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n
n
n
a i
b i
a i b i
(2.160)
i
=
1
i
=
1
i
=
1
T
with equality if and only if the two vectors a
= (
a 1 ,
a 2 ,...,
a n )
and b
=
T
(
b 1 ,
b 2 ,...,
b n )
are linearly dependent, that is, there is a nonzero real number
λ
such that a
= λ
b .FromEq.( 2.160 ), we know that c 3 (
A
,
B
)
1 with equality if
and only if there is a nonzero real number
λ
such that
μ A (
x i ) = λμ B (
x i ),
v A (
x i ) = λ
v B (
x i ), π A (
x i ) = λπ B (
x i ),
for all x i
X
(2.161)
while because
π A (
x i ) =
1
μ A (
x i )
v A (
x i ), π B (
x i ) =
1
μ B (
x i )
v B (
x i ),
for all x i
X
(2.162)
then by Eq. ( 2.161 ), we know that
λ =
1, and thus, c 3 (
A
,
B
) =
1 if and only if A
=
B .
Hence we complete the proof of the conditions (1) and (2) in Definition 2.31.
In addition, by Eq. ( 2.159 ) we know that
c 3 (
A
,
B
)
j = 1 μ A (
x j )
x j ) · μ B (
x j ) +
v A (
x j ) ·
v B (
x j ) + π A (
x j ) · π B (
=
j = 1 μ
x j ) · j = 1 μ
x j )
2
v A (
2
B (
v B (
B (
A (
x j ) +
x j ) + π
A (
x j ) +
x j ) + π
j = 1 μ B (
x j )
x j ) · μ A (
x j ) +
v B (
x j ) ·
v A (
x j ) + π B (
x j ) · π A (
=
j = 1 μ
x j ) · j = 1 μ
x j )
2
v B (
2
2
v A (
2
B (
x j ) +
x j ) + π
B (
A (
x j ) +
x j ) + π
A (
=
c 3 (
B
,
A
)
(2.163)
Thus, the condition (3) in Definition 2.31 also holds.
It's very interesting that when we add the third parameter, i.e., the indeterminacy
degree of IFSs, to c 1 (
, which
not only takes the third parameter of IFS (the hesitancy degree) into consideration,
but also satisfies all the three conditions of Definition 2.31.
In many cases, for instance, in cluster analysis, the weights of the attributes are
always different, so we should take them into account, and thus extend c 3 (
A
,
B
)
, we get a good association coefficient c 3 (
A
,
B
)
A
,
B
)
to
the following form:
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