Skip to main content

Improvement of the approximations and accuracy measure of a rough set using somewhere dense sets

Abstract

Rough set theory is a non-statistical approach to handle uncertainty and uncertain knowledge. It is characterized by two methods called classification (lower and upper approximations) and accuracy measure. The closeness of notions and results in topology and rough set theory motivates researchers to explore the topological aspects and their applications in rough set theory. To contribute to this area, this paper applies a topological concept called “somewhere dense sets” to improve the approximations and accuracy measure in rough set theory. We firstly discuss further topological properties of somewhere dense and cs-dense sets and give explicitly formulations to calculate S-interior and S-closure operators. Then, we utilize these two sets to define new concepts in rough set context such as SD-lower and SD-upper approximations, SD-boundary region, and SD-accuracy measure of a subset. We establish the fundamental properties of these concepts as well as show their relationships with the previous ones. In the end, we compare the current method of approximations with the previous ones and provide two examples to elucidate that the current method is more accurate.

This is a preview of subscription content, access via your institution.

References

  1. Abo-Tabl EA (2013) Rough sets and topological spaces based on similarity. Int J Mach Learn Cybern 4:451–458

    Article  Google Scholar 

  2. Abo-Tabl EA (2014) On links between rough sets and digital topology. Appl Math 5:941–948

    Article  Google Scholar 

  3. Abu-Donia HM (2008) Comparison between different kinds of approximations by using a family of binary relations. Knowl Based Syst 21:911–919

    Article  Google Scholar 

  4. Al-shami TM (2017) Somewhere dense sets and \(ST_1\)-spaces. Punjab University. J Math 49(2):101–111

    MathSciNet  MATH  Google Scholar 

  5. Al-shami TM (2018) Soft somewhere dense sets on soft topological spaces. Commun Korean Math Soc 33(4):1341–1356

    MathSciNet  MATH  Google Scholar 

  6. Al-shami TM (2021) An improvement of rough sets accuracy measure using containment neighborhoods with a medical application. Inform Sci 569:110–124

    MathSciNet  Article  Google Scholar 

  7. Al-shami TM, Fu WQ, Abo-Tabl EA (2021) New rough approximations based on \(E\)-neighborhoods, Complexity, 2021:6, Article ID 6666853

  8. Al-shami TM, Alshammari I, Asaad BA (2020) Soft maps via soft somewhere dense sets. Filomat 34(10):3429–3440

    MathSciNet  Article  Google Scholar 

  9. Al-shami TM, Noiri T (2019) More notions and mappings via somewhere dense sets. Afrika Matematika 30(7):1011–1024

    MathSciNet  Article  Google Scholar 

  10. Al-shami TM, El-Shafei ME, Alshammari I (2021) A comparison of two types of rough approximations based on \(N_j\)-neighborhoods. J Intell Fuzzy Syst 41(1):1393–1406

  11. Al-shami TM, Mhemdi A, Rawshdeh A, Aljarrah H (2021) Soft version of compact and Lindelöf spaces using soft somewhere dense set. AIMS Math 6(8):8064–8077

    MathSciNet  Article  Google Scholar 

  12. Allam AA, Bakeir MY, Abo-Tabl EA (2005) New approach for basic rough set concepts. In: International workshop on rough sets, fuzzy sets, data mining, and granular computing. Lecture Notes in Artificial Intelligence, 3641, Springer, Regina, 64–73

  13. Allam AA, Bakeir MY, Abo-Tabl EA (2006) New approach for closure spaces by relations. Acta Math Acad Paedagogicae Nyiregyháziensis 22:285–304

    MathSciNet  MATH  Google Scholar 

  14. Amer WS, Abbas MI, El-Bably MK (2017) On \(j\)-near concepts in rough sets with some applications. J Intell Fuzzy Syst 32(1):1089–1099

    Article  Google Scholar 

  15. Andrijevic D (1996) On \(b\)-open sets. Mat Vesnik 48:59–64

    MathSciNet  MATH  Google Scholar 

  16. Azzam A, Khalil AM, Li S-G (2020) Medical applications via minimal topological structure. J Intell Fuzzy Syst 39(3):4723–4730

    Article  Google Scholar 

  17. Chen L, Gao R, Bian Y, Di H (2021) Elliptic entropy of uncertain random variables with application to portfolio selection. Soft Comput 25:1925–1939

    Article  Google Scholar 

  18. El-Monsef MEA, El-Deeb SN, Mahmoud RA (1983) \(\beta \)-open sets and \(\beta \)-continuous mappings, Bulletin of the Faculty of Science. Assiut University 12:77–90

    Google Scholar 

  19. El-Monsef MEA, Embaby OA, El-Bably MK (2014) Comparison between rough set approximations based on different topologies. Int J Gran Comput Rough Sets Intell Syst 3(4):292–305

    Google Scholar 

  20. El-Bably MK, Al-shami TM (2021) Different kinds of generalized rough sets based on neighborhoods with a medical application. Int J Biomath. https://doi.org/10.1142/S1793524521500868

