Fuzzy hierarchical characteristics clustering

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). 

In the different steps of the clustering process the melting point (2), the boiling point (3), and the density (1) are successively separated. The two other physical properties (ionization energy and covalent radius) and all the 

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In order to identify which properties (variables) are responsible for the observed similarities and dissimilarities between the chemical elements, we also applied a very interesting algorithm, namely, fuzzy hierarchical crossclassification (FHiCC). The technique produces not only a fuzzy partitioning of the chemical elements but also a fuzzy partitioning of the physical and chemical properties considered. Moreover, the fuzzy hierarchical characteristics clustering (FHiChC) and fuzzy horizontal characteristics clustering (FHoChC) procedures showed very high similarities between the chemical and structural... 

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. 

The hierarchical cross-clustering using the same 10 physical properties, but with data normalization, gave a very similar partition as that obtained with the 10 normalized physical characteristics. Also, the membership degrees of the elements in the fuzzy sets of the fuzzy partition were similar. 

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