Hierarchical clustering techniques, selection

Median partitioning is another statistical method distinct from RR The development of this methodology was driven by the need to select representative subsets from very large compound pools. Hierarchical clustering techniques... 

The manner in which sample-to-sample resemblance is defined is a key difference between the various hierarchical clustering techniques. Sample analyses may be similar to one another in a variety of ways and reflect interest in drawing attention to different underlying processes or properties. The selection of an appropriate measure of similarity is dependent, therefore, on the objectives of the research as set forth in the problem definition. Examples of different similarity measures or coefficients that have been used in compositional studies are average Euclidean distance, correlation, and cosine. Many others that could be applied are discussed in the literature dealing with cluster analysis (15, 18, 19, 36, 37). 

The hierarchical clustering techniques have been applied to many studies of gene expression patterns with some success.57 However, the hierarchical tree cannot determine the optimal number of clusters in the dataset. The limitation of the hierarchical algorithms is that the number of classes is determined by cutting the tree structure at an ad hoc level selected by the user. Such an ad hoc level does not necessarily reflect the true nature of the underlying structure of the gene expression data. 

In hierarchical clustering one can obtain any number of clusters K,lhierarchical clustering, with the difference that there K is defined a priori by the user. The question then arises which A -clustering is significant. To introduce the problem let us first consider a technique that was proposed for the non-hierarchical method MASLOC [27], which selects so-called robust clusters. 

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. 

In the following section the power of the fractional derivative technique is demonstrated using as example the derivation of all three known patterns of anomalous, nonexponential dielectric relaxation of an inhomogeneous medium in the time domain. It is explicitly assumed that the fractional derivative is related to the dimension of a temporal fractal ensemble (in the sense that the relaxation times are distributed over a self-similar fractal system). The proposed fractal model of the microstructure of disordered media exhibiting nonexponential dielectric relaxation is constructed by selecting groups of hierarchically subordinated ensembles (subclusters, clusters, superclusters, etc.) from the entire statistical set available. 

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