Navigation Through Query Result Using Concept Order Tom

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 ADBIS 2002 1

Contents n Motivation n Example n Application of scaled context n Application of -cut concept order n Conclusion ADBIS 2002 2

Motivation SQL like specified query n Query result – a (possibly large) set of objects having acceptable parameters, not structured n Unsatisfactory for human decision n Navigation in the query result offers: n • How to find the best object • How to compare objects n Realized using hierarchical structures – ordered sets ADBIS 2002 3

Possibility of Web Searching ADBIS 2002 4

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) n price of ski-pass < 5200 CZK (s) n elevation of pistas > 1500 m a. s. (e) n ADBIS 2002 5

Numeric Values of Query Result Ski center Mayrhofen Sölden Kitzbühel Flattach Söll Zell am See Radstadt Gosau Rohrmoos Abb. Ma So Ki Fl Sl Ze Ra Go Ro d (km) 483 506 465 490 460 482 450 390 426 ADBIS 2002 s (CZK) 5196 4866 4741 4411 3664 4632 4625 3774 4565 e (m) 3250 3260 2000 3125 1835 3029 2130 1600 2700 6

Characteristics of Query Result n Set of objects with parameters n Quality of an object can is given by a relevance of values of the object’s parameters to the query specification. n Types of object’s parameter - boolean values (without problem) - numeric values n Formal Concept Analysis ADBIS 2002 7

Context n 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 limbs bird X ant X wings mammal X snake dolphin X ADBIS 2002 8

Concept Formal concept of context (O, A, I), is a pair (Q, T) where Q O, T A, Q’=T and T’ = Q n I bird ant limbs wings mammal C 1 = {bird limbs, wings} X X C 2 = {bird, ant limbs} X snake dolphin X ADBIS 2002 9

Concept Lattice Concept lattice is a set of concepts ordered by inclusion on attributes (or by inverted inclusion on objects) n C 0 = {bird, ant, snake, dolphin - no attribute} C 2 = {bird, ant - limbs} C 3 = {dolphin - mammal} C 1 = {bird - limbs, wings} Cn = {no object - limbs, wings, mammal } ADBIS 2002 10

Numeric Attributes n How to convert many-valued query result table into context table? • Attribute scaling – concept lattice of scaled context • Our method – concept order ADBIS 2002 11

Scaled Context distance (d) 550 490 470 440 Ma So Ki Fl Sl Ze Ra Go Ro x x x x x x price (s) 4. 8 4. 5 4. 4 x x x ADBISx 2002 elevation (e) > 4. 0 2. 5 3. 0 3. 2 x x x x x 12

Concept Lattice objects satisfying all attributes of ordinally scaled context containing 36 concepts Objects: Ma, Fl, Ze, Ro Attributes: d 1, e 3 2002 attributes belonging to. ADBIS all objects 13

Drawbacks of Context Scaling n Volume of scaled lattice is very large exponential dependence on number of attributes n Choice of scale user dependent n Our goal was to reduce the number of concepts and to design a scale independent structure ADBIS 2002 14

Our Solution n Creation of new context with fuzzy values n The “fuzzy context” is transformed back to several crisp -cut contexts n Concept lattice for every -cut context, called -cut concept lattice, is produced n Some concepts occur repeatedly in -cut concept lattices – we denote this concepts as significant ones ADBIS 2002 15

How Can We Get the “Fuzzy Context”? Fuzzyfication is a transformation of numerical values into interval 0, 1. n Membership function is linear n Upper and lower bounds of the membership function are the maximum/minimum values of the particular query parameter n ADBIS 2002 16

Example of “Fuzzyfication” ADBIS 2002 17

Fuzzy Context Ski Center d s e Mayrhofen Sölden Kitzbühel 0. 58 0. 12 0. 67 0. 00 0. 09 0. 17 0. 85 0. 86 0. 12 Flattach 0. 55 0. 39 0. 78 Söll Zell am See 0. 70 0. 59 0. 89 0. 24 0. 02 0. 72 Radstadt Gosau Rohrmoos 0. 75 1. 00 0. 87 0. 25 0. 82 0. 29 0. 19 0. 00 0. 53 ADBIS 2002 18

-cut Context n Let K is a fuzzy context then K = {x X, K(x) } is an -cut context for 0, 1 n We obtain a set of -concepts for this context ADBIS 2002 19

-Concept Order n We produce a new structure as a union of all -concepts n The -concepts are unified to multiset U on equality of the object set of the -concept n Repetition of an attribute in -concept denotes a more significant concept n The multiset U can be ordered according to inclusion on objects ADBIS 2002 20

