1. ADBIS 20021 Navigation Through Query Result Using Concept Order Tomáš Skopal, Václav Snášel, Daniela Ďuráková Department of Computer Science FEI, VŠB-Technical University of Ostrava Czech Republic www.cs.vsb.cz/arg arg@cs.vsb.cz

2. ADBIS 20022 Contents Motivation Example Application of scaled context Application of  -cut concept order Conclusion

3. ADBIS 20023 Motivation SQL like specified query Query result – a (possibly large) set of objects having acceptable parameters, not structured Unsatisfactory for human decision Navigation in the query result offers: How to find the best object How to compare objects Realized using hierarchical structures – ordered sets

4. ADBIS 20024 Possibility of Web Searching

5. ADBIS 20025 Example We are looking for an Alpine ski center that is near to Prague, is situated on the highest elevation and offers inexpensive ski-pass. Our requirements can be expressed by a query as follows: distance from Prague < 600 km (d) price of ski-pass < 5200 CZK (s) elevation of pistas > 1500 m a. s. (e)

6. ADBIS 20026 Numeric Values of Query Result Ski centerAbb.d (km)s (CZK)e (m) MayrhofenMa48351963250 SöldenSo50650648663260 KitzbühelKi46547412000 FlattachFl49044113125 SöllSl46036641835 Zell am SeeZe48246323029 RadstadtRa45046252130 GosauGo39037741600 RohrmoosRo42645652700

7. ADBIS 20027 Characteristics of Query Result Set of objects with parameters Quality of an object can is given by a relevance of values of the object’s parameters to the query specification. Types of object’s parameter - boolean values (without problem) - numeric values Formal Concept Analysis

8. ADBIS 20028 Context Context is a triple (O, A, I), where O is a set of objects and A is a set of attributes and a relation I  O  A I  O  A limbswingsmammal birdXX antX snake dolphinX

9. ADBIS 20029 Concept Formal concept of context (O, A, I), is a pair (Q, T) where Q  O, T  A, Q’=T and T’ = Q Ilimbswingsmammal birdXX antX snake dolphinX C1 = {bird - limbs, wings} C2 = {bird, ant - limbs}

10. ADBIS 200210 Concept Lattice Concept lattice is a set of concepts ordered by inclusion on attributes (or by inverted inclusion on objects) C0 = {bird, ant, snake, dolphin - no attribute} Cn = {no object - limbs, wings, mammal } C2 = {bird, ant - limbs} C1 = {bird - limbs, wings} C3 = {dolphin - mammal}

11. ADBIS 200211 Numeric Attributes How to convert many-valued query result table into context table? Attribute scaling – concept lattice of scaled context Our method – concept order

12. ADBIS 200212 Scaled Context distance (d)  price (s)  elevation (e) > 5504904704404.84.54.44.02.02.53.03.2 Maxx xxxx So xxxx Kixxxx Flxxxxxx Slxxxxxxx Zexxx xxx Raxxxx x Goxxxxxxxx Roxxxxxxxx

13. ADBIS 200213 of ordinally scaled context containing 36 concepts Concept Lattice objects satisfying all attributes attributes belonging to all objects Objects: Ma,Fl,Ze,Ro Attributes: d1, e3

14. ADBIS 200214 Drawbacks of Context Scaling Volume of scaled lattice is very large exponential dependence on number of attributes Choice of scale user dependent Our goal was to reduce the number of concepts and to design a scale independent structure

15. ADBIS 200215 Our Solution Creation of new context with fuzzy values The “fuzzy context” is transformed back to several crisp  -cut contexts Concept lattice for every  -cut context, called  -cut concept lattice, is produced Some concepts occur repeatedly in  -cut concept lattices – we denote this concepts as significant ones

16. ADBIS 200216 How Can We Get the “Fuzzy Context”? Fuzzyfication is a transformation of numerical values into interval  0,1 . Membership function is linear Upper and lower bounds of the membership function are the maximum/minimum values of the particular query parameter

17. Example of “Fuzzyfication”

18. ADBIS 200218 Fuzzy Context Ski Centerdse Mayrhofen0.580.000.85 Sölden0.120.090.86 Kitzbühel0.670.170.12 Flattach0.550.390.78 Söll0.700.890.02 Zell am See0.590.240.72 Radstadt0.750.250.19 Gosau1.000.820.00 Rohrmoos0.870.290.53

19. ADBIS 200219  -cut Context Let K is a fuzzy context then  K = {x  X, K(x)   } is an  -cut context for    0,1  We obtain a set of  -concepts for this context

