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Fuzziness and incremental information of disjoint regions in double-quantitative decision-theoretic rough set model

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Abstract

Double-quantitative decision-theoretic rough set (Dq-DTRS), as a new model considering double quantification to reflect the distinct degrees of quantitative information, satisfies the quantitative completeness properties and exhibits much stronger fault tolerance capabilities than decision-theoretic rough set (DTRS) and graded rough set (GRS). Since the Dq-DTRS was proposed, there have been few studies on the uncertainty analysis of the model. In this paper, we investigate the uncertainty measure of the four disjoint regions in Dq-DTRS models by introducing a fuzziness formula for rough set, and then describe the changing regularities of fuzziness of disjoint regions in DqI-DTRS model and DqII-DTRS model along with the variation of two parameters \(\alpha , \beta\) and the grade k, respectively. In addition, three kinds of incremental information for Dq-DTRS model, namely useful incremental information, useless incremental information and error-correction incremental information are presented being formed with regard to the changes of boundary regions, and also the related assessment methods for these special types of incremental information are discussed in the form of several important theorems.

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References

  1. Azam N, Yao JT (2014) Analyzing uncertainty of probabilistic rough set region with game-theoretic rough sets. Int J Approx Reason 55(1):142–155

    Article  MATH  Google Scholar 

  2. Banerjee M, Pal SK (1996) Roughness of a fuzzy set. Inf Sci 93(3–4):235–246

    Article  MathSciNet  MATH  Google Scholar 

  3. Cai M, Li Q, Ma J (2018) Knowledge reduction of dynamic covering decision information systems caused by variations of attribute values. Int J Mach Learn Cybern 8(4):1131–1144

    Article  Google Scholar 

  4. Chakrabarty K, Biswas R, Nanda S (2000) Fuzziness in rough sets. Fuzzy Sets Syst 110(2):247–251

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen H, Li T, Qiao S, Ruan D (2010) A rough sets based dynamic maintenance approach for approximations in coarsening and refining attribute values. Int J Intell Syst 25(10):1005–1026

    Article  MATH  Google Scholar 

  6. Chen H, Li T, Ruan D (2013) A rough-set-based incremental approach for updating approximations under dynamic maintenance environments. IEEE Trans Knowl Data Eng 25(2):274–284

    Article  Google Scholar 

  7. Ciucci D (2010) Classification of dynamics in rough sets. Sets and current trends in computing. Springer, Berlin, Heidelberg, pp 257–266

    Chapter  Google Scholar 

  8. Duda RO, Hart PE (1973) Pattern classification and scene analysis. Wiley, New York

    MATH  Google Scholar 

  9. I. Düntsch, G. Gediga. (1998) Uncertainty measures of rough set prediction. Artif Intell 106(1):109–137

    Article  MathSciNet  MATH  Google Scholar 

  10. Fan B, Tsang ECC, Xu W, Chen D, Li W (2018) Attribute-oriented cognitive concept learning strategy: a multi-level method. Int J Mach Learn Cynern. https://doi.org/10.1007/s13042-018-0879-5

    Article  Google Scholar 

  11. Fan B, Tsang ECC, Xu W, Yu J (2017) Double-quantitative rough fuzzy set based decisions: a logical operators method. Inf Sci 378:264–281

    Article  Google Scholar 

  12. Fan J, Xie W (1999) Distance measure and induced fuzzy entropy. Fuzzy Sets Syst 104(2):305–314

    Article  MathSciNet  MATH  Google Scholar 

  13. Greco S, Matarazzo B, Slowinski R (2008) Parameterized rough set model using rough membership and Bayesian confirmation measures. Int J Approx Reason 49(2):285–300

    Article  MathSciNet  MATH  Google Scholar 

  14. Herbert JP, Yao JT (2011) Game-theoretic rough sets. Fundam Inform 108(3):267–286

    MathSciNet  MATH  Google Scholar 

  15. Knopfmacher J (1975) On measures of fuzziness. J Math Anal Appl 49(3):529–534

    Article  MathSciNet  MATH  Google Scholar 

  16. Lang G, Miao D, Cai M (2017) Three-way decision approaches to conflict analysis using decision-theoretic rough set theory. Inf Sci 406–407:185–207

    Article  Google Scholar 

  17. Lang G, Yang T (2015) Decision-theoretic rough sets-based three-way approximations of interval-valued fuzzy sets. Fundam Inform 142:117–143

    Article  MathSciNet  MATH  Google Scholar 

  18. Li J, Huang C, Qi J (2017) Three-way cognitive concept learning via multi-granularity. Inf Sci 378:244–263

    Article  Google Scholar 

  19. Li S, Li T, Liu D (2013) Incremental updating approximations in dominance-based rough sets approach under the variation of the attribute set. Knowl Based Syst 40:17–26

    Article  Google Scholar 

  20. Li T, Ruan D, Geert W (2007) A rough sets based characteristic relation approach for dynamic attribute generalization in data mining. Knowl Based Syst 20(5):485–494

    Article  Google Scholar 

  21. Li W, Pedrycz W, Xue X, Xu W, Fan B (2018) Distance-based double-quantitative rough fuzzy sets with logic operations. Int J Approx Reason 101:206–233

    Article  MathSciNet  MATH  Google Scholar 

  22. Li W, Pedrycz W, Xue X, Zhang X, Fan B, Long B (2018) Information measure of absolute and relative quantification in double-quantitative decision-theoretic rough set model. J Eng. https://doi.org/10.1049/joe.2018.8315.

