-based rough set approach (DRSA) is an extension of rough set theory for multi-criteria decision analysis (MCDA), introduced by Greco, Matarazzo and Słowiński.
[1][2][3] The main change compared to the classical rough sets is the substitution for the indiscernibility relation by a dominance relation, which permits one to deal with inconsistencies typical to consideration of criteria and preference-ordered decision classes.
Multicriteria classification (sorting) is one of the problems considered within MCDA and can be stated as follows: given a set of objects evaluated by a set of criteria (attributes with preference-order domains), assign these objects to some pre-defined and preference-ordered decision classes, such that each object is assigned to exactly one class.
Due to the preference ordering, improvement of evaluations of an object on the criteria should not worsen its class assignment.
As an illustrative example, consider the problem of evaluation in a high school.
Each student is described by three criteria: level in Physics, Mathematics and Literature, each taking one of three possible values bad, medium and good.
Criteria are preference-ordered and improving the level from one of the subjects should not result in worse global evaluation (class).
As a more serious example, consider classification of bank clients, from the viewpoint of bankruptcy risk, into classes safe and risky.
The domains of these attributes are not simply ordered but involve a preference order since, from the viewpoint of bank managers, greater values of ROE, ROI or ROS are better for clients being analysed for bankruptcy risk .
Neglecting this information in knowledge discovery may lead to wrong conclusions.
, and that the outranking relation is a simple order between real numbers
This relation is straightforward for gain-type ("the more, the better") criterion, e.g. company profit.
For cost-type ("the less, the better") criterion, e.g. product price, this relation can be satisfied by negating the values from
For this reason, we can consider the upward and downward unions of classes, defined respectively, as: We say that
In DRSA, the knowledge being approximated is a collection of upward and downward unions of decision classes and the "granules of knowledge" used for approximation are P-dominating and P-dominated sets.
, respectively, are defined as: Analogously, the P-lower and the P-upper approximation of
, respectively, are defined as: Lower approximations group the objects which certainly belong to class union
The P-lower and P-upper approximations defined as above satisfy the following properties for all
On the basis of the approximations obtained by means of the dominance relations, it is possible to induce a generalized description of the preferential information contained in the decision table, in terms of decision rules.
The certain, possible and approximate rules represent certain, possible and ambiguous knowledge extracted from the decision table.
, related to the levels in Mathematics, Physics and Literature, respectively.
According to the decision attribute, the students are divided into three preference-ordered classes:
Thus, the following unions of classes were approximated: Notice that evaluations of objects
[1] The definitions of rough approximations are based on a strict application of the dominance principle.
However, when defining non-ambiguous objects, it is reasonable to accept a limited proportion of negative examples, particularly for large decision tables.
Such extended version of DRSA is called Variable-Consistency DRSA model (VC-DRSA)[6] In real-life data, particularly for large datasets, the notions of rough approximations were found to be excessively restrictive.
[7] Having stated the probabilistic model for ordinal classification problems with monotonicity constraints, the concepts of lower approximations are extended to the stochastic case.
Stochastic dominance-based rough sets can also be regarded as a sort of variable-consistency model.
4eMka2 Archived 2007-09-09 at the Wayback Machine is a decision support system for multiple criteria classification problems based on dominance-based rough sets (DRSA).
JAMM Archived 2007-07-19 at the Wayback Machine is a much more advanced successor of 4eMka2.