Ageneral theory of classificatory sorting strategies 1 . Hierarchical systems

Kruskal (1954, 1959), Dagnelie (1960) and Sokal and Sneath (1963), but it is doubtful whether even these extensive collections are complete. For general consideration, suppose that two groups (i) and (j) fuse to form a.group (k); then, extending (and slightly altering) the symbolism of Williams, Lambert and Lance (1966), we shall need to distinguish between three types of measure: (i)-measures, which define a property of a group, (i,j)-measures, which define a resemblance or diff."erence between two groups, and (i j, k)-measures, which define some difference between the original two groups, considered jointly, and that formed by their fusion. Of these, (i)-measures are confined to clusterin!! techniques except in so far as they may be incidentally required in the course of calculation of (ij, k)-measures. Secondly, the chosen measure has to be incorporated into a "sorting strategy" whereby groups of elemcnts are extracted. Selected sorting strategies have received some comparative study (Sokal and Michener, 1958; Soka1 and Sneath, 1963; Williams and Dale, 1965; Williams, Lambert and Lance, 1966), but until recently have bcen regarded as separate and largely unrelated syslems. In a recent brier communication (Lance and Williams, 19Mb) we have pointed out that the five best-known hierarchical strategies are, for at least one of the (i,j)-measures, variants of a single linear system which will, moreover, generate an infinite set of new strategies. It is with the further generalization of this system that we are now concerned. Ageneral theory of classificatory sorting strategies 1. Hierarchical systems