Intuitionistic Fuzzy Clustering Algorithms - Intuitionistic Fuzzy Aggregation and Clustering - page 223

Based on the idea of the traditional MST clustering algorithm and the intuitionistic
fuzzy distancematrix above, Zhao et al. (2012a) proposed an intuitionistic fuzzyMST
clustering algorithm:

Algorithm 2.9

Step 1 Construct the intuitionistic fuzzy distance matrix and the intuitionistic
fuzzy graph:
(1) Calculate the distance d ij

d A i , A j , then we get the intuitionistic fuzzy

=

= d ij n × n . distance matrix D

(2) Construct the intuitionistic fuzzy graph V D with n nodes associated to the

,

samples A i ( to be clustered which are expressed by IFSs and every

edge between A i and A j having the weight d ij , which is an element (expressed by IFV)
of the intuitionistic fuzzy distance matrix D

i = 1 , 2 ,..., n )

= ( d ij ) n × n and denotes the dissimilarity

degree between the samples A i and A j .
Step 2 Compute the MST of the intuitionistic fuzzy graph ( V , D ) by Kruskal

method (Kruskal 1956) or Prim method (Prim 1957):
(1) Arrange the edges of V

D in order from the smallest weight to the largest
one. Because the weight of each edge is an IFV, we can firstly compute the score
and the accuracy degree of each IFV, and then we use Definition 2.27 to sort all the
intuitionistic fuzzy weights.
(2) Select the edge with the smallest weight.
(3) Select the edge with the smallest weight from the rest edges which do not form
a circuit with those already chosen.
(4) Repeat the process (3) until

,

( n − 1 ) edges have been selected where n is the

number of the nodes in V D . Thus we get the MST of the intuitionistic fuzzy graph

,

V

D .
Step 3 Group the nodes (sample points) into clusters by cutting down all the edges
of the MST with weights greater than a threshold

,

is an IFV), we can get a
certain number of sub-trees (clusters) automatically. The clustering results induced
by the sub-trees do not depend on some particular MST (Gaertler 2002).
Moreover, Zhao et al. (2012a) improvedAlgorithm2.9 by changing the intuitionis-
tic fuzzy distance measure by Eq. ( 2.111 )or( 2.112 ) so as to lessen the computational
effort. They first defined another intuitionistic fuzzy distance matrix:

λ (where λ

= , ,..., =

Definition 2.29 (Zhao et al. 2012a) Let A j ( j 1 2 n ) be n IFSs. Then D

d ij n × n

d A i , A j is

=

is called an intuitionistic fuzzy distance matrix, where d ij

the distance between A i and A j , which has the following properties:
(1) 0

≤ d ij ≤ 1, for all i , j = 1 , 2 ,..., n .

(2) d ij = 0 if and only if A i =

A j .

(3) d ij =

, = , ,...,

n .
Based on Definition 2.29, Zhao et al. (2012a) developed another intuitionistic
fuzzy MST clustering algorithm:

d ji , for all i j 1 2