Average linkage method

There is a wide variety of hierarchical algorithms available and it is impossible to discuss all of them here. Therefore, we shall only explain the most typical ones, namely the single linkage, the complete linkage and the average linkage methods. 

An example of an application is shown in Fig. 30.10. This concerns the classification of 42 solvents based on three solvatochromic parameters (parameters that describe the interaction of the solvents with solutes) [13]. Different methods were applied, among which was the average linkage method, the result of which is shown in the figure. According to the method applied, several clusterings can be found. For instance, the first cluster to split off from the majority of solvents consists of solvents 36, 37, 38, 39, 40, 41, 42 (t-butanol, isopropanol, n-butanol. 

Of the four different methods of cluster analysis applied, the method of Ward described in the Clustan User Manual (10), worked best when compared to the single-, complete-, or average-linkage methods. Using Ward s method, two clusters, Gn and Gm, are fused when by pooling the variance within two existing clusters the variance of the so formed clusters increases minimally. The variance or the sum of squares within the classes will be chosen as the index h of a partition. 

In Example 5.9, aggregation of clusters was carried out by the weighted average linkage methods. In general, the distance to a new object or cluster k (labeled in the example by a star) is computed by forming the average between the individual distances of objects A and B to object i. 

There exist several methods of hierarchical clustering which use diverse measures of distance or similarity, respectively, e.g., single linkage, complete linkage, average linkage, centroid linkage, and Ward s method (Sharaf et al. [1986], Massart et al. [1988], Otto [1998] Danzer et al. [2001]). 

Hierarchical cluster analysis (Section 6.4)—with the result represented by a dendrogram—is a complementary, nonlinear, and widely used method for cluster analysis. The distance measure used was dTANi (Equation 6.5), and the cluster mode was average linkage. The dendrogram in Figure 6.6 (without the chemical structures) was obtained from the descriptor matrix X by... 

Compute the distances between all clusters using complete linkage, single linkage, average linkage, or other methods. 

Fig. 17.3. Tree displaying the relationships between the hexapeptide sequence patterns after cluster analysis of the pairwise distances between patterns using the method of average-linkage clustering...

Table 9.3 A comparison of the single-linkage, complete-linkage and average-linkage cluster methods using the data in Table 92 The figures in parentheses indicate the distance between the clusters as they are formed In this particular case Ward s clustering follows the same order of cluster formation as the group average method.

Horizontal icicle plot with clusters combined using the average linkage between groups method (called UPGMA)... 

Unweighted Average Linkage With this method, the number of objects in a cluster is used for weighting the cluster distances ... 

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