Computing and Clustering in the Environment of Order-2 Information Granules

A visible trend in representing knowledge through information granules manifests in the developments of information granules of higher type and higher order, in particular, type-2 fuzzy sets and order-2 fuzzy sets. All these constructs are aimed at the formalization and processing data at a certain level of abstraction. Along the same line, in the recent years, we have seen intensive developments in fuzzy clustering, which are not surprising in light of a growing impact of clustering on fundamentals of fuzzy sets (as supporting ways to elicit membership functions) as well as algorithms (in which clustering and clusters form an integral functional component of various fuzzy models). In this study, we investigate order-2 information granules (fuzzy sets) by analyzing their formal description and properties to cope with structural and hierarchically organized concepts emerging from data. The design of order-2 information granules on a basis of available experimental evidence is discussed and a way of expressing similarity (resemblance) of two order-2 information granules by engaging semantically oriented distance is discussed. In the sequel, the study reported here delivers highly original contributions in the realm of order-2 clustering algorithms. Formally, the clustering problem under discussion is posed as follows: given is a finite collection of reference information granules. Determine a structure in data defined over the space of such granules. Conceptually, this makes a radical shift in comparison with data defined in the p -dimensional space of real numbers Rp. In this situation, expressing distance between two data deserves prudent treatment so that such distance properly captures the semantics and consequently, the closeness between any two information granules to be determined in cluster formation. Following the proposal of the semantically guided distance (and its ensuing design process), we develop an order-2 variant of the fuzzy C-means (FCM), discuss its detailed algorithmic steps, and deliver interpretation of the obtained clustering results. Several relevant applied scenarios of order-2 FCM are identified for spatially and temporally distributed data, which deliver interesting motivating arguments and underline the practical relevance of this category of clustering. Experimental studies are provided to further elicit the performance of the clustering method and discuss essential ways of interpreting results.