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Approximation Spaces, Reducts and Representatives

  • Jaroslaw Stepaniuk
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 19)

Abstract

The main objective of this chapter is to discuss different approaches to searching for optimal approximation spaces. Basic notions concerning rough set concept based on generalized approximation spaced are presented. Different constructions of approximation spaces are described. The problems of attribute and object selection are discussed.

Keywords

Boolean Function Decision Table Approximation Space Uncertainty Function Discernibility Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Bryniarski E., Wybraniec-Skardowska U.: Generalized Rough Sets in Contextual Spaces. In: T. Y. Lin, N. Cercone (eds.), Rough sets and data mining. Analysis of imprecise data, Kluwer Academic Publishers, Boston (997) 339–354Google Scholar
  2. 2.
    Cattaneo G.: Generalized rough sets. Preclusivity fuzzy-intuitionistic (BZ) lattices. Studia Logica 58 (1997) 47–77CrossRefGoogle Scholar
  3. 3.
    Cattaneo G.: Mathematical foundations of roughness and fuzziness (manuscript). University of Milan (1997)Google Scholar
  4. 4.
    Dasarathy B. V. ed.: Nearest neighbor pattern classification techniques. IEEE Computer Society Press (1991)Google Scholar
  5. 5.
    Dubois D., Prade H.: Similarity versus preference in fuzzy set-based logics. In: E. Orlowska (ed.), Incomplete information: rough set analysis, Springer-Verlag ( Physica Verlag ), Chapter 14 (1997)Google Scholar
  6. 6.
    Funakoshi K., Ho T. B..: Information retrieval by rough tolerance relation. In: In: Tsumoto S., Kobayashi, S., Yokomori, T., Tanaka, H. (ed.), Proceedings of the Fourth International Workshop on Rough Sets, Fuzzy Sets and Machine Discovery (RSFD’96), Tokyo Nov. 6–8 (1996) 31–35Google Scholar
  7. 7.
    Gemello R., Mana F.: An Integrated characterization and discrimination scheme to improve learning efficiency in large data sets, Proceedings of the Eleventh International Joint Conference on Artificial Intelligence, Detroit MI, 20–25 August (1989) 719–724Google Scholar
  8. 8.
    Hu X., Cercone N.: Rough sets similarity-based learning from databases. In: Proceedings of the First International Conference on Knowledge Discovery and Data Mining, Montreal, Canada, August 20–21 (1995) 162–167Google Scholar
  9. 9.
    Katzberg J. D., Ziarko W.: Variable precision extension of rough sets. Fundamenta Informaticae 27 (1996) 155–168Google Scholar
  10. 10.
    Konikowska B.: A logic for reasoning about similarity. In: E. Orlowska (ed.), Incomplete information: rough set analysis, Chapter 15 (1997)Google Scholar
  11. 11.
    Krawiec K., Slowinski R, Vanderpooten D.: Construction of rough classifiers based on application of a similarity relation. In: In: Tsumoto S., Kobayashi, S., Yokomori, T., Tanaka, H. (ed.), Proceedings of the Fourth International Workshop on Rough Sets, Fuzzy Sets and Machine Discovery (RSFD’96), Tokyo Nov. 6–8 (1996) 23–30Google Scholar
  12. 12.
    Krawiec K., Slowinski R., Vanderpooten D.: Learning of decision rules from similarity based rough approximations (this book)Google Scholar
  13. 13.
    Kretowski M., Polkowski L., Skowron A., Stepaniuk J.: Data reduction based on rough set theory. In: Y. Kodratoff, G. Nakhaeizadeh, Ch. Taylor (eds.), Proceedings of the International Workshop on Statistics, Machine Learning and Knowledge Discovery in Databases, Heraklion April 25–27 (1995) 210–215Google Scholar
  14. 14.
