 Dominancebased rough set approach

Dominancebased rough set approach (DRSA) is an extension of rough set theory for multicriteria decision analysis (MCDA), introduced by Greco, Matarazzo and Słowiński. ^{[1]}^{[2]}^{[3]} The main change comparing to the classical rough sets is the substitution of the indiscernibility relation by a dominance relation, which permits to deal with inconsistencies typical to consideration of criteria and preferenceordered decision classes.
Contents
Multicriteria classification (sorting)
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 preferenceorder domains), assign these objects to some predefined and preferenceordered 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. The sorting problem is very similar to the problem of classification, however, in the latter, the objects are evaluated by regular attributes and the decision classes are not necessarily preference ordered. The problem of multicriteria classification is also referred to as ordinal classification problem with monotonicity constraints and often appears in reallife application when ordinal and monotone properties follow from the domain knowledge about the problem.
As an illustrative example, consider the problem of evaluation in a high school. The director of the school wants to assign students (objects) to three classes: bad, medium and good (notice that class good is preferred to medium and medium is preferred to bad). 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 preferenceordered 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. This may involve such characteristics as "return on equity (ROE)", "return on investment (ROI)" and "return on sales (ROS)". 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 . Thus, these attributes are criteria. Neglecting this information in knowledge discovery may lead to wrong conclusions.
Data representation
Decision table
In DRSA, data are often presented using a particular form of decision table. Formally, a DRSA decision table is a 4tuple , where is a finite set of objects, is a finite set of criteria, where is the domain of the criterion and is an information function such that for every . The set is divided into condition criteria (set ) and the decision criterion (class) . Notice, that is an evaluation of object on criterion , while is the class assignment (decision value) of the object. An example of decision table is shown in Table 1 below.
Outranking relation
It is assumed that the domain of a criterion is completely preordered by an outranking relation ; means that is at least as good as (outranks) with respect to the criterion . Without loss of generality, we assume that the domain of is a subset of reals, , and that the outranking relation is a simple order between real numbers such that the following relation holds: . This relation is straightforward for gaintype ("the more, the better") criterion, e.g. company profit. For costtype ("the less, the better") criterion, e.g. product price, this relation can be satisfied by negating the values from .
Decision classes and class unions
Let . The domain of decision criterion, consist of elements (without loss of generality we assume ) and induces a partition of into classes , where . Each object is assigned to one and only one class . The classes are preferenceordered according to an increasing order of class indices, i.e. for all such that , the objects from are strictly preferred to the objects from . For this reason, we can consider the upward and downward unions of classes, defined respectively, as:
Main concepts
Dominance
We say that dominates with respect to , denoted by , if is better than on every criterion from , . For each , the dominance relation is reflexive and transitive, i.e. it is a partial preorder. Given and , let
represent Pdominating set and Pdominated set with respect to , respectively.
Rough approximations
The key idea of the rough set philosophy is approximation of one knowledge by another knowledge. 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 Pdominating and Pdominated sets.
The Plower and the Pupper approximation of with respect to , denoted as and , respectively, are defined as:
Analogously, the Plower and the Pupper approximation of with respect to , denoted as and , respectively, are defined as:
Lower approximations group the objects which certainly belong to class union (respectively ). This certainty comes from the fact, that object belongs to the lower approximation (respectively ), if no other object in contradicts this claim, i.e. every object which Pdominates , also belong to the class union (respectively ). Upper approximations group the objects which could belong to (respectively ), since object belongs to the upper approximation (respectively ), if there exist another object Pdominated by from class union (respectively ).
The Plower and Pupper approximations defined as above satisfy the following properties for all and for any :
The Pboundaries (Pdoubtful regions) of and are defined as:
Quality of approximation and reducts
The ratio
defines the quality of approximation of the partition into classes by means of the set of criteria . This ratio express the relation between all the Pcorrectly classified objects and all the objects in the table.
Every minimal subset such that is called a reduct of and is denoted by . A decision table may have more than one reduct. The intersection of all reducts is known as the core.
Decision rules
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 decision rules are expressions of the form if [condition] then [consequent], that represent a form of dependency between condition criteria and decision criteria. Procedures for generating decision rules from a decision table use an inducive learning principle. We can distinguish three types of rules: certain, possible and approximate. Certain rules are generated from lower approximations of unions of classes; possible rules are generated from upper approximations of unions of classes and approximate rules are generated from boundary regions.
Certain rules has the following form:
if and and then
if and and then
Possible rules has a similar syntax, however the consequent part of the rule has the form: could belong to or the form: could belong to .
Finally, approximate rules has the syntax:
if and and and and and then
The certain, possible and approximate rules represent certain, possible and ambiguous knowledge extracted from the decision table.
Each decision rule should be minimal. Since a decision rule is an implication, by a minimal decision rule we understand such an implication that there is no other implication with an antecedent of at least the same weakness (in other words, rule using a subset of elementary conditions or/and weaker elementary conditions) and a consequent of at least the same strength (in other words, rule assigning objects to the same union or subunion of classes).
A set of decision rules is complete if it is able to cover all objects from the decision table in such a way that consistent objects are reclassified to their original classes and inconsistent objects are classified to clusters of classes referring to this inconsistency. We call minimal each set of decision rules that is complete and nonredundant, i.e. exclusion of any rule from this set makes it noncomplete. One of three induction strategies can be adopted to obtain a set of decision rules ^{[4]}:
 generation of a minimal description, i.e. a minimal set of rules,
 generation of an exhaustive description, i.e. all rules for a given data matrix,
 generation of a characteristic description, i.e. a set of rules covering relatively many objects each, however, all together not necessarily all objects from the decision table
The most popular rule induction algorithm for dominancebased rough set approach is DOMLEM ^{[5]}, which generates minimal set of rules.
