User-Friendly Fuzzy FCA
ICFCA 2013
Dresden, Germany
Juraj Macko, Palacky University, Olomouc, Czech Republic
User-Friendly Fuzzy FCA
User-Friendly
Fuzzy FCA
is oxymoron.
Isn’t it? 
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.
User-Friendly Fuzzy FCA
Fuzzy set
Residuated
lattice
Lukasiewicz
Fuzzy concept lattice
User R.I.P.!
Godel
Fuzzy relation
User-Friendly Fuzzy FCA
Half - red
circles
Measuring
cup
Object
Liquid color
Attribute
User-Friendly Fuzzy FCA
C
yan
Printer 1
Printer 2
M
agenta
Y
ellow
K
ey (black)
User-Friendly Fuzzy FCA
Universum X = All objects = {x1, x2, x3, x4,}
x1
x2
x3
x4
x1
x2
x3
x4
x1
x2
x3
x4
Set A = Objects which we care about = {x1, x4,}
FCA: What you share, iF you Care About.
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)
User-Friendly Fuzzy FCA
Part of the
object,
which we
care about.
Actual
amount of
RED in object
User-Friendly Fuzzy FCA
<=
Is it TRUE, that the part,
which we care about is
RED?
Yes, it is TRUE.
truth degree = 1
User-Friendly Fuzzy FCA
>=
Is it TRUE, that the part,
which we care about is
RED?
No, it is not
TRUE.
truth degree = 0
User-Friendly Fuzzy FCA
>
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?
It is TRUE
in truth degree
2/5.
Material truth
=
actual amount of
color.
User-Friendly Fuzzy FCA
>
It is TRUE
It is TRUE
in truth degree
in truth degree 1.
1 - 3/5
1/5 = 3/5
2/5
4/5
2/5
How much is TRUE, that
the part, which we care
about is RED?
X
X
X
X
X
X
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
>
How much is TRUE, that
the part, which we care
about is RED?
It is TRUE
in truth degree
2/3.
Proportional
=
what you see
/
what you care
about
User-Friendly Fuzzy FCA
Mr.
Godel
Mr.
Lukasiewicz
Mr.
Product
User-Friendly Fuzzy FCA
1
Mr.
Godel
Mr.
Lukasiewicz
Mr.
Product
User-Friendly Fuzzy FCA
0
Fuzzy
negation
Mr.
Godel
Mr.
Lukasiewicz
Mr.
Product
I(x,y)
A(x)
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)
maximality
A(x)
Sharing
=
Taking minimal
truth degree
Has an object the
amount of RED, which
we care about?
Has an object the
required amount of
RED?
I(x,y)
B(y)
How much we
care about the
attribute.
User-Friendly Fuzzy FCA
Hedge – thruth stressing
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.
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
Scaling
Godel
Scale
Lukasiewicz
Scale
Godel
Scale
User-Friendly Fuzzy FCA
Lukasiewicz
Scale
Godel
Scale
Godel
Scale
Lukasiewicz
Scale
Lukasiewicz
Scale
User-Friendly Fuzzy FCA
Scaled fuzzy context
Lukasiewicz
Data source:
Belohlavek R., FRS
User-Friendly Fuzzy FCA
Any real application?
Fuzzy FCA with measures
FCA
with measures
OLAP Cube
DB: Group by
Fuzzy OLAP Cube
DB: Fuzzy group by
User-Friendly Fuzzy FCA
Fuzzy OLAP
application?
Data source:
Belohlavek R., FRS
NASA
User-Friendly Fuzzy FCA
User-Friendly
Fuzzy FCA
Oxymoron?
User-Friendly Fuzzy FCA
C
Printer 1
Printer 2
M
Y
K