Information Technology Reference
In-Depth Information
From the above theoretical analysis, we introduce an algorithm for clustering IFSs
as follows (Xu et al. 2008):
Algorithm 2.2
Step 1 Let X
={
x 1 ,
x 2 ,...,
x n }
be a discrete universe of discourse, and let
T be the weight vector of the elements x i
w
= (
w 1 ,
w 2 ,...,
w n )
(
i
=
1
,
2
,...,
n
)
,
n , and i = 1 w i =
with w i ∈[
0
,
1
]
, i
=
1
,
2
,...,
1. Consider a collection of m IFSs
A j (
j
=
1
,
2
,...,
m
)
, where
A j ={
x
A j (
x i ),
v A j (
x i ) |
x i
X
}
(2.88)
with
m .
Step 2 Select an intuitionistic fuzzy association measure, such as
π A j (
x i ) =
1
μ A j (
x i )
v A j (
x i )
, j
=
1
,
2
,...,
c
(
A i ,
A j )
k = 1 w k
(
x k
) · μ
(
x k
) +
v A i (
x k
) ·
v A j (
x k
) + π
(
x k
) · π
(
x k
))
A i
A j
A i
A j
=
max k = 1 w k
)), k = 1 w k
2
A i (
v A i (
2
A i (
2
A j (
v A j (
2
A j (
x k
) +
x k
) + π
x k
x k
) +
x k
) + π
x k
))
(2.89)
to calculate the association coefficients of the IFSs A i and A j
(
i
,
j
=
1
,
2
,...,
m
)
.
Then construct an association matrix C
= (
c ij ) m × m , where c ij
=
c
(
A i ,
A j ),
i
,
j
=
1
,
2
,...,
m .
Step 3
If the association matrix C
= (
c ij ) n × n is an equivalent association
matrix, then we construct a
c ij ) m × m of C by using
Eq. ( 2.87 ); otherwise, we compose the a sso ciation matrix C by using Eq. ( 2.86 )to
de rive a n equivale nt association matrix C . Then we construct a
λ
-cutting matrix C
λ = ( λ
λ
-cutting matrix
C
λ = ( λ
c ij ) m × m of C by using Eq. ( 2.87 ).
Step 4 If all elements of the i th line (column) in C
(or C
λ )
are the same as
λ
the corresponding elements of the j th line (column) in C
, then the IFSs
A i and A j are of the same type. By this principle, we can classify all these m IFSs
A j (
(or C
λ )
λ
j
=
1
,
2
,...,
m
)
.
By using the cutting matrix of the equivalent association matrix, Algorithm-IFSC
classifies the IFSs under the given confidence levels. Considering that the confidence
levels have a close relationship with the elements of equivalent association matrices,
in practical applications, people can properly specify the confidence levels according
to the elements of the equivalent association matrices and the actual situations, and
thus, the algorithm has desirable flexibility and practicability. However, in some
cases, people may expect that the algorithm can automatically generate the “optimal”
clustering without any interaction with them. In other words, the algorithm should
have the ability to set the optimal
according to cluster structure. To fulfill this
requirement, here we use the Separation Index (SI), one of the relative measures for
cluster validity, which was introduced by Nasibov and Ulutagay (2007).
λ
 
 
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