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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

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Context, Concept and Concept Lattice

• Context

• Incidence matrix

Description of objects and features in incidence matrix.

C = cat q = quadrupped (four feet)M = monkey (chimpanzee) p = pilliD = dog i = intelligenceF = fish (delphinus) w = live in waterH = human h = handW = whale

Whales live in water

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Context, Concept and Concept Lattice

• Concept

Sample of formal concept: ({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 generalline 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.• 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.• 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))

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Basic interpretation of numbers

CMDFHW 13 0 1 0MFHW 11 4 0,846153846 4,727272727CDM 11 3 0,846153846 3,545454545FW 6 2 0,461538462 4,333333333MH 9 2 0,692307692 2,888888889CD 5 2 0,384615385 5,2M 9 0 0,692307692 00 0 0 0 0

)( iCv )( iC )(/)( topi CvCv ))(/)(/()( topii CvCvC

2 3 4 2 2q p i w h

C x xM x x xD x xF x xH x xW x x

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Basic interpretation of numbers

CMDFHW 15 0 1 0MFHW 10 3 0,666666667 4,5CDM 13 4 0,866666667 4,615384615FW 5 2 0,333333333 6MH 8 1 0,533333333 1,875CD 9 5 0,6 8,333333333M 8 0 0,533333333 00 0 0 0 0

)( iCv )( iC )(/)( topi CvCv ))(/)(/()( topii CvCvC

5 4 3 2 1q p i w h

C x xM x x xD x xF x xH x xW x x

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Conclusion

• Next research• Input

• Output• Evaluated, reduced concept lattice

a1 a2 a3 a4o1 x xo2 x xo3 x x

Hierarchy of attributes Incidence matrix

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Conclusion