Intuitionistic Fuzzy Clustering Algorithms - Intuitionistic Fuzzy Aggregation and Clustering - page 159

Chapter 2
Intuitionistic Fuzzy Clustering Algorithms

Since the fuzzy set theory was introduced (Zadeh 1965), many scholars have
investigated the issue how to cluster the fuzzy sets, and a lot of clustering algo-
rithms have been developed for fuzzy sets, such as the fuzzy c-means clustering
algorithm (Fan et al. 2004), the maximum tree clustering algorithm (Christopher and
Burges 1998), and the net-making clustering method (Wang 1983), etc. However,
the studies on clustering problems with intuitionistic fuzzy information are still at an
initial stage (Wang et al. 2011, 2012; Xu 2009; Xu and Cai 2012; Xu and Wu 2010;
Xu et al. 2008, 2011; Zhang et al. 2007; Zhao et al. 2012a, b). Zhang et al. (2007)
first defined the concept of the intuitionistic fuzzy similarity degree and constructed
an intuitionistic fuzzy similarity matrix, and then proposed a procedure for deriving
an intuitionistic fuzzy equivalence matrix by using the transitive closure of the intu-
itionistic fuzzy similarity matrix. After that, they presented a clustering technique
of IFSs on the basis of the -cutting matrix of the interval-valued matrix. Xu et al.

(2008) defined the concepts of the association matrix and the equivalent association
matrix, they introduced some methods for calculating the association coefficients of
IFSs, and used the derived association coefficients to construct an association matrix,
from which they derived an equivalent association matrix. Based on the equivalent
association matrix, a clustering algorithm for IFSs was developed and extended to
cluster interval-valued intuitionistic fuzzy sets (IVIFSs). Xu (2009) introduced an
intuitionistic fuzzy hierarchical algorithm for clustering IFSs, which is based on
the traditional hierarchical clustering procedure, the intuitionistic fuzzy aggregation
operator, and some basic distance measures, such as the Hamming distance, the nor-
malized Hamming distance, the Euclidean distance, and the normalized Euclidean
distance, etc. Xu andWu (2010) developed an intuitionistic fuzzy C-means algorithm
to cluster IFSs, which is based on the well-known fuzzy C-means clustering method
(Bezdek 1981) and the basic distance measures between IFSs. Then, they extended
the algorithm for clustering IVIFSs. Xu et al. (2011) extended the fuzzy closeness
degree (Wang 1983) to the intuitionistic fuzzy closeness degree, and defined an intu-
itionistic fuzzy vector, the inner and outer products of intuitionistic fuzzy vectors.
Based on the intuitionistic fuzzy closeness degree, they put forward a new method of
constructing intuitionistic fuzzy similarity matrix. Zhao et al. (2012a) developed an

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