Granular matrix-based knowledge reductions of formal fuzzy contexts

Abstract

Knowledge reduction is an important issue in formal fuzzy contexts, which can simplify the structure of concept lattices. In this paper, a novel granular matrix-based for knowledge reduction of crisp-fuzzy concept is investigated. Firstly, matrix representations of extents and intents of concepts are defined, respectively, which are used to characterize the join-irreducible elements and propose the corresponding algorithm. In this framework, granular consistent set and granular reduct are developed. Then the judgement theorem of reduction and its corresponding algorithm in formal fuzzy context are proposed. Furthermore, we generalize the matrix approach to formal fuzzy decision contexts. Finally, numerical experiments are conducted to evaluate the effectiveness of the proposed approaches. Our methods present a new framework for knowledge reduction in formal fuzzy contexts.

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References

  1. 1.

    Ganter B, Wille R (1999) Formal concept analysis. Springer, Berlin

    Google Scholar 

  2. 2.

    Wille R (1982) Restructuring lattice theory: an approach based on hierarchies of concepts, Ordered sets. Springer, Dordrecht, pp 445–470

    Google Scholar 

  3. 3.

    Wille R (2002) Why can concept lattices support knowledge discovery in databases? J Exp Theor Artif Intell 14(2–3):81–92

    MATH  Google Scholar 

  4. 4.

    Will R (2005) Formal concept analysis as mathematical theory of concepts and concept hierarchies. In: Ganter B et al (eds) Formal concept analysis. Springer, Berlin, pp 1–33

    Google Scholar 

  5. 5.

    Berry A, Sigayret A (2004) Representing a concept lattice by a graph. Discrete Appl Math 144(1):27–42

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Kuznetsov SO, Obiedkov SA (2002) Comparing performance of algorithms for generating concept lattices. J Exp Theor Artif Intell 14(2–3):189–216

    MATH  Google Scholar 

  7. 7.

    Dias SM, Vieira NJ (2015) Concept lattices reduction: definition, analysis and classification. Expert Syst Appl 42(20):7084–7097

    Google Scholar 

  8. 8.

    Dias SM, Vieira NJ (2017) A methodology for analysis of concept lattice reduction. Inf Sci 396:202–217

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Kumar CA, Srinivas S (2010) Concept lattice reduction using fuzzy K-means clustering. Expert Syst Appl 37(3):2696–2704

    Google Scholar 

  10. 10.

    Kardos̆ F, Pócs J, Pócsová J (2016) On concept reduction based on some graph properties. Knowl Based Syst 93:67–74

    Google Scholar 

  11. 11.

    Singh PK, Gani A (2015) Fuzzy concept lattice reduction using Shannon entropy and Huffman coding. J Appl Non-Classical Logics 25(2):101–119

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Zhang WX, Wei L, Qi JJ (2005) Attribute reduction theory and approach to concept lattice. Sci China Ser F Inf Sci 48(6):713–726

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Kumar CA, Dias SM, Vieira NJ (2015) Knowledge reduction in formal contexts using non-negative matrix factorization. Math Comput Simul 109:46–63

    MathSciNet  Google Scholar 

  14. 14.

    Benítez-Caballero MJ, Medina J, Ramírez-Poussa E (2017) Attribute reduction in rough set theory and formal concept analysis. In: International joint conference on rough sets. Springer, Cham, pp 513–525

    Google Scholar 

  15. 15.

    Chen JK, Mi JS, Lin YJ (2018) A graph approach for knowledge reduction in formal contexts. Knowl Based Syst 148:177–188

    Google Scholar 

  16. 16.

    Li JH, Mei CL, Lv YJ (2011) A heuristic knowledge-reduction method for decision formal contexts. Comput Math Appl 61(4):1096–1106

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Li JH, Mei CL, Lv YJ (2012) Knowledge reduction in real decision formal contexts. Inf Sci 189:191–207

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Li JH, Mei CL, Kumar CA, Zhang X (2013) On rule acquisition in decision formal contexts. Int J Mach Learn Cybern 4(6):721–731

    Google Scholar 

  19. 19.

    Shi LL, Yang HL (2018) Object granular reduction of fuzzy formal contexts. J Intell Fuzzy Syst 34(1):633–644

    Google Scholar 

  20. 20.

    Wu WZ, Leung Y, Mi JS (2009) Granular computing and knowledge reduction in formal contexts. IEEE Trans Knowl Data Eng 21(10):1461–1474

    Google Scholar 

  21. 21.

