Metodyka klasyfikacji i sortowania przy hierarchicznej
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Transcript Metodyka klasyfikacji i sortowania przy hierarchicznej
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