Abstract
Rough set theory is a useful tool for data mining. In recent yeas, ones have combined it with matroid theory to construct an excellent set-theoretical framework for empirical machine learning methods. Hence, the study of its matroidal structure is an interesting research topic, and the structure is part of the foundation of rough set theory. Few people study the combinations the second type of covering-based rough sets with matroids. x On the one hand, we establish a closure system through the fixed point family of the second type of covering lower approximation operator, and then construct a corresponding closure operator. For a covering of a universe, this closure operator is a matroidal closure operator if and only if the reduct of the covering forms a partition of the universe. On the other hand, we present two sufficient and necessary conditions for the second type of covering upper approximation operator to form a matroidal closure operator through the indiscernible neighborhood and the covering upper approximation operator.
Keywords
This is a preview of subscription content, log in via an institution.
References
Bonikowski, Z., Bryniarski, E., Skardowska, W.U.: Extensions and intentions in the rough set theory. Inf. Sci. 107, 149–167 (1998)
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)
Dai, J., Xu, Q.: Approximations and uncertainty measures in incomplete information systems. Inf. Sci. 198, 62–80 (2012)
Kryszkiewicz, M.: Rough set approach to incomplete information systems. Inf. Sci. 112, 39–49 (1998)
Lai, H.: Matroid Theory. Higher Education Press, Beijing (2001)
Li, X., Liu, S.: Matroidal approaches to rough set theory via closure operators. Int. J. Approximate Reasoning 53, 513–527 (2012)
Liang, J., Li, R., Qian, Y.: Distance: a more comprehensible perspective for measures in rough set theory. Knowl. Based Syst. 27, 126–136 (2012)
Lin, T.Y.: Neighborhood systems and relational databases. In: Proceedings of the 1988 ACM Sixteenth Annual Conference On Computer science, p. 725. ACM (1988)
Liu, G., Chen, Q.: Matroid. National University of Defence Technology Press, Changsha (1994)
Liu, Y., Zhu, W.: Matroidal structure of rough sets based on serial and transitive relations. J. Appl. Math. 2012, 16 pages (2012). Article ID 429737
Liu, Y., Zhu, W., Zhang, Y.: Relationship between partition matroid and rough set through k-rank matroid. J. Inf. Comput. Sci. 8, 2151–2163 (2012)
Miao, D., Duan, Q., Zhang, H., Jiao, N.: Rough set based hybrid algorithm for text classification. Expert Syst. Appl. 36, 9168–9174 (2009)
Min, F., He, H., Qian, Y., Zhu, W.: Test-cost-sensitive attribute reduction. Inf. Sci. 22, 4928–4942 (2011)
Pawlak, Z.: Rough sets. Int. J. Comput. Inf. Sci. 11, 341–356 (1982)
Pawlak, Z.: Fuzzy sets and rough sets. Fuzzy Sets Syst. 17, 99–102 (1985)
Qian, Y., Liang, J., Yao, Y., Dang, C.: Mgrs: A multi-granulation rough set. Inf. Sci. 180, 949–970 (2010)
Qin, K., Gao, Y., Pei, Z.: On covering rough sets. In: Yao, J.T., Lingras, P., Wu, W.-Z., Szczuka, M.S., Cercone, N.J., Ślȩzak, D. (eds.) RSKT 2007. LNCS, vol. 4481, pp. 34–41. Springer, Heidelberg (2007)
Restrepo, M., Cornelis, C., Gmez, J.: Duality, conjugacy and adjointness of approximation operators in covering-based rough sets. Int. J. Approximate Reasoning 1, 469–485 (2014)
Skowron, A., Stepaniuk, J.: Tolerance approximation spaces. Fundamenta Informaticae 27, 245–253 (1996)
Tang, J., She, K., Zhu, W.: Matroidal structure of rough sets from the viewpoint of graph theory. J. Appl. Math. 2012, 27 pages (2012). Article ID 973920
Tsumoto, S., Tanaka, H.: Algebraic specification of empirical inductive learning methods. In: Calmet, J., Campbell, J. (eds.) AISMC 1994. LNCS, vol. 958, pp. 224–243. Springer, Heidelberg (1995)
Tsumoto, S., Tanaka, H.: A common algebraic framework of empirical learning methods based on rough sets and matroid theory. Fundamenta Informaticae 27, 273–288 (1996)
Wang, G., Hu, J.: Attribute reduction using extension of covering approximation space. Fundamenta Informaticae 115, 219–232 (2012)
Wang, S., Zhu, Q., Zhu, W., Min, F.: Quantitative analysis for covering-based rough sets using the upper approximation number. Inf. Sci. 220, 483–491 (2013)
Yao, Y.: Constructive and algebraic methods of theory of rough sets. Inf. Sci. 109, 21–47 (1998)
Yao, Y.: Relational interpretations of neighborhood operators and rough set approximation operators. Inf. Sci. 111, 239–259 (1998)
Zhang, S., Wang, X., Feng, T., Feng, L.: Reduction of rough approximation space based on matroid. Int. Conf. Mach. Learn. Cybern. 2, 267–272 (2011)
Zhu, W.: Properties of the second type of covering-based rough sets. In: Workshop Proceedings of GrC&BI 2006, pp. 494–497. IEEE WI 06, Hong Kong, China, 18 December (2006)
Zhu, W.: Relationship among basic concepts in covering-based rough sets. Inf. Sci. 179, 2478–2486 (2009)
Zhu, W.: Relationship between generalized rough sets based on binary relation and covering. Inf. Sci. 179, 210–225 (2009)
Zhu, W., Wang, F.: Reduction and axiomization of covering generalized rough sets. Inf. Sci. 152, 217–230 (2003)
Zhu, W., Wang, F.: Relationships among three types of covering rough sets. In: 2006 IEEE International Conference on Granular Computing (GrC 2006), pp. 43–48 (2006)
Acknowledgments
This work is in part supported by the National Science Foundation of China under Grant Nos. 61170128, 61379049, 61379089 and 61440047.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
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.
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Liu, Y., Zhu, W. (2015). The Matroidal Structures of the Second Type of Covering-Based Rough Set. In: Ciucci, D., Wang, G., Mitra, S., Wu, WZ. (eds) Rough Sets and Knowledge Technology. RSKT 2015. Lecture Notes in Computer Science(), vol 9436. Springer, Cham. https://doi.org/10.1007/978-3-319-25754-9_21
Download citation
DOI: https://doi.org/10.1007/978-3-319-25754-9_21
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-25753-2
Online ISBN: 978-3-319-25754-9
eBook Packages: Computer ScienceComputer Science (R0)