user-friendly fuzzy fca icfca 2013 dresden, germany juraj macko, palacky university, olomouc, czech...

User-Friendly Fuzzy FCA

ICFCA 2013Dresden, Germany

Juraj Macko, Palacky University, Olomouc, Czech Republic

User-Friendly Fuzzy FCA

is oxymoron.Isn’t it?

User-Friendly Fuzzy FCA

FCA has strong mathematical foundations.FCA has a very clear meaning for users.

Fuzzy FCA has strong mathematical foundations.Fuzzy FCA does NOT have a clear meaning for users.

Residuated

lattice

Fuzzy set

Fuzzy relation

Arrow operators

User-Friendly Fuzzy FCA

Fuzzy concept lattice

Godel

LukasiewiczUser R.I.P.!

Half - redcircles

Measuring cup

Object

Liquid colorAttribute

User-Friendly Fuzzy FCA

Cyan Magenta Yellow Key (black)

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User-Friendly Fuzzy FCA

User-Friendly Fuzzy FCAx1 x2 x3 x4Universum X = All objects = {x1, x2, x3, x4,}

Set A = Objects which we care about = {x1, x4,}

x1 x2 x3 x4

x1 x2 x3 x4

Set A = for each object from X we specify either “We care about”

or “We do not care about” We care about = We care about in degree 1

We do not care about = We care about in degree 0

Fuzzy Set A(x) = for each object from X we specify How much we care about

(in which degree)

FCA: What you share, iF you Care About.

Part of the object,

which we care about.

Part of the object,

which we care about.

Actual amount of

RED in object

User-Friendly Fuzzy FCA

Part of the object,

which we care about.

=Is it TRUE, that the part, which we care about is

RED?

<

User-Friendly Fuzzy FCA

Yes, it is TRUE.truth degree = 1

=Is it TRUE, that the part, which we care about is

RED?

>

User-Friendly Fuzzy FCA

No, it is not TRUE.

truth degree = 0

It is TRUE in truth degree…How much is TRUE, that

the part, which we care about is RED?

>

?

?

?

?

?

User-Friendly Fuzzy FCA

How much is TRUE, that the part, which we care

about is RED?

Material truth =

actual amount of color.

User-Friendly Fuzzy FCA

>It is TRUE

in truth degree 2/5.

X

X

X

XX

X

How much is TRUE, that the part, which we care

about is RED?

It is TRUE in truth degree 1.

It is TRUE in truth degree 1 - 1/5 = 4/5

It is TRUE in truth degree 1 - 2/5 = 3/5

Similarity truth=

How similar it is comparing to

TRUE (truth degree 1)

Error-like truth=

How much is left to be TRUE

(truth degree 1)

User-Friendly Fuzzy FCA

>It is TRUE

in truth degree 1 - 3/5 = 2/5

How much is TRUE, that the part, which we care

about is RED?

Proportional=

what you see /

what you care about

User-Friendly Fuzzy FCA

>It is TRUE

in truth degree 2/3.

Mr. Godel

Mr. Lukasiewicz

Mr. Product

User-Friendly Fuzzy FCA

Mr. Godel

Mr. Lukasiewicz

Mr. Product1

User-Friendly Fuzzy FCA

Mr. Godel

Mr. Lukasiewicz

Mr. Product0

Fuzzy negation

User-Friendly Fuzzy FCA

A(x) B(y)I(x,y)

How much we care about the

object

How much is TRUE, that the part, which we care

about is RED?

Sharing=

Taking minimal truth degree

B(y)A(x)I(x,y)

How much we care about the

attribute.Has an object the amount of RED, which

we care about?Has an object the

required amount of RED?

Sharing=

Taking minimal truth degree

maximality

User-Friendly Fuzzy FCA

Original statement:The teeth are white.

Stronger statement:The teeth are snow-white.The teeth are very white.

It is very true, that the teeth are white.

Hedge – thruth stressing

1. The original statement: We care about the object.

2. The statement with the identity hedge (a*=a): We normally care about the object.

3. The statement with the globalization hedge (a*=0, a<1): We fully care about the object.

User-Friendly Fuzzy FCA

GodelScale

LukasiewiczScale

Scaling

User-Friendly Fuzzy FCA

GodelScale

LukasiewiczScale

GodelScale

LukasiewiczScale

GodelScale Lukasiewicz

Scale

User-Friendly Fuzzy FCA

Data source: Belohlavek R., FRS

Scaled fuzzy contextLukasiewicz

FCA with measures

User-Friendly Fuzzy FCA

Any real application?

OLAP CubeDB: Group by

Fuzzy OLAP CubeDB: Fuzzy group by

Fuzzy FCA with measures

User-Friendly Fuzzy FCA

Fuzzy OLAP application?

Data source: Belohlavek R., FRSNASA

User-Friendly Fuzzy FCA

User-Friendly Fuzzy FCA

Oxymoron?

C M Y K

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User-Friendly Fuzzy FCA