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Reduction foundation with multigranulation rough sets using discernibility

  • Anhui TanEmail author
  • Wei-Zhi Wu
  • Jinjin Li
  • Tongjun Li
Article
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Abstract

When multiple granulated knowledge in multigranulation spaces are involved in decision making, protocol principles are adopted to arrive at the final consensus. Multigranulation rough set theory utilizes a voting principle to combine the decision options derived from individual granulated knowledge. Note that those knowledge may provide different degrees of support to the final results, some are key, some are of less importance and some are even of no use. Selecting valuable knowledge and reducing worthless one are thus necessary for data processing, which can alleviate the storage occupancy and facilitate the logical and statistical analysis. However, the basic reduction foundation of multigranulation spaces has been rarely touched by researchers, which brings in many difficulties in algorithmic and real applications. This work aims to disclose the principles of multiple knowledge reduction in multigranulation spaces from the viewpoint of discernibility. First, the notions of knowledge reduction of multigranulation spaces are defined based on multigranulation rough set theory. Second, a decision function mapping each object into the decision options of its neighborhood granule is introduced. Third, several pairs of discernibility matrices and discernibility functions are successively developed using the decision function. We claim that the valuable and worthless knowledge in multigranulation spaces can be explicitly chose and eliminated respectively by using the proposed discernibility matrices and discernibility functions. That is to say, these discernibility tools provide a precise criterion for the knowledge reduction of multigranulation spaces. As a theoretical extension, a multigranulation information entropy is proposed and an approximate algorithm is constructed to compute a suboptimal reduct of a multigranulation space based on this entropy. In the end, numerical experiments are performed on public data sets to verify the effectiveness of the proposed reduction methods. This study can get us a grasp of the foundational principle of knowledge reduction and may bring a new insight for the designation of substantial reduction algorithms of multigranulation knowledge.

Keywords

Discernibility matrix Discernibility function Decision making Multigranulation rough set Knowledge reduction 

Notes

Acknowledgements

This work is supported by the grants from National Natural Science Foundation of China (61602415, 61573321, 41631179, 11871259, 61773349, and 41701447), the Natural Science Foundation of Zhejiang Province (LY18F030017) and the Natural Science Foundation of Fujian Province (2019J01748).

