Abstract
Johnson has shown that the single linkage and the complete linkage hierarchical clustering algorithms induce a metric on the data known as the ultrametric. Through the use of the Lance and Williams recurrence formula, Johnson's proof is extended to four other common clustering algorithms. It is also noted that two additional methods produce hierarchical structures which can violate the ultrametric inequality.
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Milligan, G.W. Ultrametric hierarchical clustering algorithms. Psychometrika 44, 343–346 (1979). https://doi.org/10.1007/BF02294699
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DOI: https://doi.org/10.1007/BF02294699