    Wei L, Qi JJ, Zhang WX (2008) Attribute reduction theory of concept lattice based on decision formal contexts. Sci China Ser F Inf Sci 51(7):910–923

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Li JH, Mei CL, Lv YJ (2012) Knowledge reduction in formal decision contexts based on an order-preserving mapping. Int J Gen Syst 41(2):143–161

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Li JH, Mei CL, Lv YJ (2013) Incomplete decision contexts: approximate concept construction, rule acquisition and knowledge reduction. Int J Approx Reason 54(1):149–165

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Shao MW, Leung Y, Wu WZ (2014) Rule acquisition and complexity reduction in formal decision contexts. Int J Approx Reason 55(1):259–274

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Burusco A (2000) Fuentes-González R. Concept lattices defined from implication operators. Fuzzy Sets Syst 114(3):431–436

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Krajči S (2003) Cluster based efficient generation of fuzzy concepts. Neural Netw World 13(5):521–530

    MathSciNet  Google Scholar 

  27. 27.

    Yahia SB, Arour K, Slimani A, Jaoua A (2000) Discovery of compact rules in relational databases. Inf Sci J 4(3):497–511

    Google Scholar 

  28. 28.

    Zhang WX, Ma JM, Fan SQ (2007) Variable threshold concept lattices. Inf Sci 177(22):4883–4892

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Li LF, Zhang JK (2010) Attribute reduction in fuzzy concept lattices based on the T implication. Knowl Based Syst 23(6):497–503

    MathSciNet  Google Scholar 

  30. 30.

    Mao H, Miao HR (2018) Attribute reduction based on directed graph in formal fuzzy contexts. J Intell Fuzzy Syst 34(6):4139–4148

    Google Scholar 

  31. 31.

    Li KW, Shao MW, Wu WZ (2017) A data reduction method in formal fuzzy contexts. Int J Mach Learn Cybern 8(4):1145–1155

    Google Scholar 

  32. 32.

    Shao MW, Yang HZ, Wu WZ (2015) Knowledge reduction in formal fuzzy contexts. Knowl Based Syst 73:265–275

    Google Scholar 

  33. 33.

    Shao MW, Leung Y, Wang YXZ, Wu WZ (2016) Granular reducts of formal fuzzy contexts. Knowl Based Syst 114:156–166

    Google Scholar 

  34. 34.

    Singh PK, Cherukuri AK, Li JH (2017) Concepts reduction in formal concept analysis with fuzzy setting using Shannon entropy. Int J Mach Learn Cybern 8(1):179–189

    Google Scholar 

  35. 35.

    Li LF (2016) Multi-level interval-valued fuzzy concept lattices and their attribute reduction. Int J Mach Learn Cybern 8(1):1–12

    Google Scholar 

  36. 36.

    He XL, Wei L, She YH (2018) L-fuzzy concept analysis for three-way decisions: basic definitions and fuzzy inference mechanisms. Int J Mach Learn Cybern 9(11):1857–1867

    Google Scholar 

  37. 37.

    Belohlavek R, De Baets B, Konecny J (2014) Granularity of attributes in formal concept analysis. Inf Sci 260:149–170

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Li JH, Ren Y, Mei CL, Yang XB (2016) A comparative study of multigranulation rough sets and concept lattices via rule acquisition. Knowl Based Syst 91:152–164

    Google Scholar 

  39. 39.

    Shao MW, Leung Y (2014) Relations between granular reduct and dominance reduct in formal contexts. Knowl Based Syst 65:1–11

    Google Scholar 

  40. 40.

    Trnecka M, Trneckova M (2018) Data reduction for Boolean matrix factorization algorithms based on formal concept analysis. Knowl Based Syst 158:75–80

    Google Scholar 

  41. 41.

    Falmagne JC, Doignon JP (2010) Learning spaces: interdisciplinary applied mathematics. Springer Science and Business Media, New York

    Google Scholar 

  42. 42.

    Rusch A, Wille R (1996) Knowledge spaces and formal concept analysis. Data analysis and information systems. Springer, Berlin, pp 427–436

    Google Scholar 

  43. 43.

    UCI Machine Learning Repository. http://archive.ics.uci.edu/ml/datasets.html

  44. 44.

    Scikit-feature selection repository. http://featureselection.asu.edu/index.php#myCarousel

  45. 45.

    Huang CC, Li JH, Dias SH (2016) Attribute significance, consistency measure and attribute reduction in formal concept analysis. Neural Netw World 26(6):607–623

    Google Scholar 

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Acknowledgements

This work is supported by grants from the National Natural Science Foundation of China (No. 11871259, No. 11701258, No. 61379021 and No. 61602415).

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Correspondence to Jinjin Li.

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Lin, Y., Li, J., Tan, A. et al. Granular matrix-based knowledge reductions of formal fuzzy contexts. Int. J. Mach. Learn. & Cyber. 11, 643–656 (2020). https://doi.org/10.1007/s13042-019-01022-4

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Keywords

  • Formal fuzzy contexts
  • Fuzzy relation matrix
  • Granular computing
  • Join-irreducible elements
  • Knowledge reduction