Characteristic matrices of compound operations of coverings and their relationships with rough sets

  • Aiping Huang
  • William ZhuEmail author
Original Article


As the matrix can compactly represent numeric data, simplify problem formulation and reduce time complexity, it has many applications in most of the scientific fields. For this purpose, some types of generalized rough sets have been connected with matrices. However, covering-based rough sets which play an important role in data mining and machine learning are seldom connected with matrices. In this paper, we define three composition operations of coverings and study their characteristic matrices; Moreover, the relationships between the characteristic matrices and covering approximation operators are investigated. First, for a covering, an existing matrix representation of indiscernible neighborhoods called the type-1 characteristic matrix of the covering is recalled and a new matrix representation of neighborhoods called the type-2 characteristic matrix of the covering is proposed. Second, considering the importance of knowledge fusion and decomposition, we define three types of composition operations of coverings. Specifically, their type-1 and type-2 characteristic matrices are studied. Finally, we also explore the representable properties of covering approximation operators with respect to any covering generated by each composition operation. It is interesting to find that three types of approximation operators, which are induced by each type of composition operation of coverings, can be expressed as the Boolean product of a coefficient matrix and a characteristic vector. These interesting results suggest the potential for studying covering-based rough sets by matrix approaches.


Covering-based rough set Composition operator Characteristic matrix 



This work is supported in part by the National Natural Science Foundation of China under Grant Nos. 61170128, 61379049, and 61379089, the Science and Technology Key Project of Fujian Province, China, under Grant No. 2012H0043.