Ma, So, Ki, Fl, Sl, Ze, Ra, Go, Ro -cut = 0. 4 Ma So Ki Fl Sl Ze Ra Go Co d x x x x s e x x x e: 1 d: 1 Ma, So, Fl, Ze, Ro Ma, Ki, Fl, Sl, Ze, Ra, Go, Ro d: 1, e: 1 d: 1, s: 1 Sl, Go Ma, Fl, Ze, Ro d: 1, s: 1, e: 1 x x ADBIS 2002 21

-cut = 0. 5 Ma So Ki Fl Sl Ze Ra Go Ro d x x x x s e x x Ma, So, Ki, Fl, Sl, Ze, Ra, Go, Ro e: 1 d: 1 Ma, So, Fl, Ze, Ro Ma, Ki, Fl, Sl, Ze, Ra, Go, Ro d: 2, s: 2 Sl, Go d: 1, s: 1, e: 1 Fl x x x d: 1, e: 1 Ma, Fl, Ze, Ro d: 1, s: 1, e: 1 x x ADBIS 2002 22

Ma, So, Ki, Fl, Sl, Ze, Ra, Go, Ro -cut = 0. 6 Ma So Ki Fl Sl Ze Ra Go Ro d x s e: 1 e x x x x x d: 1 Ma, Ki, Fl, Sl, Ze, Ra, Go, Ro d: 1 Ma, Ki, Sl, Ze, Ra, Go, Ro d: 1, e: 1 Ma, Fl, Ze, Ro e: 1 Ma, So, Fl, Ze d: 2, s: 2 Sl, Go d: 1, e: 1 Ma, Ze x x Ma, So, Fl, Ze, Ro d: 2, s: 2, e: 2 ADBIS 2002 23

Ma, So, Ki, Fl, Sl, Ze, Ra, Go, Ro -cut = 0. 7 d Ma So Ki Fl Sl Ze Ra Go Ro s e: 1 e x x x x Ma, So, Fl, Ze, Ro Ma, Ki, Fl, Sl, Ze, Ra, Go, Ro d: 1 Ma, Ki, Sl, Ze, Ra, Go, Ro d: 1, e: 1 Ma, Fl, Ze, Ro e: 2 Ma, So, Fl, Ze d: 1 Ki, Sl, Ra, Go, Ro x x d: 1, e: 1 Ma, Ze d: 3, s: 3 Sl, Go d: 3, s: 3, e: 3 ADBIS 2002 24

Ma, So, Ki, Fl, Sl, Ze, Ra, Go, Ro -cut = 0. 8 d Ma So Ki Fl Sl Ze Ra Go Ro s e: 1 e x x x x d: 1 Ma, So, Fl, Ze, Ro Ma, Ki, Fl, Sl, Ze, Ra, Go, Ro d: 1, e: 1 d: 1 Ma, Ki, Sl, Ze, Ra, Go, Ro Ma, Fl, Ze, Ro e: 2 Ma, So, Fl, Ze d: 1 Ki, Sl, Ra, Go, Ro d: 1 d: 3, s: 4 Go, Ro Sl, Go d: 1, e: 1 Ma, Ze d: 1, s: 1 Go e: 1 Ma, So, Fl d: 4, s: 4, e: 4 ADBIS 2002 25

Ma, So, Ki, Fl, Sl, Ze, Ra, Go, Ro -cut = 0. 9 d Ma So Ki Fl Sl Ze Ra Go Ro s e: 1 e d: 1 Ma, So, Fl, Ze, Ro Ma, Ki, Fl, Sl, Ze, Ra, Go, Ro d: 1 Ma, Ki, Sl, Ze, Ra, Go, Ro x Ki, Sl, Ra, Go, Ro d: 1 d: 3, s: 4 Go, Ro Sl, Go s: 1 Sl e: 2 Ma, So, Fl, Ze d: 1 x d: 1, e: 1 Ma, Fl, Ze, Ro d: 1, e: 1 Ma, Ze d: 2, s: 1 Go e: 1 Ma, So, Fl d: 4, s: 4, e: 4 ADBIS 2002 26

-Concept Structure n Volume reduction - resultant structure contains only fourteen -concepts n Structure suggests to the user the significant concepts n Independent on density of -cuts ADBIS 2002 27

Conclusion n Hierarchical order of query result n Lower degree of subjective choice (contrary to) context scaling n Reduction of number of concepts n Finding of the significant concepts which are independent on number of -cuts ADBIS 2002 28

Thank you for your attention. www. cs. vsb. cz/arg ADBIS 2002 29