Probabilistic hyperrough set and covering hyperrough set

Authors

  • Takaaki Fujita * Independent Researcher, Shinjuku, Shinjuku-ku, Tokyo, Japan.

https://doi.org/10.48313/uda.v2i2.70

Abstract

Rough set theory provides a mathematical framework for approximating subsets using lower and upper bounds defined by equivalence relations, effectively capturing uncertainty in classification and data analysis. Building on these foundational ideas, further generalizations such as Hyperrough Sets and Superhyperrough Sets have been developed. Probabilistic Rough Sets provide a framework for estimating uncertainty by utilizing membership probabilities, allowing for the definition of lower and upper approximations based on specified threshold values. Covering rough sets approximate information via overlapping covers, providing lower definite and upper possible boundaries when true partitions are unavailable.
In this paper, we introduce newly defined concepts of the Probabilistic HyperRough Set and Covering HyperRough Set, as well as the Probabilistic SuperHyperRough Set and Covering SuperHyperRough Set. These models extend the existing frameworks of the Probabilistic Rough Set and Covering Rough Set, respectively.

Keywords:

Rough set, Hyperrough set, Covering rough set, Superhyperrough set, Probabilistic rough set

References

  1. [1] Pawlak, Z. (1982). Rough sets. International Journal of Computer & Information Sciences, 11(5), 341–356. https://doi.org/10.1007/BF01001956
  2. [2] Pawlak, Z., Wong, S. K. M., Ziarko, W., & others. (1988). Rough sets: Probabilistic versus deterministic approach. International Journal of Man-Machine Studies, 29(1), 81–95. https://doi.org/10.1016/S0020-7373(88)80032-4
  3. [3] Pawlak, Z., Grzymala-Busse, J., Slowinski, R., & Ziarko, W. (1995). Rough sets. Communications of the ACM, 38(11), 88–95. https://doi.org/10.1145/219717.219791
  4. [4] Jech, T. (2003). Set theory: The third millennium edition, revised and expanded. Springer. https://doi.org/10.1007/3-540-44761-X
  5. [5] Pawlak, Z. (1998). Rough set theory and its applications to data analysis. Cybernetics & Systems, 29(7), 661–688. https://doi.org/10.1080/019697298125470
  6. [6] Al-Matarneh, L., Sheta, A., Bani-Ahmad, S., Alshaer, J., & Al-Oqily, I. (2014). Development of temperature-based weather forecasting models using neural networks and fuzzy logic. International Journal of Multimedia and Ubiquitous Engineering, 9(12), 343–366. https://doi.org/10.14257/ijmue.2014.9.12.31
  7. [7] Riordan, D., & Hansen, B. K. (2002). A fuzzy case-based system for weather prediction. Engineering Intelligent Systems for Electrical Engineering and Communications, 10(3), 139–146. https://doi.org/10.1049/iet-eis.2002.0027
  8. [8] Palmer, T. N. (2000). Predicting uncertainty in forecasts of weather and climate. Reports on Progress in Physics, 63(2), 71–116. https://doi.org/10.1088/0034-4885/63/2/201
  9. [9] Bauer, P., Thorpe, A., & Brunet, G. (2015). The quiet revolution of numerical weather prediction. Nature, 525(7567), 47–55. https://doi.org/10.1038/nature14956
  10. [10] Fujita, T. (2025). Short introduction to rough, hyperrough, superhyperrough, treerough. Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond: Fifth volume: Various Super-HyperConcepts (Collected Papers), 394. https://B2n.ir/qj9210
  11. [11] Fujita, T. (2025). Superhypersoft hyperrough set and superhypersoft superhyperrough set. https://B2n.ir/kd8720
  12. [12] Fujita, T. (2025). A study on hyperfuzzy hyperrough sets, hyperneutrosophic hyperrough sets, and hypersoft hyperrough sets with applications in cybersecurity. Artificial Intelligence in Cybersecurity, 2, 14–36. https://doi.org/10.14257/aics.2025.2.02
  13. [13] Fujita, T. (2025). Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond. Biblio Publishing. https://B2n.ir/yz3913
  14. [14] Fujita, T. (2023). Reconsideration and proposal of development models in projects—“quasi” development models: Quasi-waterfall and quasi-agile. European Journal of Social Sciences Studies, 9(2). https://doi.org/10.46827/ejsss.v9i2.1575
  15. [15] Das, A. K., Smarandache, F., Das, R., & Das, S. (2024). A comprehensive study on decision-making algorithms in retail and project management using double framed hypersoft sets. HyperSoft Set Methods in Engineering, 2, 62–71. https://doi.org/10.61356/j.hsse.2024.2310
  16. [16] Fujita, T. (2025). Hyperfuzzy hyperrough set, hyperneutrosophic hyperrough set, and hypersoft hyperrough set.
  17. [17] Yao, Y. (2008). Probabilistic rough set approximations. International Journal of Approximate Reasoning, 49(2), 255–271. https://doi.org/10.1016/j.ijar.2007.05.019
  18. [18] Yao, Y. (2010). Three-way decisions with probabilistic rough sets. Information Sciences, 180(3), 341–353. https://doi.org/10.1016/j.ins.2009.09.021
  19. [19] Zhang, C., Ding, J., Zhan, J., & Li, D. (2022). Incomplete three-way multi-attribute group decision making based on adjustable multigranulation Pythagorean fuzzy probabilistic rough sets. International Journal of Approximate Reasoning, 147, 40–59. https://doi.org/10.1016/j.ijar.2022.05.004
