Lance & Williams (l966a. 1967c) proposed to vary the parameter P (eq. 8.11 or 8.12) between -1 and +1 to obtain a series of intermediates solutions between single linkage chaining and the space dilation of complete linkage. The method is also called beta-flexible clustering by some authors. Lance & Williams (ibid.) have shown that, if the other parameters are constrained a follows:
the resulting clustering is ultrametric (no reversals; Section 8.6).
When p is close to 1, strong chaining is obtained. As P decreases and becomes negative, space dilation increases. The space properties are conserved for small negative values of P, near -0.25. Figure 8.14 shows the effect of varying P in the clustering of 20 objects. Like weighted centroid clustering, flexible clustering is compatible with all association measures except Pearson's r.
Pinel-Alloul et al. (1990) studied phytoplankton in 54 lakes of Québec to determine the effects of acidification, physical and chemical characteristics, and lake morphology on species assemblages. Phytoplankton was enumerated into five main taxonomic categories (microflagellates, chlorophytes, cyanophytes, chrysophytes, and pyrrophytes). The data were normalized using the generalized form of the Box-Cox method that finds the best normalizing transformation for all species (Subsection 1.5.6). A Gower (519) similarity matrix, computed among lakes, was subjected to flexible clustering with parameter P = -0.25. Six clusters were found, which were roughly distributed along a NE-SW geographic axis and corresponded to increasing concentrations of total phytoplankton, chlorophytes, cyanophytes, and microflagellates. Explanation of the phytoplankton-based lake typology was sought by comparing it to the environmental variables (Section 10.2.1).
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