Information Technology Reference
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Table 2.12
The results derived by Algorithm 2.7 with different cluster numbers on the simulated
data set
Modified data set I
c
2
3
4
5
6
7
8
9
10
V
PC
0.982
0.648
0.531
0.469
0.324
0.289
0.239
0.216
0.196
V
CE
0.073
0.750
1.163
1.465
1.750
1.975
2.153
2.335
2.395
Modified data set II
c
2
3
4
5
6
7
8
9
10
V
PC
0.982
0.648
0.531
0.381
0.324
0.289
0.266
0.248
0.235
V
CE
0.072
0.750
1.164
1.465
1.750
1.976
2.163
2.325
2.463
Note
: (1) The optimal values of the measures are highlighted in bold and italic fonts
Next, we exploit the traditional FCM algorithm on the simulated data set for the
comparison purpose. As mentioned above, the FCM algorithm does not take into
account the uncertain information. Therefore, to make sure
μ(
x
)
+
v
(
x
)
=
1 for any
x
in the simulated data set, we should modify the data set by adding
π(
x
)
to either
μ(
. We produce the two modified data sets and then exploit Algorithm 2.7
on them. The results can be found in Table
2.12
(Xu and Wu 2010).
As indicated by the
V
PC
and
V
CE
measures in Table
2.12
, Algorithm 2.7 prefers
to cluster the modified simulated data sets into two clusters, which is actually away
from the three “true” clusters in the data. In other words, the FCM algorithm cannot
identify all the three classes precisely. This further justifies the importance of the
uncertain information in IFSs.
x
)
or
v
(
x
)
2.6 Intuitionistic Fuzzy MST Clustering Algorithm
Zhao et al. (2012a) developed an intuitionistic fuzzy minimum spanning tree (MST)
clustering algorithm to deal with intuitionistic fuzzy information. To do so, they first
introduced some concepts related to the graph theory.
A graph is composed of a set of
po
ints called nodes and a set of nod
e
pairs called
edges, which can be
de
noted by
, where
V
is the set of nodes and
E
is the set of
edges. In fact, the set
E
in a normal graph is a crisp relation over
V
(
V
,
E
)
×
V
. That is to say,
if there exists an edge between
x
and
y
, then the membership degree
μ
E
(
x
,
y
)
=
1;
)
∈
V
×
V
. If we define a fuzzy relation
R
otherwise
μ
E
(
x
,
y
)
=
0, where
(
x
,
y
V
×
V
, then the membership function
over
μ
R
(
x
,
y
)
takes various values from 0 to
1, and such a graph is called a fuzzy graph.
=
V
1
,
V
2
,...,
V
n
be a collection of
n
V
Definition 2.25
(Chen et al. 2007) Let
V
. Then
V
R
is called a
nodes, and
R
=
(
r
ij
)
n
×
n
a fuzzy relation over the set
,
=
E
k
=
V
i
V
j
|∀
V
i
,
V
j
∈
V
, then
V
E
is called a basic graph of
fuzzy graph. If
E
,
V
R
.
,
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