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2.5 Intuitionistic Fuzzy C-Means Clustering Algorithms
The algorithms presented previously are straightforward, but cannot provide the
information about membership degrees of the objects to each cluster. To overcome
this drawback, Xu and Wu (2010) developed an intuitionistic fuzzy C-means algo-
rithm to cluster IFSs, which is based on the well-known fuzzy C-means clustering
method (Bezdek 1981) and the basic distance measures between IFSs. Then, they
extended the algorithm for clustering IVIFSs.
Here, we first introduce the intuitionistic fuzzy C-means (IFCM) algorithm for
IFSs. We take the normalized Euclidean distance between the IFSs
Z
i
and
Z
j
:
d
NE
(
Z
i
,
Z
j
)
n
1
2
n
2
2
2
=
1
((μ
Z
i
(
x
j
)
−
μ
Z
j
(
x
j
))
+
(
v
Z
i
(
x
j
)
−
v
Z
j
(
x
j
))
+
(π
Z
i
(
x
j
)
−
π
Z
j
(
x
j
))
j
=
(2.134)
as the proximity function of the IFCM algorithm. Then the objective function of the
IFCM algorithm can be formulated as follows:
p
c
u
ij
d
NE
(
min
J
m
(
U
,
V
)
=
A
j
,
V
i
)
(2.135)
j
=
1
i
=
1
Subject to
c
u
ij
=
1
,
1
≤
j
≤
p
i
=
1
u
ij
≥
0
,
1
≤
i
≤
c
,
1
≤
j
≤
p
p
u
ij
>
0
,
1
≤
i
≤
c
j
=
1
where
A
={
A
1
,
A
2
,...,
A
p
}
are
p
IFSs each with
n
elements,
c
is the number of
clusters (1
are the prototypical IFSs, i.e., the
centroids, of the clusters. The parameter
m
is the fuzzy factor (
m
≤
c
≤
p
)
, and
V
={
V
1
,
V
2
,...,
V
c
}
>
1
)
,
u
ij
is the
membership degree of the
j
th sample
A
j
to the
i
th cluster,
U
=
(
u
ij
)
c
×
p
is a matrix
of
c
p
.
To solve the optimization problem in Eq. (
2.135
), we employ the Lagrange mul-
tiplier method (Ito and Kunisch 2008). Let
×
p
p
c
c
u
ij
d
NE
(
L
=
A
j
,
V
i
)
−
1
ς
j
(
u
ij
−
1
)
(2.136)
j
=
1
i
=
1
j
=
i
=
1
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