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=
μ
ij
=
μ
ji
,
=
(2) (Symmetry) Since
r
ij
r
ji
, i.e.,
v
ij
v
ji
,fromEq.(
2.130
), it
r
ij
=
(λ,δ)
r
ji
.
follows that
(λ,δ)
(Sufficiency) If
R
=
(
(λ,δ)
r
ij
)
n
×
n
is an intuitionistic fuzzy similarity matrix,
(λ,δ)
then
(1) (Reflexivity) Since
r
ii
=
(
1
,
0
)
, for any 0
≤
λ, δ
≤
1
,
0
≤
λ
+
δ
≤
1,
(λ,δ)
μ
ii
≥
λ,
v
ii
≤
δ
,wehave
μ
ii
=
1
,
v
ii
=
0, i.e.,
r
ii
=
(
1
,
0
)
.
(2) (Symmetry) If there exists
r
ij
=
r
ji
, i.e.,
μ
ij
=
μ
ji
or
v
ij
=
v
ji
, in this case,
without loss of generality, suppose that
μ
ij
<μ
ji
, and let
λ
=
(μ
ij
+
μ
ji
)/
2. Then
μ
ij
<λ<μ
ji
,
λ
μ
ij
=
0
,
λ
μ
ji
=
1, and thus,
r
ij
=
(λ,δ)
r
ji
, which contradicts the
(λ,δ)
condition that
r
ij
)
n
×
n
is symmetrical.
In what follows, we introduce the orthogonal principle of intuitionistic fuzzy
cluster analysis:
Let
Y
r
ij
=
(λ,δ)
r
ji
, for any
i
,
j
. Therefore,
R
=
(
(λ,δ)
G
m
}
the set of attributes related to the considered objects. Assume that the characteristics
of the objects
y
i
(
= {
y
1
,
y
2
,...,
y
n
}
be a collection of
n
objects, and
G
={
G
1
,
G
2
,...,
i
=
1
,
2
,...,
n
)
with respect to the attributes
G
j
(
j
=
1
,
2
,...,
m
)
are represented by the IFSs, shown as follows:
y
i
=
G
,
G
j
,μ
y
i
(
G
j
),
v
y
i
(
G
j
)
|
G
j
∈
=
,
,...,
;
=
,
,...,
i
1
2
n
j
1
2
m
(2.131)
where
μ
y
i
(
G
j
)
denotes the degree that the object
y
i
should satisfy the attribute
G
j
,
v
y
i
(
G
j
)
indicates the degree that the object
y
i
should not satisfy the attribute
G
j
,
π
y
i
(
indicates the indeterminacy degree of the object
y
i
to the attribute
G
j
.ByEqs.(
2.125
) and (
2.131
), we construct the intuitionistic
fuzzy similarity matrix
R
G
j
)
=
1
−
μ
y
i
(
G
j
)
−
v
y
i
(
G
j
)
=
(
r
ij
)
n
×
n
, where
r
ij
is an IFV, and
r
ij
=
(
u
ij
,
v
ij
)
=
R
(
y
i
,
y
j
),
i
=
1
,
2
,...,
n
;
j
=
1
,
2
,...,
m
. After that, the
(λ, δ)
-cutting matrix
R
=
(
(λ,δ)
r
ij
)
n
×
n
can be determined under the confidence level
(λ, δ)
.Ifwe
(λ,δ)
denote
(λ,δ)
r
j
T
=
(
(λ,δ)
r
1
j
,
(λ,δ)
r
2
j
,...,
(λ,δ)
r
nj
)
as the vector of the
j
th column of
=
(
(λ,δ)
r
1
,
(λ,δ)
r
2
,...,
(λ,δ)
r
n
)
.
The orthogonal principle of intuitionistic fuzzy cluster analysis is to determine the
orthogonality of the column vectors of
R
, then
R
(λ,δ)
(λ,δ)
r
k
,
(λ,δ)
r
t
(λ, δ)
-cutting matrix
R
.Let
(λ,δ)
(λ,δ)
r
j
(
and
R
respectively. Then the orthogonal principles for clustering intuitionistic fuzzy infor-
mation can be classified into the following three categories:
k
,
t
,
j
=
1
,
2
,...,
n
)
denote the
k
th,
t
th and
j
th column vectors of
(λ,δ)
(λ,δ)
(1) (Direct clustering principle) If
(
1
,
1
)
;
r
k
·
(λ,δ)
r
j
=
(2.132)
(λ,δ)
(
1
,
0
),
then
(λ,δ)
r
k
and
(λ,δ)
r
j
are non-orthogonal. In this case,
y
k
and
y
j
are clustered into
one class.
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