Granular Prototyping in Fuzzy Clustering

We introduce a logic-driven clustering in which prototypes are formed and evaluated in a sequential manner. The way of revealing a structure in data is realized by maximizing a certain performance index (objective function) that takes into consideration an overall level of matching (to be maximized) and a similarity level between the prototypes (the component to be minimized). The prototypes identified in the process come with the optimal weight vector that serves to indicate the significance of the individual features (coordinates) in the data grouping represented by the prototype. Since the topologies of these groupings are in general quite diverse the optimal weight vectors are reflecting the anisotropy of the feature space, i.e., they show some local ranking of features in the data space. Having found the prototypes we consider an inverse similarity problem and show how the relevance of the prototypes translates into their granularity.