Intuitionistic Fuzzy Clustering Algorithms - Intuitionistic Fuzzy Aggregation and Clustering - page 177

2.2 Clustering Algorithms Based on Association Matrices

In the above section, we have introduced an intuitionistic fuzzy clustering algorithm,
which is on the basis of the intuitionistic fuzzy similarity matrix. In this clustering
technique, all the given intuitionistic fuzzy information is first transformed into the
interval-valued fuzzy information. The intuitionistic fuzzy similarity degrees derived
by using distance measures are interval numbers, and both the intuitionistic fuzzy
similarity matrix and the intuitionistic fuzzy equivalence matrix are also interval-
valued matrices. As a result, this clustering technique requires much computational
effort and cannot be extended to cluster IVIFSs, and more importantly, it produces
the loss of too much information in the process of calculating intuitionistic fuzzy
similarity degrees, which implies a lack of precision in the final results. To overcome
this drawback, Xu et al. (2008) proposed a straightforward and practical clustering
algorithm for IFSs, and extended the algorithm to cluster IVIFSs.
Xu and Chen (2008) gave an overview of the existing association measures for
IFSs (or IVIFSs). Based on the association measures, in the following, we introduce
the concept of association matrix:

Definition 2.9 (Xu et al. 2008) Let A j ( j = 1 , 2 ,..., m ) be m IFSs. Then

= (

c ij ) m × m is called an association matrix, where c ij

= (

A i , A j )

C is the asso-

ciation coefficient of A i and A j (which can be derived by one of the intuitionistic
fuzzy association measures introduced by Xu and Chen (2008)), and has the follow-
ing properties:

c

(1) 0 ≤ c ij ≤ 1 , i , j = 1 , 2 ,..., m .

(2) c ij = 1 if and only if A i = A j .

(3) c ij = c ji , i , j = 1 , 2 ,..., m .

Definition 2.10 ( Xu et al. 2008) Let C = ( c ij ) m × m be an association matrix. If

C 2 c ij ) m × m , then C 2 is called the composition matrix of C , where = C ◦ C = (

c ij = ma k { min { c ik , c kj }} , i , j = 1 , 2 ,..., m (2.80)

According to Definition 2.9, we have

Theorem 2.8 (Xu et al. 2008) Let C c ij ) m × m be an association matrix. Then the

composition matrix C 2 is also an association matrix.

= (

Proof (1) Since C is an association matrix, then for any i , j = 1 , 2 ,..., m ,wehave

0 ≤ c ij ≤ 1. Thus

0 ≤ c ij = ma k { min { c ik , c kj }} ≤ 1 , i , j = 1 , 2 ,..., m (2.81)