Skip to main content

Approximation Spaces, Reducts and Representatives

  • Chapter
Rough Sets in Knowledge Discovery 2

Part of the book series: Studies in Fuzziness and Soft Computing ((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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  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) 339–354

    Google Scholar 

  2. Cattaneo G.: Generalized rough sets. Preclusivity fuzzy-intuitionistic (BZ) lattices. Studia Logica 58 (1997) 47–77

    Article  Google Scholar 

  3. Cattaneo G.: Mathematical foundations of roughness and fuzziness (manuscript). University of Milan (1997)

    Google Scholar 

  4. Dasarathy B. V. ed.: Nearest neighbor pattern classification techniques. IEEE Computer Society Press (1991)

    Google Scholar 

  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. 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) 31–35

    Google Scholar 

  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) 719–724

    Google Scholar 

  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) 162–167

    Google Scholar 

  9. Katzberg J. D., Ziarko W.: Variable precision extension of rough sets. Fundamenta Informaticae 27 (1996) 155–168

    Google Scholar 

  10. Konikowska B.: A logic for reasoning about similarity. In: E. Orlowska (ed.), Incomplete information: rough set analysis, Chapter 15 (1997)

    Google Scholar 

  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) 23–30

    Google Scholar 

  12. Krawiec K., Slowinski R., Vanderpooten D.: Learning of decision rules from similarity based rough approximations (this book)

    Google Scholar 

  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) 210–215

    Google Scholar 

  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) 169–180

    Google Scholar 

  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) 355–372

    Chapter  Google Scholar 

  16. Marcus S.: Tolerance rough sets, Cech topologies, learning processes. Bull.Polish Acad. Sci. Ser. Sci. Tech. 42 /3 (1994) 471–487

    Google Scholar 

  17. Michalewicz Z.: Genetic algorithms + data structures = evolution programs, Springer-Verlag, Berlin (1996)

    Book  Google Scholar 

  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. 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) 265–276

    Google Scholar 

  20. Nieminen J.: Rough tolerance equality. Fundamenta Informaticae 11 (1988) 289296

    Google Scholar 

  21. Pawlak Z.: Rough sets. International Journal of Computer and Information Science 11 (1982) 341–356

    Article  Google Scholar 

  22. Pawlak Z.: Rough sets: theoretical aspects of reasoning about data, Kluwer Academic Publishers, Dordrecht (1991)

    Google Scholar 

  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) 251–271

    Google Scholar 

  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) 55–58

    Google Scholar 

  25. Pomykala J. A.: Approximation operations in approximation space, Bull. Polish Acad.Sci.Ser. Sci. Math. 35 653–662

    Google Scholar 

  26. Pomykala J. A.: On definability in the nondeterministic information system. Bull. Polish Acad. Sci.Ser. Sci. Math., 36 193–210

    Google Scholar 

  27. Skowron A.: Data filtration: a rough set approach. In: W. Ziarko (ed.), Rough sets, fuzzy sets and knowledge discovery, Springer-Verlag, Berlin (1994) 108–118

    Chapter  Google Scholar 

  28. Skowron A.: Extracting laws from decision tables. Computational Intelligence 11/2 (1995) 371–388

    Google Scholar 

  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) 259–299

    Chapter  Google Scholar 

  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) 11–17

    Google Scholar 

  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) 331–362

    Chapter  Google Scholar 

  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) 156–163

    Google Scholar 

  33. Skowron A., Stepaniuk J.: Generalized approximation apaces. In: T.Y.Lin, A.M.Wildberger (eds.), Soft computing, Simulation Councils, San Diego (1995) 18–21

    Google Scholar 

  34. Skowron A., Stepaniuk J.: Tolerance approximation spaces. Fundamenta Informaticae 27 (1996) 245–253

    Google Scholar 

  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) 65–78

    Google Scholar 

  36. Slowinski R Strict and weak indiscernibility of objects described by quantitative attributes with overlapping norms. Foundations of Computing and Decision Sciences 18 (1993) 361–369

    Google Scholar 

  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. 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) 62–73

    Google Scholar 

  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) 18–22

    Google Scholar 

  40. Stanfill C., Waltz D.: Toward memory-based reasoning. Communications of the ACM 29 (1986) 1213–1228

    Article  Google Scholar 

  41. Tentush I.: On minimal absorbent sets for some types of tolerance relations. Bull. Polish Acad. Sci. Ser. Sci. Tech. 43/1 (1995) 79–88

    Google Scholar 

  42. Yao Y. Y., Lin T. Y.: Generalization of rough sets using modal logic. Intelligent Automation and Soft Computing 2 (1996) 103–120

    Google Scholar 

  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) 47–75

    Chapter  Google Scholar 

  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 16

    Google Scholar 

  45. Wilson D. A., Martinez T. R.: Improved heterogeneous distance functions. Journal of Artificial Intelligence Research 6 (1997) 1–34

    Google Scholar 

  46. Wybraniec-Skardowska U.: On a generalization of approximation space. Bull. Polish Acad. Sci. Ser. Sci. Math. 37 (1989) 51–61

    Google Scholar 

  47. Zadeh L. A.: Similarity relations and fuzzy orderings. Information Sciences 3 (1971) 177–200

    Article  Google Scholar 

  48. Ziarko W.: Variable precision rough sets model. Journal of Computer and Systems Sciences 46/1 (1993) 39–59

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Stepaniuk, J. (1998). Approximation Spaces, Reducts and Representatives. In: Polkowski, L., Skowron, A. (eds) Rough Sets in Knowledge Discovery 2. Studies in Fuzziness and Soft Computing, vol 19. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-1883-3_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-7908-1883-3_6

  • Publisher Name: Physica, Heidelberg

  • Print ISBN: 978-3-7908-2459-9

  • Online ISBN: 978-3-7908-1883-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics