Relational Generalizations of Cluster Validity Indices

Numerous computational schemes have arisen over the years that attempt to learn information about objects based upon the similarity or dissimilarity of one object to another. One such scheme, clustering, looks for self-similar groups of objects. To use clustering algorithms, an investigator must often have
a priori
knowledge of the number of clusters, i.e.,
$c$
, to search for in the data. Moreover, it is often convenient to have ways to rank the returned results, either for a single value of
$c$
, a range of
$c$
’s different clustering methods, or any combination thereof. However, the task of assessing the quality of the results, so that
$c$
may be determined objectively, is currently ill-defined for object–object relationships. To bridge this gap, we generalize three well-known validity indices: the modified Hubert’s Gamma, Xie–Beni, and the generalized Dunn’s indices, to relational data. In doing so, we develop a framework to convert many other validity indices to a relational form. Numerical examples on 12 datasets (samples from four normal mixtures, four real-world object datasets, and four real-world “pure relational” datasets) using the relational duals of the hard, fuzzy, and possibilistic
$c$
-means cluster algorithms are offered to illustrate and evaluate the new indices.