Hierarchical Cross-Classification

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 simultaneous clustering procedure for X and Y may generalize to a pair (C, D). This procedure may be described as follows  

Compute a binary partition P = A, of the fuzzy set C using the GFNM algorithm (or other appropriate procedure). 

A fuzzy partition Q = B, Bj of the fuzzy set D is obtained using GFNM procedure. Thus we have 

At the next classification level the fuzzy classes and 2 of the partition P have to be split. We may compute a binary fuzzy partition of Ai, i = 1,2, as usual using GFNM algorithm. To obtain a stronger correla- 

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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... 

FIGURE 5 The first decomposition level of hierarchical cross-classification. 

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-... 

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

A number of chemometric tools have been employed for these classifications, including partial least squares - hierarchical cluster analysis (PLS-HCA) for Viagra tablets [98] and antimalarial artesunate tablets [99]. de Peinder et al. used partial least squares discriminant analysis (PLS-DA) models to distinguish genuine from counterfeit Lipitor tablets even when the real API was present [100]. The counterfeit samples also were found to have poorer API distribution than the genuine ones based on spectra collected in a cross pattern on the tablet. 

Wang et al. 2012. Cross-subject workload classification with a hierarchical bayes model. Neuroimage 59 (1) 64-69. 

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