  20. [20] Yao, Y., Greco, S., & Słowiński, R. (2015). Probabilistic rough sets. In Springer Handbook of Computational Intelligence (pp. 387–411). Springer. https://doi.org/10.1007/978-3-319-11680-8_15
  21. [21] Azam, N., & Yao, J. (2014). Analyzing uncertainties of probabilistic rough set regions with game-theoretic rough sets. International Journal of Approximate Reasoning, 55(1), 142–155. https://doi.org/10.1016/j.ijar.2013.03.015
  22. [22] Yao, Y. (2011). Two semantic issues in a probabilistic rough set model. Fundamenta Informaticae, 108(3–4), 249–265. https://doi.org/10.3233/FI-2011-422
  23. [23] Ziarko, W. (2005). Probabilistic rough sets. In Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing: 10th International Conference, RSFDGrC 2005, Regina, Canada, August 31–September 3, 2005, Proceedings, Part I (Vol. 10, pp. 283–293). Springer. https://doi.org/10.1007/11548669_30
  24. [24] Liu, C., Miao, D., & Qian, J. (2014). On multi-granulation covering rough sets. International Journal of Approximate Reasoning, 55(6), 1404–1418. https://doi.org/10.1016/j.ijar.2014.01.004
  25. [25] Zhu, W., & Wang, F.-Y. (2006). A new type of covering rough set. In 2006 3rd International IEEE Conference on Intelligent Systems (pp. 444–449). IEEE. https://doi.org/10.1109/IS.2006.249
  26. [26] Zhu, W. (2007). Topological approaches to covering rough sets. Information Sciences, 177(6), 1499–1508. https://doi.org/10.1016/j.ins.2006.06.009
  27. [27] Zhu, W., & Wang, F.-Y. (2007). On three types of covering-based rough sets. IEEE Transactions on Knowledge and Data Engineering, 19(8), 1131–1144. https://doi.org/10.1109/TKDE.2007.1062
  28. [28] Han, S.-E. (2019). Covering rough set structures for a locally finite covering approximation space. Information Sciences, 480, 420–437. https://doi.org/10.1016/j.ins.2018.12.049
  29. [29] Hsiao, C.-C., Chuang, C.-C., Jeng, J.-T., & Su, S.-F. (2013). A weighted fuzzy rough sets based approach for rule extraction. In The SICE Annual Conference 2013 (pp. 104–109). IEEE. https://doi.org/10.1109/SICE.2013.6736152
  30. [30] Lenz, O. U., Cornelis, C., & Peralta, D. (2022). Fuzzy-rough-learn 0.2: A Python library for fuzzy rough set algorithms and one-class classification. In 2022 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE) (pp. 1–8). IEEE. https://doi.org/10.1109/FUZZ45936.2022.9822755
  31. [31] Wang, X., & Wang, Q. (2022). Uncertainty measurement of variable precision fuzzy soft rough set model. In CECNet 2022. https://doi.org/10.1109/CECNet54114.2022.9775690
  32. [32] Shabir, M., Ali, M. I., & Shaheen, T. (2013). Another approach to soft rough sets. Knowledge-Based Systems, 40, 72–80. https://doi.org/10.1016/j.knosys.2012.11.012
  33. [33] Zhao, H., & Zhang, H. (2019). On hesitant neutrosophic rough set over two universes and its application. Artificial Intelligence Review, 53, 4387–4406. https://doi.org/10.1007/s10462-019-09795-4
  34. [34] Bo, C., Zhang, X., Shao, S., & Smarandache, F. (2018). Multi-granulation neutrosophic rough sets on a single domain and dual domains with applications. Symmetry, 10, 296. https://doi.org/10.3390/sym10070296
  35. [35] Raghavan, R., & Tripathy, B. K. (2011). On some topological properties of multigranular rough sets. Advances in Applied Science Research, 2(3), 536–543. https://B2n.ir/uu4328
  36. [36] Fujita, T., & Smarandache, F. (2025). Forestfuzzy, forestneutrosophic, forestplithogenic, and forestrough set. Infinite Study. https://doi.org/10.13140/RG.2.2.15515.78888
  37. [37] Fujita, T. (2025). Iterative treefuzzy set, iterative treeneutrosophic set, and iterative treesoft set. Preprint. https://doi.org/10.13140/RG.2.2.15515.78888
  38. [38] Ma, T., & Tang, M. (2006). Weighted rough set model. In Sixth International Conference on Intelligent Systems Design and Applications (Vol. 1, pp. 481–485). IEEE. https://doi.org/10.1109/ISDA.2006.118
  39. [39] Sateesh, N., Srinivasa Rao, P., & Rajya Lakshmi, D. (2023). Optimized ensemble learning-based student’s performance prediction with weighted rough set theory enabled feature mining. Concurrency and Computation: Practice and Experience, 35(7), e7601. https://doi.org/10.1002/cpe.7601
  40. [40] Yang, J., Wang, X., Wang, G., Zhang, Q., Zheng, N., & Wu, D. (2024). Fuzziness-based three-way decision with neighborhood rough sets under the framework of shadowed sets. IEEE Transactions on Fuzzy Systems. https://doi.org/10.1109/TFUZZ.2024.1234567
  41. [41] Hu, Q., Yu, D., Liu, J., & Wu, C. (2008). Neighborhood rough set based heterogeneous feature subset selection. Information Sciences, 178(18), 3577–3594. https://doi.org/10.1016/j.ins.2008.05.024

Downloads

Published

2025-06-15

Issue

Vol. 2 No. 2 (2025): Uncertainty Discourse and Applications

Section

Articles

How to Cite

Fujita, T. (2025). Probabilistic hyperrough set and covering hyperrough set. Uncertainty Discourse and Applications, 2(2), 124-138. https://doi.org/10.48313/uda.v2i2.70

Similar Articles

11-20 of 20

You may also start an advanced similarity search for this article.