International
Tables for Crystallography Volume H Powder diffraction Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk © International Union of Crystallography 2018 
International Tables for Crystallography (2018). Vol. H, ch. 3.8, p. 327
Section 3.8.3.1. Dendrograms^{a}Department of Chemistry, University of Glasgow, University Avenue, Glasgow, G12 8QQ, UK 
Using d and s, agglomerative, hierarchical cluster analysis is now carried out, in which the patterns are put into clusters as defined by their distances from each other. [Gordon (1981, 1999) and Everitt et al. (2001) provide excellent and detailed introductions to the subject. Note that the two editions of Gordon's monograph are quite distinct and complementary.] The method begins with a situation in which each pattern is considered to be in a separate cluster. It then searches for the two patterns with the shortest distance between then, and joins them into a single cluster. This continues in a stepwise fashion until all the patterns form a single cluster. When two clusters (C_{i} and C_{j}) are merged, there is the problem of defining the distance between the newly formed cluster and any other cluster C_{k}. There are a number of different ways of doing this, and each one gives rise to a different clustering of the patterns, although often the difference can be quite small. A general algorithm has been proposed by Lance & Williams (1967), and is summarized in a simplified form by Gordon (1981). The distance from the new cluster formed by merging C_{i} and C_{j} to any other cluster C_{k} is given byThere are many possible clustering methods. Table 3.8.1 defines six commonly used clustering methods, defined in terms of the parameters α, β and γ. All these methods can be used with powder data; in general, the groupaveragelink or singlelink formalism is the most effective, although differences between the methods are often slight.

The results of cluster analysis are usually displayed as a dendrogram, a typical example of which is shown in Fig. 3.8.6(a), where a set of 13 powder patterns is analysed using the centroid method. Each pattern begins at the bottom of the plot as a separate cluster, and these amalgamate in stepwise fashion linked by horizontal tie bars. The height of the tie bar represents a similarity measure as measured by the relevant distance. As an indication of the differences that can be expected in the various algorithms used for dendrogram generation, Fig. 3.8.6(e) shows the same data analysed using the singlelink method: the resulting clustering is slightly different: the similarity measures are larger, and, in consequence, the tie bars are higher on the graph. [For further examples see Barr et al. (2004b,c) and Barr, Dong, Gilmore & Faber (2004).]
References
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