Fuzzy divisive hierarchical clustering

It is easy to see that Cj,C2 is a hard partition of the classical set C. Thus, from these assignment rules we finally obtain the classical (or hard) hierarchy corresponding to the respective fuzzy hierarchy. Accordingly, the fuzzy divisive hierarchical clustering (FDHC) procedure may be used to obtain the optimal cluster structure of the data set and a hierarchical relationship between clusters and subclusters. The method is especially useful when the number of clusters is unknown. In most cases the number of clusters to be expected in the data set is unknown. 

In order to develop the classifications presented in this section, we will apply the fuzzy divisive hierarchical clustering (FDHiC) procedure described in the theoretical section to different characteristic sets considered here. The hierarchical procedure obtained in this way is called fuzzy hierarchical characteristics clustering (FHiChC). 

Divisive hierarchical simultaneous clustering procedures build a fuzzy hierarchy of objects and a fuzzy hierarchy of characteristics. Each node of the corresponding tree is labeled by a pair (C, D), where C is a fuzzy class of objects and D is a fuzzy class of characteristics. At the first level a binary fuzzy partition of data set X and the corresponding binary partition of characteristics set Y are computed. The classes that emerge are subdivided until no pair of real clusters can be obtained. 

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