Petr Gajdoš _ VŠB-TU Ostrava _ 2004Page - 1 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 _ 2004Page - 2 Context, Concept and Concept Lattice Context Incidence matrix Description of objects and features in incidence matrix. C = catq = quadrupped (four feet) M = monkey (chimpanzee) p = pilli D = dog i = intelligence F = fish (delphinus) w = live in water H = human h = hand W = whale Whales live in water

Petr Gajdoš _ VŠB-TU Ostrava _ 2004Page - 3 Context, Concept and Concept Lattice Concept Sample of formal concept: ({C,M,D},{p})

Petr Gajdoš _ VŠB-TU Ostrava _ 2004Page - 4 Context, Concept and Concept Lattice Concept lattice

Petr Gajdoš _ VŠB-TU Ostrava _ 2004Page - 5 Concept Lattice Valuation Diversity

Petr Gajdoš _ VŠB-TU Ostrava _ 2004Page - 6 Concept Lattice Valuation CMI

Petr Gajdoš _ VŠB-TU Ostrava _ 2004Page - 7 Concept Lattice Valuation

Petr Gajdoš _ VŠB-TU Ostrava _ 2004Page - 8 Concept Lattice Valuation Weighting by CMI

Petr Gajdoš _ VŠB-TU Ostrava _ 2004Page - 9 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.

Petr Gajdoš _ VŠB-TU Ostrava _ 2004Page - 10 Concept Lattice Valuation

Petr Gajdoš _ VŠB-TU Ostrava _ 2004Page - 11 Concept Lattice Valuation Splitting into hierarchies

Petr Gajdoš _ VŠB-TU Ostrava _ 2004Page - 12 Splitting into hierarchies

Petr Gajdoš _ VŠB-TU Ostrava _ 2004Page - 13 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(C top ) … 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), (C i ) 3. v(Ci) / v(C top ) … upon normalization we get a number that represents measure of covering of skills according to their values. 4. (C i ) / (v(Ci) / v(C top ))

Petr Gajdoš _ VŠB-TU Ostrava _ 2004Page - 14 Basic interpretation of numbers

Petr Gajdoš _ VŠB-TU Ostrava _ 2004Page - 15 Basic interpretation of numbers

Petr Gajdoš _ VŠB-TU Ostrava _ 2004Page - 16 Conclusion Next research Input Output Evaluated, reduced concept lattice Hierarchy of attributesIncidence matrix

Petr Gajdoš _ VŠB-TU Ostrava _ 2004Page - 17 Conclusion