Three-way distances

In this paper, dissimilarity relations are defined on triples rather than on dyads. We give a definition of a three-way distance analogous to that of the ordinary two-way distance. It is shown, as a straightforward generalization, that it is possible to define three-way ultrametric, three-way star, and three-way Euclidean distances. Special attention is paid to a model called the semi-perimeter model. We construct new methods analogous to the existing ones for ordinary distances, for example: principal coordinates analysis, the generalized Prim (1957) algorithm, hierarchical cluster analysis.