Database Reference
In-Depth Information
3.2.1 Formal Concept Analysis
The formal concept analysis (shortly FCA, introduced by
Rudolf Wille
in 1980) is a
well-known method for object-attribute data analysis. The input of FCA is called
formal context C
, which can be described as
C
(
G
,
M
,
I
) - a triplet consisting of a
set of objects
G
and set of attributes
M
,with
I
as relation of
G
and
M
. The elements of
G
are defined as objects and the elements of
M
as attributes of the context.
For a set
A
¼
G
of objects, we define
A
0
as the set of attributes common to the
objects in
A
. Correspondingly, for a set
B
M
of attributes, we define
B
0
as the set
of objects which have all attributes in
B
.A
formal concept
of the context (
G
,
M
,
I
)is
a pair (
A
,
B
) with
A
M
,
A
0
¼
B
, and
B
0
¼
G
,
B
A
.
B
ð
G
;
M
;
I
Þ
denotes the set of
all concepts of context
and forms a complete lattice (so-called
Galois
lattice
). For more details, see [
19
,
20
].
The Galois lattice may be represented by the Hasse diagram. In this diagram,
every node represents one formal concept from the lattice. Nodes are usually labeled
by attributes (above the node) and objects (below the node) possessed by a concept.
For the sake of clarity, sometimes reduced labeling is used (see Fig.
3.9
for illustra-
tion), which means that attributes are shown only at the first node (concept) they
appear in. This holds reciprocally for objects. These two labelings are equivalent.
For this environment in particular, where the FCA meets social networks (and
consequently also the graph theory), we have proposed a modified concept lattice
drawing method. The result of this method can be seen in Fig.
3.16
. The first
difference is that the size of presented nodes is linked to their semantic value;
therefore, nodes representing concepts covering more objects (with respect to
ð
G
;
M
;
I
Þ
Fig. 3.16 Modified visualization of concept lattice at rank 5
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