
Research Article


Set Approximation in Incomplete Data


Renpu Li



ABSTRACT

The problem of set approximation in incomplete data is addressed.
Different with complete data where the upper/lower approximation of an object
set is certain and can be given by one set, for incomplete data upper/lower
approximation of a set is uncertain and needs to be bracketed by a set pair.
From the completion view of incomplete data, the semantic interpretations of
four boundaries used to approximate a set in incomplete data are given. It is
illustrated that existing definitions based on tolerance class or covering are
not enough to describe precisely the set approximation in incomplete data. Based
on a concept of interval granule, new methods are presented for incomplete data
to compute the four approximation boundaries of a set. This study provides a
new view of granular computing on set approximation in incomplete data and is
helpful for computing the uncertainty of a set more accurately.








REFERENCES 
Couso, I. and D. Dubois, 2011. Rough sets, coverings and incomplete information. Fundamenta Inform., 108: 223247. CrossRef 
GrzymalaBusse, J.W., 2005. Incomplete Data and Generalization of Indiscernibility Relation, Definability and Approximations. In: Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing, Slezak, D., G. Wang, M. Szczuka, I. Duntsch and Y. Yao (Eds.). Springer, Berlin, Germany, ISBN13: 9783540286530, pp: 244253.
Jarvinen, J. and J.A. Kortelainen, 2004. Note on Definability in Rough Set Theory. In: Current Issues in Data and Knowledge Engineering, De Baets, B., R. De Caluwe, G. De Tre, J. Fodor, J. Kacprzyk and S. Zadrozny (Eds.). Akademicka Oficyna Wydawnicza Exit, Warsaw, Poland, pp: 272277.
Kryszkiewicz, M., 1998. Rough set approach to incomplete information systems. Inform. Sci., 112: 3949. CrossRef 
Kryszkiewicz, M., 1999. Rules in incomplete information systems. Inform. Sci., 113: 271292. CrossRef 
Lipski Jr., W., 1979. On semantic issues connected with incomplete information databases. ACM Trans. Database Syst., 4: 262296. CrossRef 
Nakamura, A., 1996. A rough logic based on incomplete information and its application. Int. J. Approximate Reason., 15: 367378. CrossRef 
Orlowska, E., 1998. Introduction: What You Always Wanted to Know about Rough Sets. In: Incomplete Information: Rough Set Analysis, Orlowska, E. (Ed.). PhysicaVerlag, Heidelberg, Germany, pp: 120.
Pawlak, Z., 1981. Information systems theoretical foundations. Inform. Syst., 6: 205218. CrossRef 
Pawlak, Z., 1991. Rough Sets: Theoretical Aspects of Reasoning about Data. 1st Edn., Kluwer Academic Publishers, London, UK., ISBN13: 9780792314721.
Pawlak, Z., J. GrzymalaBusse, R. Slowinski and W. Ziarko, 1995. Rough sets. Commun. ACM, 38: 8895. CrossRef 
Yao, Y.Y. and B. Zhou, 2007. A logic language of granular computing. Proceedings of the 6th IEEE International Conference on Cognitive Informatics, August 68, 2007, Lake Tahoo, CA., USA., pp: 178185.
Yao, Y.Y., 1998. Relational interpretations of neighborhood operators and rough set approximation operators. Inform. Sci., 111: 239259. Direct Link 
Yao, Y.Y., 2001. Information granulation and rough set approximation. Int. J. Intell. Syst., 16: 87104. Direct Link 
Yao, Y.Y., 2003. On generalizing rough set theory. Proceedings of the 9th International Conference on Rough Sets, Fuzzy Sets, Data Mining and Granular Computing, May 2629, 2003, Chongqing, China, pp: 4451.
Yao, Y.Y., 2007. A Note on Definability and Approximations. In: Transactions on Rough Sets VII, Peters, J.F., A. Skowron, V.W. Marek, E. Orlowska, R. Slowinski and W. Ziarko (Eds.). Springer, Berlin, Germany, pp: 274282.



