Single-link clustering

Figure 4.8L Single link clustering for a five-sample, two-measurement data set.

Single-link clustering and Euclidean distance are used in the calculations. 

In single-link clustering, the proximity between two clusters is the minimum distance between any pair of items (one from each cluster), that is, the closest pair of points between each cluster. In contrast, in complete-link clustering, the proximity between two clusters is the maximum distance between any pair of items, that is, the farthest pair of points between each cluster. Single-link and complete-link represent the extremes of SAHN clustering. In... 

When bidentate solvents are present, the clusters may crystallize forming linear polymers by linking clusters together, as in complexes 292-295. This further diminishes solubility and reactivity of the alkyllithium compound, making the unassociated RLi species the most probable reactive form in synthetic processes. MeLi dissolved in diethoxymethane shows at room temperature a single Li NMR peak, that resolves into four distinct peaks at —80°C, pointing to temperature-dependent monomer/oligomer/polymer equilibria . 

It is not possible to make a general statement as to whether the single- or centroid-link approach will perform better. The centroid method results in a better measure of the distance betw een clusters when the clusters are well separated. This is because it measures the distance between the centers of the clusters rather than the distances from the edges as in single link. On the other hand, the centroid method is sensitive to outliers because these unusual points... 

The single-link approach links clusters based on the distance between the nearest neighbors contained in the clusters, as illustrated in Figure 4.8. In step i the samples are initially in five individual clusters, and samples A and B... 

The single and centroid linking methods do not always join clusters in. the same order as in the previous example. Figure 4.12 shows an example where these methods yield different dendrograms. If the single-link method is used, the clustering proceeds as follows (solid lines in Figure 4.12) ... 

Figure 4.12. Single-link and centroid-link techniques lead to different clustering schemes. Solid, single link dashed, centroid link.

The H-atom chain is to solid-state structures as the diatomic molecule is to polyatomic molecules, e.g., clusters. The geometric-structure problems for H2 and Hoc are so simple that one can focus on the electronic structure problem exclusively. However, real solid-state structures, e.g., a solid with linked clusters or even bulk elemental Al, are not found in the form of a linear chain, square sheet or simple cubic structures so we need a way to treat solids with more complex structures, i.e., define a repeat unit that is more than a single atom. 

Single link (or nearest neighbour) method, d(Ci, Cj) is the smallest distance between the observations for both clusters (d(Ci,Cj) = mind(dj j),... 

Agglomerative methods, such as single link and complete link, are stepwise procedures. The formulation in (5)-(7) allows us to define the hierarchical clustering problem in terms of combinatorial optimization. To do this, however, we need an appropriate internal clustering criterion. The most obvious is squared error. 

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