Dominance-Bases Rough Set Approach:
Features, Extensions and Application
Krzysztof Dembczyński
Institute of Computing Science,
Poznań University of Technology, Poland
Salvatore Greco
Faculty of Economy,
University of Catania, Italy
Roman Słowiński
Institute of Computing Science,
Poznań University of Technology, Poland
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
2
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.
3
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
4
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.
HIGH
LOW
5
The Philosophy of
Dominance-Based Rough Set Approach

Greco, Matarazzo and Słowiński have proposed DominanceBased 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
6
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
7
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
8
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
9
Preliminaries of DRSA

An Example
First 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
10
Preliminaries of DRSA

An Example
c1
+
40
BEST
+
+
+
o
o
o
o
20
0
WORST
+
+
o
- -
o
o
+
o
+
+
+
+
+
+
+
o
o
o
o
-
c2
20
40
11
Preliminaries of DRSA

Granules of Knowledge: Dominating and Dominated Sets
c1
+
40
BEST
+
+
+
o
o
o
o
20
0
WORST
+
+
o
- -
o
o
+
o
+
+
+
+
+
+
+
o
o
o
o
-
c2
20
40
12
Preliminaries of DRSA

Granules of Knowledge: Dominating and Dominated Sets
c1
+
40
BEST
+
+
+
o
o
o
o
20
0
WORST
+
+
o
- -
+
o
o
o
+
+
+
+
+
+
+
o
o
o
o
-
c2
20
40
13
Preliminaries of DRSA

Lower and Upper Approximation of the class unions
c1
+
40
BEST
+
+
+
o
o
o
o
20
0
WORST
+
+
o
- -
o
o
+
o
+
+
+
+
+
+
+
o
o
o
o
-
c2
20
40
14
Preliminaries of DRSA

Inducing of Decision Rules
c1
+
40
BEST
+
+
+
o
o
o
o
20
0
WORST
+
+
o
- -
o
o
+
o
+
+
+
+
+
+
+
o
o
o
o
-
c2
20
40
15
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
16
Preliminaries of DRSA

Inducing of Decision Rules with Hyperplanes
c1
+
40
BEST
+
+
+
o
o
o
o
20
0
WORST
+
+
o
- -
o
o
+
o
+
+
+
+
+
+
+
o
o
o
o
-
c2
20
40
17
Preliminaries of DRSA

Inducing of Decision Rules with Hyperplanes
c1
+
40
BEST
+
+
+
o
o
o
o
20
0
WORST
+
+
o
- -
o
o
+
o
+
+
+
+
+
+
+
o
o
o
o
-
c2
20
40
18
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
19
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
20
Variable-Consistency DRSA

Lower Approximation consists of limited counterexamples
controlled by
c1 pre-defined level of certainty
BEST
+
+
40
+ +
+
+
+
o
+
+
+
o
+
o
+
20
o
+
o
+
o
o
o
o
o
o
o
- c
2
0
WORST
20
40
21
Multi-Valued DRSA


Interval order
C1
C2
Decision
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
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
Form of the rules:
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
if u(x)  21 then,
x is at least High
22
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
23
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
24
Continuous Decision Criterion

What we can do?
C1
C2
D1
DC
Pre-discretization of decision
criterion
34.4
23.4
High
34.5
30.3
22.1
High
31.5
Or
25
19
High
25.4
Analyzing data with continues
decision
20
17
High
22.1
22
19.5
Medium
21.5
Large number of classes and
unions of classes?
12
25
Medium
20.1
15.5
16.8
Medium
17.4
This is more inconsistencies
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
Looking for good association
on the conditional part of the
decision table
25
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
26
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
27
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
28