    Kretowski M., Stepaniuk J.: Selection of objects and attributes, a tolerance rough set approach. In: Proceedings of the Poster Session of Ninth International Symposium on Methodologies for Intelligent Systems, Zakopane Poland,June 10–13 (1996) 169–180Google Scholar
  15. 15.
    Kryszkiewicz M.: Maintenance of reducts in the variable precision rough set model. In: T. Y. Lin, N. Cercone (eds.), Rough sets and data mining analysis of imprecise data, Kluwer Academic Publishers, Dordrecht (1997) 355–372CrossRefGoogle Scholar
  16. 16.
    Marcus S.: Tolerance rough sets, Cech topologies, learning processes. Bull.Polish Acad. Sci. Ser. Sci. Tech. 42 /3 (1994) 471–487Google Scholar
  17. 17.
    Michalewicz Z.: Genetic algorithms + data structures = evolution programs, Springer-Verlag, Berlin (1996)CrossRefGoogle Scholar
  18. 18.
    Michalski R. S., Larson J. B.: Selection of most representative training examples and incremental generation of VL1 hypotheses. Report 867 Department of Computer Science University of Illinois at Urbana-Champaign (1978)Google Scholar
  19. 19.
    Nguyen S. H.,Skowron A.: Searching for relational patterns in data. In: Proceedings of the First European Symposium on Principles of Data Mining and Knowledge Discovery (PKDD’97) Trondheim, Norway, June 25–27 Lecture Notes in Artificial Intelligence 1263 (1997) 265–276Google Scholar
  20. 20.
    Nieminen J.: Rough tolerance equality. Fundamenta Informaticae 11 (1988) 289296Google Scholar
  21. 21.
    Pawlak Z.: Rough sets. International Journal of Computer and Information Science 11 (1982) 341–356CrossRefGoogle Scholar
  22. 22.
    Pawlak Z.: Rough sets: theoretical aspects of reasoning about data, Kluwer Academic Publishers, Dordrecht (1991)Google Scholar
  23. 23.
    Pawlak Z., Skowron A.: Rough membership functions. In: M. Fedrizzi, J.Kacprzyk, R. R. Yager (eds.), Advances in the Dempster-Shafer theory of evidence, John Wiley and Sons, New York (1994) 251–271Google Scholar
  24. 24.
    Polkowski L., Skowron A., Zytkow J.: Tolerance based rough sets. In: T.Y.Lin, A.M.Wildberger (eds.), Soft Computing Simulation Councils, San Diego (1995) 55–58Google Scholar
  25. 25.
    Pomykala J. A.: Approximation operations in approximation space, Bull. Polish Acad.Sci.Ser. Sci. Math. 35 653–662Google Scholar
  26. 26.
    Pomykala J. A.: On definability in the nondeterministic information system. Bull. Polish Acad. Sci.Ser. Sci. Math., 36 193–210Google Scholar
  27. 27.
    Skowron A.: Data filtration: a rough set approach. In: W. Ziarko (ed.), Rough sets, fuzzy sets and knowledge discovery, Springer-Verlag, Berlin (1994) 108–118CrossRefGoogle Scholar
  28. 28.
    Skowron A.: Extracting laws from decision tables. Computational Intelligence 11/2 (1995) 371–388Google Scholar
  29. 29.
    Skowron A., Polkowski L.: Synthesis of decision systems from data tables. In: T. Y. Lin, N. Cercone (eds.), Rough sets and data mining. Analysis of imprecise data, Kluwer Academic Publishers, Boston (1997) 259–299CrossRefGoogle Scholar
  30. 30.
    Skowron A., Polkowski L., Komorowski J.: Learning tolerance relations by Boolean descriptors: automatic feature extraction from data tables. In: Tsumoto S., Kobayashi, S., Yokomori, T., Tanaka, H. (ed.), Proceedings of the Fourth International Workshop on Rough Sets, Fuzzy Sets and Machine Discovery (RSFD’96), Tokyo Nov. 6–8 (1996) 11–17Google Scholar
  31. 31.