References

  1. Che XY, Mi JS, Chen DG (2018) Information fusion and numerical characterization of a multi-source information system. Knowl Based Syst 145:121–133 CrossRefGoogle Scholar
  2. Chen DG, Wang CZ, Hu QH (2007) A new approach to attribute reduction of consistent and inconsistent covering decision systems with covering rough sets. Inf Sci 177:3500–3518MathSciNetCrossRefzbMATHGoogle Scholar
  3. Chen DG, Zhao SY, Zhang L et al (2012) Sample pair selection for attribute reduction with rough set. IEEE Trans Knowl Data Eng 24:2080–2093CrossRefGoogle Scholar
  4. Feng T, Mi JS (2016) Variable precision multigranulation decision-theoretic fuzzy rough sets. Knowl Based Syst 91:93–101CrossRefGoogle Scholar
  5. Feng QR, Zhou Y (2014) Soft discernibility matrix and its applications in decision making. Appl Soft Comput 24:749–756CrossRefGoogle Scholar
  6. Grecoa S, Matarazzoa B, Slowinski R (2001) Rough sets theory for multicriteria decision analysis. Eur J Oper Res 129:1–47CrossRefGoogle Scholar
  7. Hu J, Pedrycz W, Wang GY et al (2016) Rough sets in distributed decision information systems. Knowl Based Syst 94:13–22CrossRefGoogle Scholar
  8. Jensen R, Shen Q (2004) Semantics-preserving dimensionality reduction: rough and fuzzy-rough-based approaches. IEEE Trans Knowl Data Eng 16:1457–1471CrossRefGoogle Scholar
  9. Kaneiwa K (2011) A rough set approach to multiple dataset analysis. Appl Soft Comput 11:2538–2547CrossRefGoogle Scholar
  10. Khan M (2016) Formal reasoning in preference-based multiple-source rough set model. Inf Sci 334–335:122–143CrossRefzbMATHGoogle Scholar
  11. Khan M, Banerjee M (2008) Formal reasoning with rough sets in multiple-source approximation systems. Int J Approx Reason 49:466–477MathSciNetCrossRefzbMATHGoogle Scholar
  12. Kong QZ, Zhang XW, Xu WH (2019) Attribute reducts of multi-granulation information system. Artif Intell Rev.  https://doi.org/10.1007/s10462-019-09699-3 Google Scholar
  13. Kumar S, Inbarani H (2015) Optimistic multi-granulation rough set based classification for medical diagnosis. Proc Comput Sci 47:374–382CrossRefGoogle Scholar
  14. Li JH, Ren Y, Mei CL et al (2016) A comparative study of multigranulation rough sets and concept lattices via rule acquisition. Knowl Based Syst 91:152–164CrossRefGoogle Scholar
  15. Liang JY, Wang F, Dang CY et al (2012) An efficient rough feature selsction algorithm with a multi-granulation view. Int J Approx Reason 53:1080–1093CrossRefGoogle Scholar
  16. Liang JY, Wang F, Dang CY et al (2014) A group incremental approach to feature selection applying rough set technique. IEEE Trans Knowl Data Eng 26:294–308CrossRefGoogle Scholar
  17. Lin TY (1997) From rough sets and neighborhood systems to information granulation and computing in words. In: Proceeding Europe congress intelligent techniques and soft computing, 8–12 Sept 1997, pp 1602–1606Google Scholar
  18. Lin GP, Liang JY, Qian YH (2015) An information fusion approach by combining multigranulation rough sets and evidence theory. Inf Sci 314:184–199MathSciNetCrossRefzbMATHGoogle Scholar
  19. Liu CH, Miao DQ, Qian J (2014) On multi-granulation covering rough sets. Int J Approx Reason 55:1404–1418MathSciNetCrossRefzbMATHGoogle Scholar
  20. Mi JS, Leung Y, Wu WZ (2011) Dependence-space-based attribute reduction in consistent decision tables. Soft Comput 15:261–268CrossRefzbMATHGoogle Scholar
  21. Pawlak Z (1991) Rough sets: theoretical aspects of reasoning about data. Kluwer Academic Publishers, DordrechtCrossRefzbMATHGoogle Scholar
  22. Pedrycz W (2002) Relational and directional aspects in the construction of information granules. IEEE Trans Syst Man Cybern Part A 32:605–614CrossRefGoogle Scholar
  23. Qian YH, Liang JY, Yao YY et al (2010a) MGRS: a multi-granulation rough set. Inf Sci 180:949–970MathSciNetCrossRefzbMATHGoogle Scholar
  24. Qian YH, Liang JY, Yao YY et al (2010b) Incomplete mutigranulation rough set. IEEE Trans Syst Man Cybern A Syst Hum 20:420–430CrossRefGoogle Scholar
  25. Qian YH, Li SY, Liang JY et al (2014a) Pessimistic rough set based decisions: a multigranulation fusion strategy. Inf Sci 264:196–210MathSciNetCrossRefzbMATHGoogle Scholar
  26. Qian YH, Zhang H, Sang YL et al (2014b) Multigranulation decision-theoretic rough sets. Int J Approx Reason 55:225–237MathSciNetCrossRefzbMATHGoogle Scholar
  27. Qian YH, Liang JY, Lin GP et al (2015) Fuzzy granular structure distance. IEEE Trans Fuzzy Syst 23:2245–2259CrossRefGoogle Scholar
  28. Rauszer CM (1992) Rough logic for multi-agent systems. In: International conference on logic at work. Springer, Berlin, pp 161–181Google Scholar
  29. She YH, He XL (2012) On the structure of the multigranulation rough set model. Knowl Based Syst 36:81–92CrossRefGoogle Scholar
  30. Skowron A (1993) Boolean reasoning for decision rules generation. In: Proceedings of the international symposium on methodologies for intelligent systems, pp 295–305Google Scholar
