Skip to main content
SpringerLink
Log in

Rough Set Analysis of Imprecise Classes

  • Chapter
  • First Online:
Thriving Rough Sets

Part of the book series: Studies in Computational Intelligence ((SCI,volume 708))

Abstract

Lower approximations of single decision classes have been mainly treated in the classical rough set approaches. Attribute reduction and rule induction have been developed based on the lower approximations of single classes. In this chapter, we propose to use the lower approximations of unions of k decision classes instead of the lower approximations of single classes. We first show various kinds of attribute reduction are obtained by the proposed approach. Then we consider set functions associated with attribute reduction and demonstrate that the attribute importance degrees defined from set functions are very different depending on k. Third, we consider rule induction based on the lower approximations of unions of k decision classes and show that the classifiers with rules for unions of k decision classes can perform better than the classifiers with rules for single decision classes. Finally, utilization of rules for unions of k decision classes in privacy protection is proposed. Throughout this chapter, we demonstrate that the consideration of lower approximations of unions of k classes enriches the applicability of rough set approaches.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
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

Notes

  1. 1.

    We use capital letter K because we already used lower case letter k to show the number of decision classes to be combined by union.

References

  1. Błaszczyński, J., Greco, S., Słowiński, R., Szelg, M.: Monotonic variable consistency rough set approaches. Int. J. Approx. Reason. 50(7), 979–999 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Dubois, D., Prade, H.: Rough fuzzy sets and fuzzy rough sets. Int. J. General Syst. 17, 191–209 (1990)

    Article  MATH  Google Scholar 

  3. Flores, V., et al. (eds.): Rough Sets: International Joint Conference, IJCRS 2016, Proceedings, LNAI 9920. Springer, Cham (2016)

    Google Scholar 

  4. Grabisch, M.: Set Functions. Games and Capacities in Decision Making. Springer, Swizerland (2016)

    MATH  Google Scholar 

  5. Greco, S., Matarazzo, B., Słowiński, R.: Fuzzy measure technique for rough set analysis. In: Proceedings of EUFIT’98, pp. 99–103 (1998)

    Google Scholar 

  6. Greco, S., Matarazzo, B., Słowiński, R.: Rough sets theory for multi-criteria decision analysis. Eur. J. Oper. Res. 129(1), 1–7 (2001)

    Article  MATH  Google Scholar 

  7. Greco, S., Matarazzo, B., Słowiński, R.: Decision rule approach. In: Figueira, J., Greco, S., Ehrgott, M. (eds.) Multiple Criteria Decision Analysis, pp. 507–561. Springer, New York (2005)

    Google Scholar 

  8. Grzymala-Busse, J.W.: LERS–a system for learning from examples based on rough sets. In: Słowiński, R. (ed.) Intelligent Decision Support: Handbook of Application and Advances of Rough Set Theory, pp. 3–18. Kluwer Academic Publishers, Dordrecht (1992)

    Chapter  Google Scholar 

  9. Grzymala-Busse, J.W.: MLEM2—discretization during rule induction. In: Proceedings of the IIPWM2003, pp. 499–508 (2003)

    Google Scholar 

  10. Hamakawa, T., Inuiguchi, M.: On the utility of imprecise rules induced by MLEM2 in classification. In: Kudo Y., Tsumoto, S. (Eds.) Proceedings of 2014 IEEE International Conference on Granular Computing (GrC), pp. 76–81 (2014)

    Google Scholar 

  11. Inuiguchi, M.: Approximation-oriented fuzzy rough set approaches. Fundamenta Informaticae 142(1–4), 21–51 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Inuiguchi, M.: Variety of rough set based attribute importance. Proc. SCIS-ISIS 2016, 548–551 (2016)

    Google Scholar 

  13. Inuiguchi, M.: Rough set approaches to imprecise modeling. In: Flores, V., et al. (eds.) Rough Sets: International Joint Conference, IJCRS 2016, Proceedings, LNAI 9920, pp. 54–64. Springer, Cham (2016)

    Chapter  Google Scholar 

  14. Inuiguchi, M.: Attribute importance degrees corresponding to several kinds of attribute reduction in the setting of the classical rough sets. In: Torra, V., Dahlbom, A., Narukawa, Y. (Eds.) Fuzzy Sets, Rough Sets, Multisets and Clustering. Springer (in press)

    Google Scholar 

  15. Inuiguchi, M., Hamakawa, T.: The utilities of imprecise rules and redundant rules for classifiers. In: Huynh, V.-N., Denoeux, T., Tran, D.H., Le, A.C., Pham, S.B. (Eds.) Knowledge and Systems Engineering: Proceedings of the Fifth International Conference KSE 2013, vol. 2, AISC 245, pp. 45–56. Springer, Cham (2013)

    Google Scholar 

  16. Inuiguchi, M., Hamakawa, T., Ubukata, S.: Utilization of imprecise rules induced by MLEM2 algorithm. In: Proceedings of the 10th Workshop on Uncertainty Processing (WUPES’15), pp. 73–84 (2015)

    Google Scholar 

  17. Inuiguchi, M., Hamakawa, T., Ubukata, S.: Imprecise rules for data privacy. In: Ciucci, D., Wang, G., Mitra, S., Wu, W.-Z. (eds.) Rough Sets and Knowledge Technology 10th International Conference, RSKT 2015, LNCS 9436, pp. 129–139. Springer, Cham (2015)

