Hierarchical fuzzy classification

HIERARCHICAL FUZZY CLASSIFICATION OF CHEMICAL ELEMENTS BASED ON TEN PHYSICAL PROPERTIES... 

We are now able to state the hierarchical cross-classification procedure. The root node of the partition tree corresponds to the pair (X,Y). At the first level a fuzzy partition P = A, A2 of X is computed. Let Y = y, y, ...,y be the characteristics set induced by A and A2- We then have... 

The fuzzy hierarchical cross-classification algorithm was used to classify eight mud samples. Each sample was characterized as a vector with 23 components representing chemical analysis. The fuzzy partition tree obtained by using simultaneous classification of muds and their characteristics is shown in Fig. 8. There are six final fuzzy classes in this hierarchy. The classical partition corresponding to the final fuzzy classes of the muds is 111 Krinides Lisbori Ai 2z> Argilla Solare A 2i> Pnkolimni ... 

The fuzzy cross-classification algorithm produces both a fuzzy partition and a fuzzy partition of characteristics compatible with the former. The advantages of this algorithm include the ability to observe not only the fuzzy classes obtained and their relationship, but also the characteristics corresponding to each final class of objects. Each object class may be well described using the corresponding characteristics. These are the characteristics that have contributed to the separation of the respective fuzzy class. Fuzzy divisive hierarchical cross-classification of therapeutic muds based on their physicochemical characteristics allowed an objective interpretation of their origin and maturation and helped in their classification. It also permitted quantitative and qualitative identification of the compo-... 

VI. FUZZY HIERARCHICAL CLASSIFICATION TECHNIQUES IN ANALYTICAL CHEMISTRY... 

Dumitrescu and Kekedy used the FHDC procedure for the classification of mineral waters. Each sample was characterized by the analytical chemical data for eight major components CO2, HCO, Cl , Ca, Mg ", Fe, Na, and mineralization. Tentatively, classifications were made with a smaller number of components. The components present in greater concentration were successively omitted so as to investigate the possible influence on the classification of components present in smaller concentrations. Thus fuzzy hierarchical clustering was performed by considering successively only seven components (mineralization omitted), six constituents (mineralization and HCOj content omitted), or only five constituents (the previous and COj content omitted), respectively. With eight components the FHDC method yielded three clusters. To characterize the three distinct classes of water found, the total hardness (in German... 

This classification must use primary physical and chemical properties of the elements. Their selection is rather difficult since they have to be expressed numerically and have to be known for all the investigated elements. The method applied for the classification uses fuzzy set theory in order to obtain a clustering of the set of chemical elements. This method is hierarchical and does not suppose a preliminary knowledge with respect to the number of classes in the data set, nor does it suppose a certain structure for the classes. Both the number of classes and their mutual relationships will be detected by the classification algorithm used. 

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). 

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