Ward s method

A fourth hierarchical method that is quite popular is Ward s method [Ward 1963]. This method merges those two clusters whose fusion minimises the information toss due to the fusion. Information loss is defined in terms of a function rvhich fdr each cluster i corresponds to the total sum of squared deviations from the mean of the cluster ... 

Fig. 8.8. Result of cluster analysis of 88 German wines according to Ward s method (Thiel et al. [2004])...

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]). 

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. 

Table I. Means and Standard Deviations of the Data for 23 Differentially Substituted Pyridazinones Before and After Analysis by Ward s Method.

Figure 4. Dendrogram showing sub-division of main chemical group according to cluster analysis (Euclidean distance Ward s method).

Figure 7.3 Authentication of monovarietal virgin olive oils results of applying multidimensional scaling to volatile compounds. The distance was Manhattan (city block) and the amalgamation was Ward s method. Note A, cv. Arbequina C, cv. Coratina K, cv. Koroneiki P, cv. Picual (source SEXIA Group-Instituto de la Grasa, Seville, Spain).

Cluster Analysis. The general technique used for grouping ore and artifact samples is cluster analysis. The computational method used was Ward s method (13),... 

Ward s method cluster analysis. The resultant dendrogram grouped all data into one undiflFerentiated cluster. 

These results show clearly the importance of the optimization criterion to clustering. The computationally simple Ward s method performs better than the simulated annealing approach with a simplistic criterion. However, a criterion that more correctly accounts for the hierarchy, by minimizing the sum of squared error at each level, performs much better. As with partitional clustering the application of simulated annealing to hierarchical clustering requires careful selection of the internal clustering criterion. 

Another procedure, Ward s method, is commonly encountered in chemo-metrics. A centroid point is calculated for all combinations of two clusters and the distance between this point and all other objects calculated. In practice this technique generally favours the production of small clusters. 

The process is repeated between BC and other objects, and the iteration starts again with the new distance matrix until a single cluster is produced. The dendrogram from applying Ward s method is illustrated in Figure 11. 

Figure 11 Dendrogram of the data from Table 7(a) using Ward s method...

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