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Dominance-Bases Rough Set Approach: Features, Extensions and Application
Krzysztof Dembczyński Institute of Computing Science,Poznań University of Technology, Poland
Salvatore GrecoFaculty of Economy, University of Catania, Italy
Roman SłowińskiInstitute of Computing Science,Poznań University of Technology, Poland
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Topics
Philosophy of Dominance-Based Rough Set Approach (DRSA)
Preliminaries of DRSA
Extensions of DRSA
Variable Consistency DRSA
Multi-Valued DRSA
Continuous Decision Criterion and DRSA
Conclusion
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The Philosophy of Dominance-Based Rough Set Approach
The aim of the decision analysis is to answer two questions:
To explain decisions in terms of the circumstances in which they were made.
To give a recommendation how to make a good decision under specific circumstances.
One of decision problems is the multicriteria sorting
Multicriteria sorting concerns an assignment of the objects to pre-defined classes (concepts) that are preference-ordered.
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The Philosophy of Dominance-Based Rough Set Approach
Analyzed objects are described using criteria
Criteria are attributes with preference-ordered domain
Decision criterion shows the class of any object
Multicriteria decision problem has no solution unless a preference model is defined
Functional
Relational
Decision rules
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The Philosophy of Dominance-Based Rough Set Approach
Data are very often inconsistent with dominance principle that requires that an object having a better (not worse) evaluation on considered criteria cannot be assigned to a worse class.
H I G H
L O W
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The Philosophy of Dominance-Based Rough Set Approach
Greco, Matarazzo and Słowiński have proposed Dominance-Based Rough Set Approach
The Classical Rough Set Approach, proposed by Pawlak, has been proved as excellent tool for data analysis, however, it was falling for multicriteria sorting problem
The analyzed objects may be considered only in the perspective of available information
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The Philosophy of Dominance-Based Rough Set Approach
The rough set approaches features:
Information has granular structure
Approximation of one knowledge by another knowledge
Analysis of uncertain and inconsistent data
Inducing of “if…, then” decision rules
In DRSA the set of decision rules plays a role of comprehensive preference model
The rules syntax is concordant with Dominance Principle
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Topics
Philosophy of Dominance-Based Rough Set Approach (DRSA)
Preliminaries of DRSA
Extensions of DRSA
Variable Consistency DRSA
Multi-Valued DRSA
Continuous Decision Criterion and DRSA
Conclusion
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Preliminaries of DRSA
Basic notions
Outranking relation
x is at least so good as y with respect to criterion q
Dominance relation (reflexive and transitive)
x dominates y when on all criteria x outranks y (x is at least so good then y)
Data are often presented as a table
Because of preference order of classes it is possible to consider upward and downward unions of classes
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Preliminaries of DRSA
An ExampleFirst Criterion Second Criterion Decision Criterion
34.4 23.4 High
30.3 22.1 High
25 19 High
20 17 High
22 19.5 Medium
12 25 Medium
15.5 16.8 Medium
17.1 17.6 Medium
8.9 10.1 Low
11 13.5 Low
9 7 Low
12.5 4 Low
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Preliminaries of DRSA
An Example
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Preliminaries of DRSA
Granules of Knowledge: Dominating and Dominated Sets
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Preliminaries of DRSA
Granules of Knowledge: Dominating and Dominated Sets
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Preliminaries of DRSA
Lower and Upper Approximation of the class unions
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Preliminaries of DRSA
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Inducing of Decision Rules
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Preliminaries of DRSA
Form of Decision Rules
if f(x, c1) 25 and f(x, c2) 19, then x is at least High
if f(x, c1) 20 and f(x, c2) 17, then x could be at least High
if f(x, c1) 20 and f(x, c2) 17 and f(x, c1) 22 and f(x, c2) 19.5,
then x belongs to High or Medium
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Preliminaries of DRSA
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Inducing of Decision Rules with Hyperplanes
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Preliminaries of DRSA
Inducing of Decision Rules with Hyperplanes
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Preliminaries of DRSA
Features
Analysis of multicriteria sorting problems with inconsistent information
It is possible to analyze objects described by criteria and regular attributes
Continuous domain of criteria (discretization is not needed)
Sorting of new objects
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Topics
Philosophy of Dominance-Based Rough Set Approach (DRSA)
Preliminaries of DRSA
Extensions of DRSA
Variable Consistency DRSA
Multi-Valued DRSA
Continuous Decision Criterion and DRSA
Conclusion
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Variable-Consistency DRSA
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Lower Approximation consists of limited counterexamples controlled by pre-defined level of certainty
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Multi-Valued DRSA
C1 C2 Decision
34.4 23.4 High
30.3 22.1 High
25-21 19 High
20-17 17 High
18-15 19.5 Medium
12 25 Medium
15.5 16.8 Medium
17.1 17.6 Medium
8.9 10.1 Low
11 13.5 Low
9 7 Low
12.5 4 Low
Interval order
object x is not worse than
y with respect to a single criterion, if there exist a value describing x that is not worse than at least one value describing y
Form of the rules:
if u(x) 21 then,
x is at least High
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Extensions of DRSA
VC-DRSA and MV-DRSA are only examples of extensions of DRSA.
Another example is the methodology that allows deal with missing values
There exist different strategies of induction of decision rules
It is also possible to induces decision trees using rough approximations
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Topics
Philosophy of Dominance-Based Rough Set Approach (DRSA)
Preliminaries of DRSA
Extensions of DRSA
Variable Consistency DRSA
Multi-Valued DRSA
Continuous Decision Criterion and DRSA
Conclusion
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Continuous Decision Criterion
C1 C2 D1 DC
34.4 23.4 High 34.5
30.3 22.1 High 31.5
25 19 High 25.4
20 17 High 22.1
22 19.5 Medium 21.5
12 25 Medium 20.1
15.5 16.8 Medium 17.4
17.1 17.6 Medium 16
8.9 10.1 Low 10.7
11 13.5 Low 9.5
9 7 Low 4.3
12.5 4 Low 3.5
What we can do?
Pre-discretization of decision criterion
Or
Analyzing data with continues decision
Large number of classes and unions of classes?
This is more inconsistencies
Looking for good association on the conditional part of the decision table
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Continuous Decision Criterion
Decision Rules
if f(x, c1) 34.4, then x is at least 34.5
if f(x, c2) 25, then x is at least 25.4
if f(x, c1) 20, then x is at least 21.5
if f(x, c1) 8.9, then x is at least 4.3
if f(x, c1) 17.1, then x is at most 20.1
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Topics
Philosophy of Dominance-Based Rough Set Approach (DRSA)
Preliminaries of DRSA
Extensions of DRSA
Variable Consistency DRSA
Multi-Valued DRSA
Discussion about Continuous Decision Criterion and DRSA
Conclusion
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Conclusion
It is proven that:
The preference model in the form of rules derived from examples is more general then the classic functional or relational model and it is more understandable for the users because of its natural syntax.
It fulfils both explanation and recommendation tasks that are principal aims of decision analysis.
DRSA is still developing
DRSA in the Malaria Vulnerability Case Study in IIASA during YSSP