20. ADBIS 200220  -Concept Order We produce a new structure as a union of all  -concepts The  -concepts are unified to multiset U on equality of the object set of the  -concept Repetition of an attribute in  -concept denotes a more significant concept The multiset U can be ordered according to inclusion on objects

21. ADBIS 200221 dse Maxx Sox Kix Flxx Slxx Zexx Rax Goxx Coxx  -cut  = 0.4 d:1 d:1,s:1 Sl, Go e:1 d:1,e:1 d:1,s:1,e:1 Ma,So,Ki,Fl,Sl,Ze,Ra,Go,Ro Ma,Ki,Fl,Sl,Ze,Ra,Go,Ro Ma,So,Fl,Ze,Ro Ma,Fl,Ze,Ro

22. ADBIS 200222 dse Maxx Sox Kix Flxx Slxx Zexx Rax Goxx Roxx  -cut  = 0.5 d:1 d:2,s:2 Sl,Go e:1 d:1,e:1 d:1,s:1,e:1 Fl Ma,Fl,Ze,Ro d:1,s:1,e:1 Ma,Ki,Fl,Sl,Ze,Ra,Go,Ro Ma,So,Fl,Ze,Ro Ma,So,Ki,Fl,Sl,Ze,Ra,Go,Ro

23. ADBIS 200223 dse Maxx Sox Kix Flx Slxx Zexx Rax Goxx Rox Ma,So,Ki,Fl,Sl,Ze,Ra,Go,Ro  -cut  = 0.6 d:1 d:2,s:2 Sl,Go e:1 d:1,e:1 Ma,Fl,Ze,Ro d:2,s:2,e:2 Ma,So,Fl,Ze,Ro Ma,So,Fl,Ze e:1 Ma,Ki,Fl,Sl,Ze,Ra,Go,Ro Ma,Ki,Sl,Ze,Ra,Go,Ro d:1 d:1,e:1 Ma,Ze

24. ADBIS 200224 dse Max Sox Kix Flx Slxx Zex Rax Goxx Rox  -cut  = 0.7 Ma,So,Ki,Fl,Sl,Ze,Ra,Go,Ro d:1 d:3,s:3 Sl,Go e:1 d:1,e:1 Ma,Fl,Ze,Ro Ma,So,Fl,Ze,Ro Ma,So,Fl,Ze e:2 Ma,Ki,Fl,Sl,Ze,Ra,Go,Ro Ma,Ki,Sl,Ze,Ra,Go,Ro d:1 d:1,e:1 Ma,Ze Ki,Sl,Ra,Go,Ro d:1 d:3,s:3,e:3

25. ADBIS 200225 dse Max Sox Ki Flx Slx Ze Ra Goxx Rox  -cut  = 0.8 Ma,So,Ki,Fl,Sl,Ze,Ra,Go,Ro d:1 d:3,s:4 Sl,Go e:1 d:1,e:1 Ma,Fl,Ze,Ro Ma,So,Fl,Ze,Ro Ma,So,Fl,Ze e:1 Ma,Ki,Fl,Sl,Ze,Ra,Go,Ro d:1 d:1,e:1 Ma,Ze d:1 d:4,s:4,e:4 Ma,So,Fl Ma,Ki,Sl,Ze,Ra,Go,Ro Ki,Sl,Ra,Go,Ro e:2 d:1,s:1 Go Go,Ro d:1

26. ADBIS 200226 dse Ma So Ki Fl Slx Ze Ra Gox Ro  -cut  = 0.9 Ma,So,Ki,Fl,Sl,Ze,Ra,Go,Ro d:1 d:3,s:4 Sl,Go e:1 d:1,e:1 Ma,Fl,Ze,Ro Ma,So,Fl,Ze,Ro Ma,So,Fl,Ze e:1 Ma,Ki,Fl,Sl,Ze,Ra,Go,Ro d:1 d:1,e:1 Ma,Ze d:1 d:4,s:4,e:4 Ma,So,Fl Ma,Ki,Sl,Ze,Ra,Go,Ro Ki,Sl,Ra,Go,Ro e:2 d:2,s:1 Go Go,Ro d:1 s:1 Sl

27. ADBIS 200227  -Concept Structure Volume reduction - resultant structure contains only fourteen  -concepts Structure suggests to the user the significant concepts Independent on density of  -cuts

28. ADBIS 200228 Conclusion Hierarchical order of query result Lower degree of subjective choice (contrary to) context scaling Reduction of number of concepts  -cuts Finding of the significant concepts which are independent on number of  -cuts

29. ADBIS 200229 Thank you for your attention. www.cs.vsb.cz/arg