Rough Set on Two Universal Sets Based on Multigranulation

Rough Set on Two Universal Sets Based on Multigranulation

D. P. Acharjya (VIT University, India) and Mary A. Geetha (VIT University, India)
Copyright: © 2014 |Pages: 21
DOI: 10.4018/978-1-4666-6086-1.ch009
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The fundamental concept of crisp set has been extended in many directions in the recent past. The notion of rough set by Pawlak is noteworthy among them. The rough set philosophy is based on the concept that there is some information associated with each object of the universe. There is a need to classify objects of the universe based on the indiscernibility relation among them. In the view of granular computing, rough set model is researched by single granulation. It has been extended to multigranular rough set model in which the set approximations are defined by using multiple equivalence relations on the universe simultaneously. However, in many real life scenarios, an information system establishes the relation with different universes. This gave the extension of multigranulation rough set on single universal set to multigranulation rough set on two universal sets. This chapter defines multigranulation rough set for two universal sets U and V. In addition, the algebraic properties, measures of uncertainty and topological characterization that are interesting in the theory of multigranular rough sets are studied. This helps in describing and solving real life problems more accurately.
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Information technology revolution in the recent past has brought radical change in the way data are collected or generated for ease of decision making. The huge data collected has no relevance unless it provides certain meaningful information pertaining to the interest of an organization. Therefore, the real challenge lies in converting huge data into knowledge. This leads to classification and clustering. The earliest to handle classification is classical set. In addition, knowledge associated with classical set is very limited and it fails to process ill-posed objects. But, the objects associated in the information system contains uncertainties and are imprecise in nature. Therefore, the rudimentary concept of classical sets has been extended in many directions as far as modeling of real life situations is concerned. The earliest is the notion of Fuzzy set by L. A. Zadeh (1965) that captures impreciseness in information. On the other hand rough sets of Z. Pawlak (1982, 1991) capture indiscernibility among objects to model imperfect knowledge. The basic philosophy is that human knowledge about a universe depends upon their capability to classify its objects. So, classification of a universe and indiscernibility relations defined on it are known to be interchangeable notions. The basic idea of rough set is based upon the approximation of sets by pair of sets known as lower approximation and upper approximation. Here, the lower and upper approximation operators are based on equivalence relations. However, the requirement of equivalence relations is a restrictive one and failure in many real life situations. In order to achieve this, rough set is generalized to binary relations (Yao, 1998; Kondo, 2006; Pawlak & Skowron, 2007a), fuzzy proximity relations (Tripathy & Acharjya (2008, 2010)), intuitionistic fuzzy proximity relations (Tripathy, 2006; Tripathy & Acharjya (2009, 2011)), Boolean algebras (Liu, 2005; Pawlak & Skowron, 2007b), fuzzy lattices (Liu, 2008), completely distributive lattices (Chen et. al., 2006) and neighborhood systems (Lin, 1989). Development of these techniques and tools is studied under different domains like knowledge discovery in database, computational intelligence, knowledge representation, granular computing etc. (Saleem Durai et al., 2012; Acharjya et al. (2011, 2012); Tripathy et al., 2011).

Granular computing is an upcoming conceptual and computing pattern of information processing. It has been strongly encouraged by the urgent need for processing practical data in an intelligent manner (Pedrycz, 2007; Pedrycz et al, 2002). Such processing need is now commonly available in vast quantities into a humanly manageable abstract knowledge. On the contrary, granular computing offers a platform to transit from the current machine-centric to human-centric approach to gather information and knowledge. Granular computing as opposed to numeric computing is knowledge oriented. Numeric computing is data oriented. The origin of granular computing is in the context of fuzzy sets (Zadeh, 1965). But, there are many other theories like interval analysis, rough set theory and probabilistic approach, which follow this approach.

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