    Google Scholar 

  18. Inuiguchi, M., Kusunoki, Y., Inoue, M.: Rule induction considering implication relations between conclusions. Ind. Eng. Manage. Syst. Int. J. 10(1), 66–74 (2011)

    Google Scholar 

  19. Inuiguchi, M., Tsurumi, M.: Measures based on upper approximations of rough sets for analysis of attribute importance and interaction. Int. J. Innov. Comput. Inf. Control 2(1), 1–12 (2006)

    Google Scholar 

  20. Inoue, K. (ed.): Application of Rough Sets to Kansei Engineering. Kaibundo, Tokyo (2009) (in Japanese)

    Google Scholar 

  21. Klir, G.J.: Where do we stand on measures of uncertainty, ambiguity, fuzziness, and the like? Fuzzy Sets Syst. 24(2), 141–160 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kryszkiewicz, M.: Rough set approach to incomplete information systems. Inf. Sci. 112, 39–49 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  23. Murofushi, T., Soneda, S.: Techniques for reading fuzzy measures (iii): interaction index. In Proceedings of the 9th Fuzzy Systems Symposium, pp. 693–696 (1993) (in Japanese)

    Google Scholar 

  24. Murofushi, T., Sugeno, M.: A theory of fuzzy measures: representations, the Choquet integral and null sets. J. Math. Anal. Appl. 159, 532–549 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  25. Pawłak, Z.: Rough sets. Int. J. Comput. Inf. Sci. 11(5), 341–356 (1982)

    Article  MATH  Google Scholar 

  26. Pawłak, Z.: Rough Sets: Theoretical Aspects of Reasoning About Data. Kluwer Academic Publishing, Dordrecht (1991)

    MATH  Google Scholar 

  27. Pawłak, Z., Skowron, A.: Rough sets and Boolean reasoning. Inf. Sci. 177(1), 41–73 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Polkowski, L., Skowron, A. (eds.): Rough Sets in Knowledge Discovery 2: Applications. Case Studies and Software Systems. Physica-Verlag, Heidelberg (2010)

    Google Scholar 

  29. Shapley, L.S.: A value for \(n\)-person games. In: Kuhn, H.W., Tucker, A.W. (eds.) Contributions to the Theory of Games II, pp. 307–317. Princeton, Princeton University Press (1953)

    Google Scholar 

  30. Ślȩzak, D.: Various approaches to reasoning with frequency based decision reducts: a survey. In: Polkowski, L., Tsumoto, S., Lin, T.Y. (eds.) Rough Set Methods and Applications, pp. 235–285. Physica-Verlag, Heidelberg (2000)

    Google Scholar 

  31. Ślȩzak, D.: On generalized decision functions: Reducts, networks and ensembles. In: Yao, Y., Hu, Q., Yu, H., Grzymala-Busse, J.W. (eds.) Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing, 15th International Conference, RSFDGrC 2015, LNCS 9437, pp. 13–23. Springer, Cham (2015)

    Google Scholar 

  32. Skowron, A., Rauser, C.M.: The discernibility matrix and function in information systems. In: Słowiński, R. (ed.) Intelligent Decision Support: Handbook of Application and Advances of Rough Set Theory, pp. 331–362. Kluwer Academic Publishers, Dordrecht (1992)

    Chapter  Google Scholar 

  33. Słowiński, R., Vanderpooten, D.: A generalized definition of rough approximations based on similarity. IEEE Trans. Data Knowl. Eng. 12(2), 331–336 (2000)

    Article  Google Scholar 

  34. Sweeney, L.: K-anonymity: a model for protecting privacy. Int. J. Uncertain. Fuzziness. Knowl.-Based Syst. 10(5), 557–570 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  35. UCI Machine Learning Repository. http://archive.ics.uci.edu/ml/

  36. Yao, Y.Y., Zhao, Y.: Attribute reduction in decision theoretic rough set models. Inf. Sci. 178(17), 3356–3373 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ziarko, W.: Variable precision rough sets model. J. Comput. Syst. Sci. 46(1), 39–59 (1993)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was partially supported by JSPS KAKENHI Grant Number 26350423. This chapter is the extended version of [13] with new data and discussions.

Author information

Authors and Affiliations

  1. Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka, 560-8531, Japan

    Masahiro Inuiguchi

Authors
  1. Masahiro Inuiguchi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Masahiro Inuiguchi .

Editor information

Editors and Affiliations

  1. Chongqing Key Laboratory of Computational Intelligence, Chongqing University of Posts and Telecommunications, Chongqing, China

    Guoyin Wang

  2. Institute of Mathematics, University of Warsaw Institute of Mathematics, Warsaw, Poland

    Andrzej Skowron

  3. University of Regina, Department of Computer Science University of Regina, Regina, Saskatchewan, Canada

    Yiyu Yao

  4. Institute of Informatics, University of Warsaw, Warsaw, Poland

    Dominik Ślęzak

  5. Polish-Japanese Academy IT , Warsaw, Poland

    Lech Polkowski

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Inuiguchi, M. (2017). Rough Set Analysis of Imprecise Classes. In: Wang, G., Skowron, A., Yao, Y., Ślęzak, D., Polkowski, L. (eds) Thriving Rough Sets. Studies in Computational Intelligence, vol 708. Springer, Cham. https://doi.org/10.1007/978-3-319-54966-8_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-54966-8_8

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-54965-1

  • Online ISBN: 978-3-319-54966-8

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation