Generalized proximity function comparisons

Generalizations are given for a simple cross‐product statistic that has been used to compare two proximity functions defined on the Cartesian product of a given set S. The extensions developed here are based on a similar cross‐product form but rely on two functions defined either on S × S × S or on S × S × S × S. Typically, the latter are obtained in some way from the original proximity measures. A variety of these indices, suggested by the recent literature on cluster analysis and seriation, are mentioned explicitly along with a discussion of an approximate permutation test for assessing their relative size. Finally, a representative application of two generalized cross‐product statistics is presented in terms of a data analysis problem that requires the comparison of a partition of S against a proximity function denned on S × S.