Information Technology Reference
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ϑ(
C
i
,
C
j
)(
=
)
For two clusters
C
i
and
C
j
,let
i
j
be the inter-cluster similarity
ϑ
(
C
i
)
degree of
C
i
and
C
j
, and let
be the intra-cluster similarity degree of
C
i
. Then
the similarity-based SI can be defined as:
max
i
j
ϑ(
C
i
,
C
j
)
=
SI
sim
=
(2.90)
ϑ
(
min
i
C
i
)
where
ϑ(
C
i
,
C
j
)
=
max
C
j
ϑ(
A
,
B
)
(2.91)
A
∈
C
i
,
B
∈
ϑ
(
C
i
)
=
min
A
C
i
ϑ(
A
,
B
)
(2.92)
,
B
∈
As a relative measure, SI does not depend on the cluster number, but on the structure
of clusters. Therefore, the optimal
λ
can be selected as:
λ
=
argmin
λ
SI
sim
(λ)
(2.93)
where
SI
sim
(λ)
is the SI of the resultant clusters with
λ
being the confidence level of
the equivalent association matrix.
In the following, we shall extend the algorithm for clustering IVIFSs. Before
doing so, we first introduce the basic concepts related to IVIFSs:
Atanassov and Gargov (1989) defined the concept of IVIFS:
Definition 2.13
(Atanassov and Gargov 1989) Let
X
be a fixed set. Then
A
={
x
, μ
A
(
x
),
˜
v
A
(
x
)
|
x
∈
X
}
(2.94)
is called an interval-valued intuitionistic fuzzy set (IVIFS), where
μ
A
(
x
)
⊂[
0
,
1
]
and
v
˜
A
(
x
)
⊂[
0
,
1
]
,
x
∈
X
, with the condition:
sup
μ
A
(
x
)
+
sup
v
˜
A
(
x
)
≤
1
,
x
∈
X
(2.95)
, then the IVIFS
A
μ
A
(
)
=
μ
A
(
)
v
A
(
˜
)
=
˜
v
A
(
)
Clearly, if inf
x
sup
x
and inf
x
sup
x
reduces to a traditional IFS.
Atanassov and Gargov (1989) further gave some basic operational laws of IVIFSs:
A
Definition 2.14
(Atanassov and Gargov 1989) Let
={
x
, μ
A
(
x
),
˜
v
A
(
x
)
|
x
A
1
A
2
∈
X
}
,
={
x
, μ
A
1
(
x
),
˜
v
A
1
(
x
)
|
x
∈
X
}
and
={
x
, μ
A
2
(
x
),
˜
v
A
2
(
x
)
|
x
∈
X
}
be three IVIFSs. Then
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