    Article  Google Scholar 

  21. Hosny M (2018) On generalization of rough sets by using two different methods. J Intell Fuzzy Syst 35(1):979–993

    Article  Google Scholar 

  22. Hosny RA, Asaad BA, Azzam AA, Al-shami TM (2021) Various topologies generated from \(E_j\)-neighbourhoods via ideals, Complexity, Volume 2021, Article ID 4149368, 11 pages

  23. Kondo M, Dudek WA (2006) Topological structures of rough sets induced by equivalence relations. J Adv Comput Intell Intell Inform 10(5):621–624

    Article  Google Scholar 

  24. Kozae AM, Khadra AAA, Medhat T (2007) Topological approach for approximation space (TAS). Proceeding of the 5th International Conference INFOS 2007 on Informatics and Systems. Information Cairo University, Cairo, Egypt, Faculty of Computers, pp 289–302

    Google Scholar 

  25. Lashin EF, Kozae AM, Khadra AAA, Medhat T (2005) Rough set theory for topological spaces. Int J Approx Reason 40:35–43

  26. Levine N (1963) Semi-open sets and semi-continuity in topological spaces. Am Math Month 70:36–41

    MathSciNet  Article  Google Scholar 

  27. Li Z, Xie T, Li Q (2012) Topological structure of generalized rough sets. Comput Math Appl 63:1066–1071

    MathSciNet  Article  Google Scholar 

  28. Mareay R (2016) Generalized rough sets based on neighborhood systems and topological spaces. J Egypt Math Soc 24:603–608

    MathSciNet  Article  Google Scholar 

  29. Mashhour AS, El-Monsef MEA, El-Deeb SN (1982) On precontinuous and weak precontinuous mappings. Proc Math Phys Soc Egypt 53:47–53

    MathSciNet  MATH  Google Scholar 

  30. Njastad O (1965) On some classes of nearly open sets. Pac J Math 15:961–970

    MathSciNet  Article  Google Scholar 

  31. Pawlak Z (1982) Rough sets. Int J Comput Inform Sci 11(5):341–356

    Article  Google Scholar 

  32. Pawlak Z (1991) Rough sets. Kluwer Acadmic Publishers Dordrecht, Theoretical Aspects of Reasoning About Data

    Book  Google Scholar 

  33. Salama AS (2020) Sequences of topological near open and near closed sets with rough applications. Filomat 34(1):51–58

    MathSciNet  Article  Google Scholar 

  34. Salama AS (2020) Bitopological approximation apace with application to data reduction in multi-valued information systems. Filomat 34(1):99–110

    MathSciNet  Article  Google Scholar 

  35. Salama AS (2018) Generalized topological approximation spaces and their medical applications. J Egypt Math Soc 26(3):412–416

    MathSciNet  Article  Google Scholar 

  36. Salama AS (2010) Topological solution for missing attribute values in incomplete information tables. Inform Sci 180:631–639

    MathSciNet  Article  Google Scholar 

  37. Salama AS, Mhemdi A, Elbarbary OG, Al-shami TM (2019) Topological approaches for rough continuous functions with applications, Complexity, Volume 2021, Article ID 5586187, 12 pages

  38. Skowron A (1988) On topology in information system. Bull Polish Acad Sci Math 36:477–480

    MathSciNet  MATH  Google Scholar 

  39. Sun S, Li L, Hu K (2019) A new approach to rough set based on remote neighborhood systems, Mathematical Problems in Engineering, Volume 2019, Article ID 8712010, 8 pages

  40. Wang R, Nan G, Chen L, Li M (2020) Channel integration choices and pricing strategies for competing dual-channel retailers. IEEE Trans Eng Manag. https://doi.org/10.1109/TEM.2020.3007347

    Article  Google Scholar 

  41. Wiweger A (1989) On topological rough sets. Bull Polish Acad Sci Math 37:89–93

    MathSciNet  MATH  Google Scholar 

  42. Wu QE, Wang T, Huang YX, Li JS (2008) Topology theory on rough sets. IEEE Trans Syst Man Cybern Part B (Cybernetics) 38(1):68–77

    Article  Google Scholar 

  43. Xiao Q, Chen L, Xie M, Wang C (2021) Optimal contract design in sustainable supply chain: Interactive impacts of fairness concern and overconfidence. J the Op Res Soc 72(7):1505–1524

    Article  Google Scholar 

  44. Yao YY (1996) Two views of the theory of rough sets in finite universes. Int J Approx Reason 15:291–317

    MathSciNet  Article  Google Scholar 

  45. Yao YY (1998) Generalized rough set models, Rough sets in knowledge Discovery 1, L. Polkowski, A. Skowron (Eds.), Physica Verlag, Heidelberg, 286–318

  46. Zhu W (2007) Topological approaches to covering rough sets. Inform Sci 177:1499–1508

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The author would like to thank the referees for their valuable comments, which helped us to improve the manuscript.

Funding

Not applicable.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Tareq M. Al-shami.

Ethics declarations

Conflict of interest

The author declares that there is no conflict of interests regarding the publication of this article.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Al-shami, T.M. Improvement of the approximations and accuracy measure of a rough set using somewhere dense sets. Soft Comput 25, 14449–14460 (2021). https://doi.org/10.1007/s00500-021-06358-0

Download citation

Keywords

  • Somewhere dense set
  • Lower and upper approximations
  • Accuracy measure
  • Interior and closure operators