Advertisement

The Lower Approximation Number in Covering-Based Rough Set

  • Hui LiuEmail author
  • William ZhuEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9436)

Abstract

Covering-based rough set has attracted much research interest with significant achievements. However, there are few analysis that have been conducted to quantify covering-based rough set. The approximation number is viewed as a quantitative tool for analyzing the covering-based rough set. In this paper, we focus on the lower approximation number. Firstly, we investigate some key properties of the lower approximation number. Secondly, we establish a lattice and two semilattice structures in covering-based rough set with the lower approximation number. Finally, based on the lower approximation number, a pair of matroid approximation operators is constructed. Moreover, we investigate the relationship between the pair of matroid approximation operators and a pair of lattice approximation operators.

Keywords

Covering Rough set The lower approximation number Granular computing 

Notes

Acknowledgments

This work is in part supported by The National Nature Science Foundation of Chi- na under Grant Nos. 61170128, 61379049 and 61379089, the Key Project of Education Department of Fujian Province under Grant No. JA13192, the Project of Education De- partment of Fujian Province under Grant No. JA14194, the Zhangzhou Municipal Nat- ural Science Foundation under Grant No. ZZ2013J03, and the Science and Technology Key Project of Fujian Province, China Grant No. 2012H0043.

References

  1. 1.
    Pawlak, Z.: Rough sets. Int. J. Comput. Inf Sci. 11, 341–356 (1982)CrossRefGoogle Scholar
  2. 2.
    Pawlak, Z.: Rough classification. Int. J. Man-Mach. Stud. 20, 469–483 (1984)CrossRefGoogle Scholar
  3. 3.
    Yao, Y., Chen, Y.: Rough set approximations in formal concept analysis. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets V. LNCS, vol. 4100, pp. 285–305. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  4. 4.
    Drwal, G.: Rough, and fuzzy-rough classification methods implemented in RClass system. In: Ziarko, W.P., Yao, Y. (eds.) RSCTC 2000. LNCS (LNAI), vol. 2005, pp. 152–159. Springer, Heidelberg (2001) CrossRefGoogle Scholar
  5. 5.
    Bianucci, D., Cattaneo, G., Ciucci, D.: Entropies and co-entropies of coverings with application to incomplete information systems. Fundamenta Informaticae 75, 77–105 (2007)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Chen, D., Wang, C., Hu, Q.: A new approach to attribute reduction of consistent and inconsistent covering decision systems with covering rough sets. Inf. Sci. 177, 3500–3518 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Li, F., Yin, Y.: Approaches to knowledge reduction of covering decision systems based on information theory. Inf. Sci. 179, 1694–1704 (2009)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Zhu, F., He, H.: Logical properties of rough sets. In: The Fourth International Conference on High Performance Computing in the Asia-Pacific Region, pp. 670–671. IEEE Press (2000)Google Scholar
  9. 9.
    Zhu, F., He, H.: The axiomization of the rough set. Chin. J. Comput. 23, 330–333 (2000)Google Scholar
  10. 10.
    Zhu, W.: Relationship among basic concepts in covering-based rough sets. Inf. Sci. 179, 2478–2486 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ma, L.: On some types of neighborhood-related covering rough sets. Int. J. Approx. Reasoning 53, 901–911 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Yao, Y., Yao, B.: Covering based rough set approximations. Inf. Sci. 200, 91–107 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Bonikowski, Z., Bryniarski, E., Wybraniec-Skardowska, U.: Extensions and intentions in the rough set theory. Inf. Sci. 107, 149–167 (1998)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Bartol, W., Miró, J., Pióro, K., Rosselló, F.: On the coverings by tolerance classes. Inf. Sci. 166, 193–211 (2004)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Liu, G., Sai, Y.: A comparison of two types of rough sets induced by coverings. Int. J. Approx. Reasoning 50, 521–528 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Zhu, W., Wang, F.: The fourth type of covering-based rough sets. Inf. Sci. 201, 80–92 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Liu, G., Sai, Y.: Invertible approximation operators of generalized rough sets and fuzzy rough sets. Inf. Sci. 180, 2221–2229 (2010)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Wang, S., Zhu, W., Zhu, Q., Min, F.: Covering base. J. Inf. Comput. Sci. 9, 1343–1355 (2012)Google Scholar
  19. 19.
    Wang, S., Zhu, Q., Zhu, W., Min, F.: Matroidal structure of rough sets and its characterization to attribute reduction. Knowl.-Based Syst. 36, 155–161 (2012)CrossRefGoogle Scholar
  20. 20.
    Min, F., Zhu, W.: Attribute reduction of data with error ranges and test costs. Inf. Sci. 211, 48–67 (2012)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Qian, Y., Liang, J., Li, D., Wang, F., Ma, N.: Approximation reduction in inconsistent incomplete decision tables. Knowl.-Based Syst. 23, 427–433 (2010)CrossRefGoogle Scholar
  22. 22.
    Wang, S., Zhu, W.: Matroidal structure of covering-based rough sets through the upper approximation number. Int. J. Granular Comput., Rough Sets Intel. Syst. 2, 141–148 (2011)CrossRefGoogle Scholar
  23. 23.
    Birhoff, G.: Lattice Theory. American Mathematical Society, Rhode Island (1995)Google Scholar
  24. 24.
    Zhang, W., Yao, Y., Liang, Y.: Rough set and concept lattice. Xi’an Jiaotong University Press (2006)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Open Access This chapter is licensed under the terms of the Creative Commons Attribution-NonCommercial 2.5 International License (http://creativecommons.org/licenses/by-nc/2.5/), which permits any noncommercial use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

Authors and Affiliations

  1. 1.Lab of Granular ComputingMinnan Normal UniversityZhangzhouChina

Personalised recommendations