Fuzzy cross-classification

Chemical data can sometimes be very complex. Each object (or sample) in the data set is then described by a large number of features (characteristics, attributes). Usually the characteristics are not of equal importance for the clusters in the data set. Each cluster may be characterized in terms of its relevant features, the latter being those that are responsible for holding together the points of the cluster. The objects in a class will have comparable values of the common relevant characteristics. 

Knowledge of the cluster structure of a data set is complete only if we are able to detect the clusters as well as the corresponding feature classes. Simultaneous classification of the objects and their characteristics is necessary in many of the practical problems encountered in analytical chemistry and related fields. Such problems include the following  

Each column i, i = 1,2. s, in this table represents the values of the i-th characteristic, and y denotes the corresponding p component vector. If now Y = y, . y is the characteristics set where y is from then yf is the value of the fc-th set of characteristics with respect to the object xL As indicated, some practical problems require simultaneous classification (or cross-classification) of the objects and their corresponding characteristics. Therefore we need a fuzzy partition of X compatible with a fuzzy partition of the set Y of characteristics. 

If A is a set of materials (or substances) and T is a set of analyses of these materials, we require a partition of X and a partition of Y such that each material in a given class i may be analyzed by using the analyses of 

Let Y = y y, . ybe the set of modified characteristics, where y is a vector of dimension n and Q = B, 82. is the fuzzy partition of Y that fits the partition P. The prototype of the fuzzy class B, is v from R . The fuzzy partition Q may be obtained by minimizing the criterion function, 

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FIGURE 8 Fuzzy cross-classification hierarchy of therapeutic muds. 

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-... 

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 ... 

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

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