  1. 1.
    Pawlak Z (1991) Rough sets: theoretical aspects of reasoning about data. Kluwer Academic Publishers, BostonCrossRefzbMATHGoogle Scholar
  2. 2.
    Dai J, Xu Q (2012) Approximations and uncertainty measures in incomplete information systems. Inf Sci 198:62–80MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Qian Y, Liang J, Pedrycz W, Dang C (2010) Positive approximation: an accelerator for attribute reduction in rough set theory. Artif Intell 174(9–10):597–618MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Skowron A, Stepaniuk J, Swiniarski R (2012) Modeling rough granular computing based on approximation spaces. Inf Sci 184:20–43CrossRefzbMATHGoogle Scholar
  5. 5.
    Wang S, Zhu Q, Zhu W, Min F (2013) Quantitative analysis for covering-based rough sets through the upper approximation number. Inf Sci 220:483–491MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Lin TY, Liu Q (1994) Rough approximate operators: axiomatic rough set theory. In: Ziarko W (ed) Rough sets, fuzzy sets and knowledge discovery. Springer, Berlin, pp 256–260CrossRefGoogle Scholar
  7. 7.
    Liu G (2008) Axiomatic systems for rough sets and fuzzy rough sets. Int J Approx Reason 48(3):857–867MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Zhang Y, Li J, Wu W (2010) On axiomatic characterizations of three pairs of covering based approximation operators. Inf Sci 180(2):274–287MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Zhu W, Wang F (2007) On three types of covering rough sets. IEEE Trans Knowl Data Eng 19(8):1131–1144CrossRefGoogle Scholar
  10. 10.
    Hu Q, Yu D, Xie Z (2008) Numerical attribute reduction based on neighborhood granulation and rough approximation. J Softw 19(3):640–649 (in Chinese)CrossRefzbMATHGoogle Scholar
  11. 11.
    Meng Z, Shi Z (2009) A fast approach to attribute reduction in incomplete decision systems with tolerance relation-based rough sets. Inf Sci 179(16):2774–2793MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Min F, Zhu W (2012) Attribute reduction of data with error ranges and test costs. Inf Sci 211:48–67MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Tsang ECC, Chen D, Yeung DS (2008) Approximations and reducts with covering generalized rough sets. Comput Math Appl 56(1):279–289MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Du Y, Hu Q, Zhu P, Ma P (2011) Rule learning for classification based on neighborhood covering reduction. Inf Sci 181(24):5457–5467MathSciNetCrossRefGoogle Scholar
  15. 15.
    Inuiguchi M, Tanino T (2002) Generalized rough sets and rule extraction. In: Rough sets and current trends in computing, vol 2475 of LNCS, pp 105–112Google Scholar
  16. 16.
    Lee S, Propes N, Zhang G, Zhao Y, Vachtsevanos GJ (2002) Rough set feature selection and diagnostic rule generation for industrial applications. In: Rough sets and current trends in computing, vol 2475 of LNCS, pp 568–571Google Scholar
  17. 17.
    Wang G, Liu F (2000) The inconsistency in rough set based rule generation. In: Rough sets and current trends in computing, vol 2005 of LNCS, pp 370–377Google Scholar
  18. 18.
    Huang A, Zhao H, Zhu W (2013) Nullity-based matroid of rough sets and its application to attribute reduction. Inf Sci 263:153–165MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Huang A, Zhu W (2012) Geometric lattice structure of covering-based rough sets through matroids. J Appl Math. Article ID 236307:1–25Google Scholar
  20. 20.
    Huang A, Zhu W (2014) Geometric lattice structure of covering and its application to attribute reduction through matroids. J Appl Math. Article ID 183621:8Google Scholar
  21. 21.
    Huang A, Zhu W. Connectedness of graphs and its application to matroids through covering-based rough sets. Soft Comput. doi: 10.1007/s00500-015-1859-2
  22. 22.
    Wang S, Zhu W (2011) Matroidal structure of covering-based rough sets through the upper approximation number. Int J Granul Comput Rough Sets Intell Syst 2(2):141–148Google Scholar
  23. 23.
    Eldén L (2007) Matrix methods in data mining and pattern recognition. The Society for Industrial and Applied MathematicsGoogle Scholar
  24. 24.
    Wang S, Zhu W, Min QZF (2013) Characteristic matrix of covering and its application to boolean matrix decomposition. Inf Sci 263:186–197MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Zhang J, Li T, Chen H (2014) Composite rough sets for dynamic data mining. Inf Sci 257:81–100MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    lang G, Li Q, Cai M, Yang T (2015) Characteristic matrixes-based knowledge reduction in dynamic covering decision information systems. Knowl Based Syst 85:1–26Google Scholar
  27. 27.
    Luo C, Li T, Yi Z, Fujita H (2016) Matrix approach to decision-theoretic rough sets for evolving data. Knowl Based Syst 99:123–134CrossRefGoogle Scholar
  28. 28.
    Liu G (2008) Generalized rough sets over fuzzy lattices. Inf Sci 178(6):1651–1662MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Liu G (2010) Closures and topological closures in quasi-discrete closure. Appl Math Lett 23(7):772–776MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Liu G, Sai Y (2010) Invertible approximation operators of generalized rough sets and fuzzy rough sets. Inf Sci 180(11):2221–2229MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Miao D, Gao C, Zhang N, Zhang Z (2011) Diverse reduct subspaces based co-training for partially labeled data. Int J Approx Reason 52(8):1103–1117MathSciNetCrossRefGoogle Scholar
  32. 32.
    Skowron A, Rauszer C (1992) The discernibility matrices and functions in information systems. In: Intelligent Decision SupportGoogle Scholar
  33. 33.
    Yao Y, Zhao Y (2009) Discernibility matrix simplification for constructing attribute reducts. Inf Sci 179(7):867–882MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Chen D, Hu Q, Yang Y (2011) Parameterized attribute reduction with gaussian kernel based fuzzy rough sets. Inf Sci 181(23):5169–5179CrossRefzbMATHGoogle Scholar
  35. 35.
    Hu Q, Yu D, Pedrycz W, Chen D (2011) Kernelized fuzzy rough sets and their applications. IEEE Trans Knowl Data Eng 23(11):1649–1667CrossRefGoogle Scholar
  36. 36.
    Pomykala JA (1987) Approximation operations in approximation space. Bull Polish Acad Sci 35(9–10):653–662MathSciNetzbMATHGoogle Scholar
  37. 37.
    Zhu W (2009) Relationship between generalized rough sets based on binary relation and covering. Inf Sci 179(3):210–225MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Tan Kah Kee CollegeXiamen UniversityZhangzhouChina
  2. 2.Lab of Granular ComputingMinnan Normal UniversityZhangzhouChina

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