ADBIS 2002 1

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

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Contents Motivation Example Application of scaled context Application of -cut concept order Conclusion

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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

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Possibility of Web Searching

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Example

We are looking for an Alpine ski center that is near to Prague, is situated on the highestelevation 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)

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Numeric Values of Query Result Ski center Abb. d (km) s (CZK) e (m)

Mayrhofen Ma 483 5196 3250

Sölden So 506 4866 3260

Kitzbühel Ki 465 4741 2000

Flattach Fl 490 4411 3125

Söll Sl 460 3664 1835

Zell am See Ze 482 4632 3029

Radstadt Ra 450 4625 2130

Gosau Go 390 3774 1600

Rohrmoos Ro 426 4565 2700

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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

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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

limbs wings mammal

bird X X

ant X

snake

dolphin X

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Concept Formal concept of context (O, A, I), is a pair (Q, T) where Q O, T A, Q’=T and T’ = Q

I limbs wings mammal

bird X X

ant X

snake

dolphin X

C1 = {bird - limbs, wings}

C2 = {bird, ant - limbs}

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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}

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Numeric Attributes

How to convert many-valued query result table into context table?• Attribute scaling

– concept lattice of scaled context• Our method

– concept order

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Scaled Context distance (d) price (s) elevation (e) >550 490 470 440 4.8 4.5 4.4 4.0 2.0 2.5 3.0 3.2

Ma

x x x x x x

So

x x x x

Ki

x x x x

Fl x x x x x x

Sl x x x x x

x x

Ze

x x x x x x

Ra

x x x x x

Go

x x x x x x x x

Ro

x x x x x x x x

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of ordinally scaled contextcontaining 36 concepts

Concept Lattice objects satisfying all attributes

attributes belonging to all objects

Objects: Ma,Fl,Ze,RoAttributes: d1, e3

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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

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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

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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

Example of “Fuzzyfication”

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Fuzzy Context

Ski Center d s e

Mayrhofen 0.58 0.00 0.85

Sölden 0.12 0.09 0.86

Kitzbühel 0.67 0.17 0.12

Flattach 0.55 0.39 0.78

Söll 0.70 0.89 0.02

Zell am See 0.59 0.24 0.72

Radstadt 0.75 0.25 0.19

Gosau 1.00 0.82 0.00

Rohrmoos 0.87 0.29 0.53

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-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

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-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

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d s e

Ma x x

So x

Ki x

Fl x x

Sl x x

Ze x x

Ra x

Go x x

Co x x

--cutcut = 0. = 0.44 d:1

d:1,s:1Sl, 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

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d s e

Ma x x

So x

Ki x

Fl x x

Sl x x

Ze x x

Ra x

Go x x

Ro x x

--cutcut = 0. = 0.55 d:1

d:2,s:2Sl,Go

e:1

d:1,e:1

d:1,s:1,e:1Fl

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

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d s e

Ma x x

So x

Ki x

Fl x

Sl x x

Ze x x

Ra x

Go x x

Ro x

Ma,So,Ki,Fl,Sl,Ze,Ra,Go,Ro--cutcut = 0. = 0.66

d:1

d:2,s:2Sl,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:1Ma,Ze

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d s e

Ma x

So x

Ki x

Fl x

Sl x x

Ze x

Ra x

Go x x

Ro x

--cutcut = 0. = 0.77 Ma,So,Ki,Fl,Sl,Ze,Ra,Go,Ro

d:1

d:3,s:3Sl,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:1Ma,ZeKi,Sl,Ra,Go,Ro

d:1

d:3,s:3,e:3

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d s e

Ma x

So x

Ki

Fl x

Sl x

Ze

Ra

Go x x

Ro x

--cutcut = 0. = 0.88 Ma,So,Ki,Fl,Sl,Ze,Ra,Go,Ro

d:1

d:3,s:4Sl,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:1Ma,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:1Go

Go,Ro

d:1

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d s e

Ma

So

Ki

Fl

Sl x

Ze

Ra

Go x

Ro

--cutcut = 0. = 0.99 Ma,So,Ki,Fl,Sl,Ze,Ra,Go,Ro

d:1

d:3,s:4Sl,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:1Ma,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:1Go

Go,Ro

d:1

s:1Sl

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-Concept Structure

Volume reduction - resultant structure contains only fourteen -concepts

Structure suggests to the user the significant concepts

Independent on density of -cuts

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Conclusion

Hierarchical order of query result Lower degree of subjective choice

(contrary to) context scaling Reduction of number of concepts Finding of the significant concepts

which are independent on number of -cuts-cuts

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Thank you for your attention.

www.cs.vsb.cz/arg