Granule structures, distances and measures in neighborhood systems

Abstract

High-dimensional, quantity, uncertain and diverse data sets bring serious challenges to the development of intelligent systems. Granular computing is a theoretical approach to deal with uncertain and massive data, including rough sets, fuzzy sets, quotient spaces, covering rough sets, neighborhood rough sets and etc. In this paper, by introducing the neighborhood rough set model, some structured data named neighborhood granules are formed to achieve the cognition of a neighborhood system. Then, a three-level structure of granules in the neighborhood system is proposed: the neighborhood granule, the neighborhood granule swarm and the neighborhood granule library. The size measures of neighborhood granules and neighborhood granule swarms are also presented. Furthermore, we define a variety of distance measures for the neighborhood granules and the neighborhood granule swarms, and discuss their properties and relationships. Finally, considering the uncertainties of neighborhood systems, we propose the uncertainty measures of various neighborhood granules from the perspectives of algebra and entropy, and prove the monotonicity principle of these measures. Theoretical analysis and examples show that the granule structures, distances and measures in neighborhood systems are effective tools for complex data measuring and classifying.