Example
Consider the following problem of high school students evaluations:

Table 1: Example—High School Evaluations object (student) q_{1}
(Mathematics)q_{2}
(Physics)q_{3}
(Literature)d
(global score)x_{1} medium medium bad bad x_{2} good medium bad medium x_{3} medium good bad medium x_{4} bad medium good bad x_{5} bad bad medium bad x_{6} bad medium medium medium x_{7} good good bad good x_{8} good medium medium medium x_{9} medium medium good good x_{10} good medium good good
Each object (student) is described by three criteria , related to the levels in Mathematics, Physics and Literatire, respectively. According to the decision attribute, the students are divided into three preferenceordered classes: Cl_{1} = {bad}, Cl_{2} = {medium} and Cl_{3} = {good}. Thus, the following unions of classes were approximated:
 i.e. the class of (at most) bad students,
 i.e. the class of at most medium students,
 i.e. the class of at least medium students,
 i.e. the class of (at least) good students.
Notice that evaluations of objects and are inconsistent, because has better evaluations on all three criteria than but worse global score.
Therefore, lower approximations of class unions consist of the following objects:
Thus, only classes and cannot be approximated precisely. Their upper approximations are as follows:
while their boundary regions are:
Of course, since and are approximated precisely, we have , and
The following minimal set of 10 rules can be induced from the decision table:
 if then
 if and and then
 if then
 if and then
 if and then
 if and then
 if and then
 if then
 if then
 if and then
The last rule is approximate, while the rest are certain.
Extensions
Multicriteria choice and ranking problems
The other two problems considered within multicriteria decision analysis, multicriteria choice and ranking problems, can also be solved using dominancebased rough set approach. This is done by converting the decision table into pairwise comparison table (PCT)^{[1]}.
Variableconsistency DRSA
The definitions of rough approximations are based on a strict application of the dominance principle. However, when defining nonambiguous objects, it is reasonable to accept a limited proportion of negative examples, particularly for large decision tables. Such extended version of DRSA is called VariableConsistency DRSA model (VCDRSA)^{[6]}
Stochastic DRSA
In reallife data, particularly for large datasets, the notions of rough approximations were found to be excessively restrictive. Therefore an extension of DRSA, based on stochastic model (Stochastic DRSA), which allows inconsistencies to some degree, has been introduced^{[7]}. Having stated the probabilistic model for ordinal classification problems with monotonicity constraints, the concepts of lower approximations are extended to the stochastic case. The method is based on estimating the conditional probabilities using the nonparametric maximum likelihood method which leads to the problem of isotonic regression.
Stochastic dominancebased rough sets can also be regarded as a sort of variableconsistency model.
Software
4eMka2 is a decision support system for multiple criteria classification problems based on dominancebased rough sets (DRSA). JAMM is a much more advanced successor of 4eMka2. Both systems are freely available for nonprofit purposes on the Laboratory of Intelligent Decision Support Systems (IDSS) website.
See also
 Rough sets
 Soft computing
 Granular computing
 Multicriteria Decision Analysis (MCDA)
References
 ^ ^{a} ^{b} Greco, S., Matarazzo, B., Słowiński, R.: Rough sets theory for multicriteria decision analysis. European Journal of Operational Research, 129, 1 (2001) 1–47
 ^ Greco, S., Matarazzo, B., Słowiński, R.: Multicriteria classification by dominancebased rough set approach. In: W.Kloesgen and J.Zytkow (eds.), Handbook of Data Mining and Knowledge Discovery, Oxford University Press, New York, 2002
 ^ Słowiński, R., Greco, S., Matarazzo, B.: Rough set based decision support. Chapter 16 [in]: E.K. Burke and G. Kendall (eds.), Search Methodologies: Introductory Tutorials in Optimization and Decision Support Techniques, SpringerVerlag , New York (2005) 475–527
 ^ Stefanowski, J.: On rough set based approach to induction of decision rules. In Skowron, A., Polkowski, L. (eds.): Rough Set in Knowledge Discovering, Physica Verlag, Heidelberg (1998) 500529
 ^ Greco S., Matarazzo, B., Słowiński, R., Stefanowski, J.: An Algorithm for Induction of Decision Rules Consistent with the Dominance Principle. In W. Ziarko, Y. Yao (eds.): Rough Sets and Current Trends in Computing. Lecture Notes in Artificial Intelligence 2005 (2001) 304313. SpringerVerlag
 ^ Greco, S., B. Matarazzo, R. Slowinski and J. Stefanowski: Variable consistency model of dominancebased rough set approach. In W.Ziarko, Y.Yao (eds.): Rough Sets and Current Trends in Computing. Lecture Notes in Artificial Intelligence 2005 (2001) 170–181. SpringerVerlag
 ^ Dembczyński, K., Greco, S., Kotłowski, W., Słowiński, R.: Statistical model for rough set approach to multicriteria classification. In Kok, J.N., Koronacki, J., de Mantaras, R.L., Matwin, S., Mladenic, D., Skowron, A. (eds.): Knowledge Discovery in Databases: PKDD 2007, Warsaw, Poland. Lecture Notes in Computer Science 4702 (2007) 164–175.
External links
 The International Rough Set Society
 Laboratory of Intelligent Decision Support Systems (IDSS) at Poznań University of Technology.
 Extensive list of DRSA references on the Roman Słowiński home page.
Categories: Theoretical computer science
 Decision theory
 Machine learning algorithms
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