Hierarchical divisive methods

The hierarchical methods so far discussed are called agglomerative. Good results can also be obtained with hierarchical divisive methods, i.e., methods that first divide the set of all objects in two so that two clusters result. Then each cluster is again divided in two, etc., until all objects are separated. These methods also lead to a hierarchy. They present certain computational advantages [21,22]. 

Little has been reported on the use of hierarchical divisive methods for processing chemical data sets (other than the inclusion of the minimum-diameter method in some of the comparative studies mentioned above). Recursive partitioning, which is a supervised classification technique very closely related to monothetic divisive clustering, has, however, been used at the GlaxoSmithKline and Organon companies. 

There are two main types of clustering techniques hierarchical and nonhierarchical. Hierarchical cluster analysis may follow either an agglomerative or a divisive scheme agglomerative techniques start with as many clusters as objects and, by means of repeated similarity-based fusion steps, they reach a final situation with a unique cluster containing all of the objects. Divisive methods follow exactly the opposite procedure they start from an all-inclusive cluster and then perform a number of consecutive partitions until there is a bijective correspondence between clusters and objects (see Fig. 2.12). In both cases, the number of clusters is defined by the similarity level selected. 

There are many other benefits of pluralism, some of which have already been discussed in detail by Hasok Chang (2012, chap. 5.2) or mentioned before. First of all, if we acknowledge that biology, physics, chemistry, geology and so on (as well as correspondingly their various research fields) have different subject matter and different research aims that require different methods, it is obvious that monism can pick only a single aspect and disregard the rest. Pluralism instead allows a non-hierarchical division of labor, which in most fields of society is the most effective and successful approach. Moreover, as new kinds of issues arise, either out of the research process or by societal demand, science can flexibly adjust by... 

The hierarchical clustering methods are either agglomerative (i.e., the methods successively merge small clusters into larger ones), or divisive (i.e., the methods successively split large clusters into smaller ones). The methods differ in the rule used for merging or splitting clusters, the possibility to use... 

Structured micro-assemblies built from nanoparticle primary subunits can be classified into several categories arranged, hierarchical, and oriented nanoparticle assemblies. In addition to this, one-, two-, and three-dimensional systems exist under each of these divisions. Methods of preparation can differ vastly depending on the material being synthesized or processed, i here is hardly a universal method that can be applied for the majority of chemistries, but there are several general approaches involving wet chemical techniques that are commonly used. Examples of commonly used techniques include solvothermal, sol-gel processing, surfactant-assisted synthesis, or solvent-controlled synthesis [73]. 

E. Marengo and R. Todeschini, Linear discriminant hierarchical clustering a modeling and cross-validable divisive clustering method. Chemom. Intell. Lab. Syst., 19 (1993) 43-51. 

It is easy to see that Cj,C2 is a hard partition of the classical set C. Thus, from these assignment rules we finally obtain the classical (or hard) hierarchy corresponding to the respective fuzzy hierarchy. Accordingly, the fuzzy divisive hierarchical clustering (FDHC) procedure may be used to obtain the optimal cluster structure of the data set and a hierarchical relationship between clusters and subclusters. The method is especially useful when the number of clusters is unknown. In most cases the number of clusters to be expected in the data set is unknown. 

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