  31. Skowron A, Rauszer C (1992) The discernibility matrices and functions in information systems. In: Slowiński R (ed) Intelligent decision support. Handbook of applications and advances of the rough sets theory. Kluwer, DordrechtGoogle Scholar
  32. Słezak D (2002) Approximate entropy reducts. Fund Inform 53:365–390MathSciNetzbMATHGoogle Scholar
  33. Slowinski R, Vanderpooten D (2000) A generalized definition of rough approximations based on similarity. IEEE Trans Knowl Data Eng 12:331–336CrossRefGoogle Scholar
  34. Sun BZ, Ma WM (2014) Soft fuzzy rough sets and its application in decision-making. Artif Intell Rev 41:67–80CrossRefGoogle Scholar
  35. Sun BZ, Ma WM (2015) An approach to consensus measurement of linguistic preference relations in multi-attribute group decision making and application. Omega 51:83–92CrossRefGoogle Scholar
  36. Sun BZ, Ma WM (2016) An approach to evaluation of emergency plans for unconventional emergency events based on soft fuzzy rough set. Kybernetes 45:1–26MathSciNetGoogle Scholar
  37. Tan AH, Wu WZ, Li JJ et al (2016) Evidence-theory-based numerical characterization of multigranulation rough sets in incomplete information systems. Fuzzy Sets Syst 294:18–35MathSciNetCrossRefzbMATHGoogle Scholar
  38. Tan AH, Wu WZ, Tao YZ (2017) On the belief structures and reductions of multigranulation spaces with decisions. Int J Approx Reason 88:39–52MathSciNetCrossRefzbMATHGoogle Scholar
  39. Tan AH, Wu WZ, Qian YH et al (2019) Intuitionistic fuzzy rough set-based granular structures and attribute subset selection. IEEE Trans Fuzzy Syst 27:527–539CrossRefGoogle Scholar
  40. Teng SH, Lu M, Yang AF et al (2016) Efficient attribute reduction from the viewpoint of discernibility. Inf Sci 326:297–314MathSciNetCrossRefzbMATHGoogle Scholar
  41. Tsang ECC, Chen DG, Yeung DS et al (2008) Attributes reduction using fuzzy rough sets. IEEE Trans Fuzzy Syst 16:1130–1141CrossRefGoogle Scholar
  42. Wei W, Liang JY (2019) Information fusion in rough set theory: an overview. Inform Fusion 48:107–118CrossRefGoogle Scholar
  43. Wu WZ, Leung Y (2013) Optimal scale selection for multi-scale decision tables. Int J Approx Reason 54:1107–1129MathSciNetCrossRefzbMATHGoogle Scholar
  44. Xu WH, Yang JH (2017) A novel approach to information fusion in multi-source datasets: a granular computing viewpoint. Inf Sci 378:410–423CrossRefGoogle Scholar
  45. Xu WH, Li WT, Zhang XT (2017) Generalized multigranulation rough sets and optimal granularity selection. Granul Comput 2:271–288CrossRefGoogle Scholar
  46. Yang XB, Xu SP, Dou HL et al (2017) Multigranulation rough set: a multiset based strategy. Int J Comput Int Syst 10:277–292CrossRefGoogle Scholar
  47. Yao YY (1999) Granular computing: basis issues and possible solutions. In: Proceedings of the 5th joint conference on information science 1999, pp 186–189Google Scholar
  48. Yao YY (2005) Perspectives of granular computing. In: IEEE international conference on granular computing, vol 1, pp 85–90Google Scholar
  49. Yao YY, She YH (2016) Rough set models in multigranulation spaces. Inf Sci 327:40–56MathSciNetCrossRefzbMATHGoogle Scholar
  50. Yao YY, Zhao Y (2009) Discernibility matrix simplification for constructing attribute reducts. Inf Sci 179:867–882MathSciNetCrossRefzbMATHGoogle Scholar
  51. Young T (2000) Data mining and machine oriented modeling: a granular computing approach. Appl Intell 13:113–124CrossRefGoogle Scholar
  52. Zadeh LA (1997) Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic. Fuzzy Sets Syst 90:111–127MathSciNetCrossRefzbMATHGoogle Scholar
  53. Zadeh LA (1998) Some reflections on soft computing, granular and their roles in the conception, design and utilization of information/intelligent systems. Soft Comput 2:23–25CrossRefGoogle Scholar
  54. Zhan JM, Alcantud JCR (2019) A novel type of soft rough covering and its application to multicriteria group decision making. Artif Intell Rev.  https://doi.org/10.1007/s10462-018-9617-3 Google Scholar
  55. Zhan JM, Sun BZ (2019) Covering-based intuitionistic fuzzy rough sets and applications in multi-attribute decision-making. Artif Intell Rev.  https://doi.org/10.1007/s10462-018-9674-7 Google Scholar
  56. Zhang QH, Zhang T (2016) Binary classification of multigranulation searching algorithm based on probabilistic decision. Math Probl Eng 2:1–14MathSciNetzbMATHGoogle Scholar
  57. Zhang QH, Zhang Q, Wang GY (2016) The uncertainty of probabilistic rough sets in multi-granulation spaces. Int J Approx Reason 77:38–54MathSciNetCrossRefzbMATHGoogle Scholar
  58. Zhao Y, Yao YY (2007) Data analysis based on discernibility and indiscernibility. Inf Sci 177:4959–4976CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Anhui Tan
    • 1
    • 2
    Email author
  • Wei-Zhi Wu
    • 1
    • 2
  • Jinjin Li
    • 3
  • Tongjun Li
    • 1
    • 2
  1. 1.School of Mathematics, Physics and Information ScienceZhejiang Ocean UniversityZhoushanChina
  2. 2.Key Laboratory of Oceanographic Big Data Mining and Application of Zhejiang ProvinceZhoushanChina
  3. 3.School of Mathematics and StatisticsMinnan Normal UniversityZhangzhouChina

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