    Skowron A, Rauszer C.: The Discernibility matrices and functions in information systems. In: R. Slowinski (ed.), Intelligent decision support. Handbook of applications and advances of rough sets theory, Kluwer Academic Publishers, Dordrecht (1992) 331–362CrossRefGoogle Scholar
  32. 32.
    Skowron A., Stepaniuk J.: Generalized approximation spaces. In: Proceedings of the Third International Workshop on Rough Sets and Soft Computing, San Jose, November 10–12 (1994) 156–163Google Scholar
  33. 33.
    Skowron A., Stepaniuk J.: Generalized approximation apaces. In: T.Y.Lin, A.M.Wildberger (eds.), Soft computing, Simulation Councils, San Diego (1995) 18–21Google Scholar
  34. 34.
    Skowron A., Stepaniuk J.: Tolerance approximation spaces. Fundamenta Informaticae 27 (1996) 245–253Google Scholar
  35. 35.
    Slowinski R.: A Generalization of the indiscernibility relation for rough sets analysis of quantitative information. Revista di Matematica per le Scienze Economiche e Sociali 15/1 (1992) 65–78Google Scholar
  36. 36.
    Slowinski R Strict and weak indiscernibility of objects described by quantitative attributes with overlapping norms. Foundations of Computing and Decision Sciences 18 (1993) 361–369Google Scholar
  37. 37.
    Slowinski R, Vanderpooten D.: Similarity relation as a basis for rough approximations. Warsaw University of Technology, Institute of Computer Science Research Report 53 (1995)Google Scholar
  38. 38.
    Stepaniuk J., Kretowski M.: Decision system based on tolerance rough sets. In: Proceedings of the Fourth International Workshop on Intelligent Information Systems, Augustow, Poland, June 5–9 (1995) 62–73Google Scholar
  39. 39.
    Stepaniuk J.: Similarity based rough sets and learning. In: Tsumoto S., Kobayashi, S., Yokomori, T., Tanaka, H. (ed.), Proceedings of the Fourth International Workshop on Rough Sets, Fuzzy Sets and Machine Discovery (RSFD’96), Tokyo Nov. 6–8 (1996) 18–22Google Scholar
  40. 40.
    Stanfill C., Waltz D.: Toward memory-based reasoning. Communications of the ACM 29 (1986) 1213–1228CrossRefGoogle Scholar
  41. 41.
    Tentush I.: On minimal absorbent sets for some types of tolerance relations. Bull. Polish Acad. Sci. Ser. Sci. Tech. 43/1 (1995) 79–88Google Scholar
  42. 42.
    Yao Y. Y., Lin T. Y.: Generalization of rough sets using modal logic. Intelligent Automation and Soft Computing 2 (1996) 103–120Google Scholar
  43. 43.
    Yao Y. Y., Wong S. K. M., Lin T. Y.: A review of rough set models. In: T. Y. Lin, N. Cercone (eds.), Rough sets and data mining. Analysis of imprecise data, Kluwer Academic Publishers, Boston (1997) 47–75CrossRefGoogle Scholar
  44. 44.
    Vakarelov D.: Information systems, similarity relations and modal logic. In: E. Orlowska (ed.), Incomplete information: Rough set analysis, Springer - Verlag (Physica Verlag), Berlin (1997) Chapter 16Google Scholar
  45. 45.
    Wilson D. A., Martinez T. R.: Improved heterogeneous distance functions. Journal of Artificial Intelligence Research 6 (1997) 1–34Google Scholar
  46. 46.
    Wybraniec-Skardowska U.: On a generalization of approximation space. Bull. Polish Acad. Sci. Ser. Sci. Math. 37 (1989) 51–61Google Scholar
  47. 47.
    Zadeh L. A.: Similarity relations and fuzzy orderings. Information Sciences 3 (1971) 177–200CrossRefGoogle Scholar
  48. 48.
    Ziarko W.: Variable precision rough sets model. Journal of Computer and Systems Sciences 46/1 (1993) 39–59Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Jaroslaw Stepaniuk
    • 1
  1. 1.Institute of Computer ScienceBialystok University of TechnologyBialystokPoland

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