Fuzzy horizontal characteristics clustering

The data normalization does not change the overall pattern of the clustering. The classification hierarchy differs only by the order of successive separations of the characteristics, the first now being the density (1), then the melting point (2), and the boiling point (3) and without any splitting for the other seven characteristics to the final clusters. As is also apparent from Table 4, most characteristics have values near unity and zero, except for the melting point, with an MD of 0.28 to the class 1 (of the density). 

In order to develop the classification presented in this section, we applied the well-known Fuzzy n-Means (FNM) algorithm.The classification procedure obtained in this way is called fuzzy horizontal characteristics clustering (FHoChC). 

The characteristic clustering for the 10 physical properties, without data normalization and with a predefined number of five classes, distribute the characteristics in three clusters only the other two remain empty. Cluster 1 contains the atomic mass (1), with an MD of 0.33 this characteristic appears with equal MDs (0.33) to the two empty clusters, 3 and 4. In cluster 2, we find the density (MD = 1), while cluster 5 includes all the remaining eight physical properties, without any separation. All the MDs, except those for atomic mass, are either 1 or 0. 

Let us remark that, even though five fuzzy classes were designed, the clustering actually produced only three cluster 2, cluster 5, and another 

The results are very much the same for the 10 physical, structural, and chemical characteristics, without data normalization. For the five predefined number of classes, one remains vacant the MDs of all the characteristics for this class are zero. From the other four classes, again three contain one characteristic each, while the last includes the rest of the seven properties. All the MDs are 1 or 0. The three properties separated are, again, the density (class 1), the melting point (class 2), and the boiling point (class 3). 

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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... 

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