Transcript Concepts Valuation by Conjugate Moebius Inverse

Concepts Valuation by Conjugate Möebius Function
• Background
• Context, Concept and Concept Lattice
• Diversity Function
• Conjugate Möebius Inverse
• Concept Lattice Valuation
• Diversity, Weight, CMI
• Dissimilarity, hierarchy
• Splitting into hierarchy
• Basic interpretation of numbers
• Conclusion
• Next research
• References
Petr Gajdoš _ VŠB-TU Ostrava _ 2004
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Context, Concept and Concept Lattice
• Context
• Incidence matrix
Description of objects and features in incidence matrix.
Whales live in water
Petr Gajdoš _ VŠB-TU Ostrava _ 2004
C = cat
M = monkey (chimpanzee)
D = dog
F = fish (delphinus)
H = human
W = whale
q = quadrupped (four feet)
p = pilli
i = intelligence
w = live in water
h = hand
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Context, Concept and Concept Lattice
• Concept
Sample of formal concept:
Petr Gajdoš _ VŠB-TU Ostrava _ 2004
({C,M,D},{p})
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Context, Concept and Concept Lattice
• Concept lattice
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Concept Lattice Valuation
• Diversity
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Concept Lattice Valuation
• CMI
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Concept Lattice Valuation
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Concept Lattice Valuation
• Weighting by CMI
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Concept Lattice Valuation
• Dissimilarity
There are two models in Theory of Diversity. Hierarchical a more general
line model. Concept lattice are hierarchical ordered. But, weighting of concepts is
a difficult task. We can assign value to concepts only in small simly lattice because
of next condition.
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Concept Lattice Valuation
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Concept Lattice Valuation
• Splitting into hierarchies
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Splitting into hierarchies
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Basic interpretation of numbers
•
What represent the numbers (diversity, weight)
•
For example, we have a set of different people with different skills. We
are looking for teams of people (concepts), which can cover most of
required skills.
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1. We assign value to each attribute. Higher value represents more important
attribute.
2. We compute diversities of concepts = v(Ci).
3. v(Ci) / v(Ctop) … upon normalization we get a number that represents
measure of covering of skills according to their values.
We want to find „compact“ teams (concepts) whose members have
general knowledge. Compact = most of skills of pleople in the team are
shared.
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1. We assign value to each attribute. Higher value represents more important
attribute
2. We compute diversities and weights of concepts = v(Ci), (Ci)
3. v(Ci) / v(Ctop) … upon normalization we get a number that represents
measure of covering of skills according to their values.
4. (Ci) / (v(Ci) / v(Ctop))
Petr Gajdoš _ VŠB-TU Ostrava _ 2004
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Basic interpretation of numbers
C
M
D
F
H
W
2
q
x
3
p
x
x
x
x
13
11
11
6
9
5
9
0
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2
w
x
x
x
x
v(Ci )
CMDFHW
MFHW
CDM
FW
MH
CD
M
0
4
i
2
h
x
x
x
x
 (Ci ) v(Ci ) / v(Ctop )  (Ci ) /(v(Ci ) / v(Ctop ))
0
4
3
2
2
2
0
0
1
0,846153846
0,846153846
0,461538462
0,692307692
0,384615385
0,692307692
0
0
4,727272727
3,545454545
4,333333333
2,888888889
5,2
0
0
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Basic interpretation of numbers
C
M
D
F
H
W
5
q
x
4
p
x
x
x
x
3
i
2
w
x
x
x
x
1
h
x
x
x
x
v(Ci )  (Ci ) v(Ci ) / v(Ctop )  (Ci ) /(v(Ci ) / v(Ctop ))
CMDFHW
MFHW
CDM
FW
MH
CD
M
0
15
10
13
5
8
9
8
0
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0
3
4
2
1
5
0
0
1
0,666666667
0,866666667
0,333333333
0,533333333
0,6
0,533333333
0
0
4,5
4,615384615
6
1,875
8,333333333
0
0
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Conclusion
• Next research
• Input
Hierarchy of attributes
Incidence matrix
o1
o2
o3
a1
x
a2
a3
x
x
x
a4
x
x
• Output
• Evaluated, reduced concept lattice
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Conclusion
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