Concept lattices constrained by Concept lattices constrained by attribute dependenciesattribute dependencies

Radim Bělohlávek, Vladimír Sklenář, Jiří Zacpal

radim.belohlavek@upol.czvladimir.sklenar@upol.cz

jiri.zacpal@upol.cz

Formal concept anaysisFormal concept anaysis

Finding interesting patterns in data – formal concepts

Formal CFormal Contextontext

Triplet (X, Y, I)

X – set of Objects

Y – set of Attributes

I X Y

Object x has attribute y

FormFormal Conceptal Concept

Pair (A,B)A X – set of all objects sharing all

attributes from B

B Y - set of all attributes shared by all objects from B

Concept LatticeConcept Lattice

Set of all concepts

Partial order

(A1,B1) (A2,B2) iff A1 A2

(equivalently B2 B1)

Additional informationAdditional information

Attribute x is more important than attribute y

AD - AD - FormFormulaula

yyyn

...1

Constrained Concept LatticeConstrained Concept Lattice

Concept (A,B) satisfies an AD-formula

if y then y1 or … or yn

ExampleExample

Objects cars

Attributescar properties (sedan, diesel engine, ABS, …)

Engine is most importantEngine is most important

sedanhatchbackairbag

dieselgasolinesedan

sedanhatchbackABS

sedanhatchbackingpowerstear sedanhatchbackngconditioniair

dieselgasolinehatchback

Type of car is most importantType of car is most important

dieselgasolineairbag

sedanhatchbackdiesel

dieselgasolineABS

dieselgasolineingpowerstear dieselgasolinengconditioniair

sedanhatchbackgasoline

Safety is most importantSafety is most important

ABSairbag

dieselgasolineingpowerstear dieselgasolinengconditioniair

sedanhatchbackdiesel sedanhatchbackgasoline

airbagABS

ABSsedanABShatchback

Another approachAnother approach

Concept lattice constrained by equivalence relations