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The aim of this paper is to study methods of twofold membership clustering using the nearest prototype and nearest neighbor. The former uses the K -means, whereas the latter extends the single linkage in agglomerative hierarchical clustering. The concept of inductive clustering is moreover used for the both methods, which means that natural classification rules are derived as the results of clustering, a typical example of which is the Voronoi regions in K -means clustering. When the rule of nearest prototype allocation in K -means is replaced by nearest neighbor classification, we have inductive clustering related to the single linkage in agglomerative hierarchical clustering. The former method uses K -means or fuzzy c -means with noise clusters, whereby twofold memberships are derived; the latter method also derives two memberships in a different manner. Theoretical properties of the both methods are studied. Illustrative examples show implications and significances of this concept.

Nearest Prototype and Nearest Neighbor Clustering with Twofold Memberships Based on Inductive Property