    Article  Google Scholar 

  23. Li W, Xu W (2015) Double-quantitative decision-theoretic rough set. Inf Sci 316:54–67

    Article  MathSciNet  MATH  Google Scholar 

  24. Li W, Xu W (2015) Multigranulation decision-theoretic rough set in ordered information system. Fundam Inform 139:67–89

    Article  MathSciNet  MATH  Google Scholar 

  25. Li W, Xu W (2014) Probabilistic rough set model based on dominance relation, In: Proceedings of rough sets and knowledge technology, lecture notes in artificial intelligence, vol 8818, pp 856-863

    Chapter  Google Scholar 

  26. Liang D, Liu D (2015) Deriving three-way decisions from intuitionistic fuzzy decision-theoretic rough sets. Inf Sci 300:28–48

    Article  MathSciNet  MATH  Google Scholar 

  27. Liang D, Pedrycz W, Liu D (2017) Determining three-way decisions with decision-theoretic rough sets using a relative value approach. IEEE Trans Syst Man Cybern Syst 47(8):1785–1799

    Article  Google Scholar 

  28. Liang J, Shi Z (2004) The information entropy, rough entropy and knowledge granulation in rough set theory. Int J Uncertain Fuzziness Knowl Based Syst 12(1):37–46

    Article  MathSciNet  MATH  Google Scholar 

  29. Lingras P, Chen M, Miao D (2014) Qualitative and quantitative combinations of crisp and rough clustering schemes using dominance relations. Int J Approx Reason 55:238–258

    Article  MathSciNet  MATH  Google Scholar 

  30. Liu C, Miao D, Zhang N (2012) Graded rough set model based on two universes and its properties. Knowl Based Syst 33:65–72

    Article  Google Scholar 

  31. Liu D, Li T, Li H (2012) A multiple-category classification approach with decision-theoretic rough sets. Fundam Inform 115:173–188

    MathSciNet  MATH  Google Scholar 

  32. Liu D, Li T, Ruan D (2009) An incremental approach for inducing knowledge from dynamic information systems. Fundam Inform 94(2):245–260

    MathSciNet  MATH  Google Scholar 

  33. Luo C, Li T, Chen H (2013) Incremental approaches for updating approximations in set-valued ordered information systems. Knowl Based Syst 50:218–233

    Article  Google Scholar 

  34. Ma W, Sun B (2012) Probabilistic rough set over two universes and rough entropy. Int J Approx Reason 53(4):608–619

    Article  MathSciNet  MATH  Google Scholar 

  35. Ma Z, Mi J (2016) Boundary region-based rough sets and uncertainty measures in the approximation space. Inf Sci 370:239–255

    Article  Google Scholar 

  36. Mi J, Wu W, Zhang W (2004) Approaches to knowledge reduction based on variable precision rough set model. Inf Sci 159:255–272

    Article  MathSciNet  MATH  Google Scholar 

  37. Pawlak Z (1982) Rough sets. J Comput Inf Sci 11(5):341–356

    Article  MATH  Google Scholar 

  38. Pawlak Z (1995) Vagueness and uncertainty: a rough set perspective. Comput Intell 11(2):227–232

    Article  MathSciNet  Google Scholar 

  39. Qian Y, Liang X, Lin G, Guo Q, Liang J (2017) Local multigranulation decision-theoretic rough sets. Int J Approx Reason 82:119–137

    Article  MathSciNet  MATH  Google Scholar 

  40. Sang B, Guo Y, Shi D, Xu W (2018) Decision-theoretic rough set model of multi-source decision systems. Int J Mach Learn Cybern 9(11):1941–1954

    Article  Google Scholar 

  41. Shao MW, Guo L, Wang CZ (2018) Connections between two-universe rough sets and formal concepts. Int J Mach Learn Cybern 9(11):1869–1877

    Article  Google Scholar 

  42. Słowiński R, Stefanowski J (1994) Handling various types of uncertainty in the rough set approach. In: Ziarko WP (ed) Rough sets, fuzzy sets and knowledge discovery. Workshops in computing. Springer, London, pp 366–376

    Chapter  MATH  Google Scholar 

  43. Sun B, Ma W, Li B, Li X (2018) Three-way decisions approach to multiple attribute group decision making with linguistic information-based decision-theoretic rough fuzzy set. Int J Approx Reason 93:424–442

    Article  MathSciNet  MATH  Google Scholar 

  44. Sun B, Ma W, Xiao X (2017) Three-way group decision making based on multigranulation fuzzy decision-theoretic rough set over two universes. Int J Approx Reason 81:87–102

    Article  MathSciNet  MATH  Google Scholar 

  45. Sun B, Ma W, Zhao H (2014) Decision-theoretic rough fuzzy set model and application. Inf Sci 283:180–196

    Article  MathSciNet  MATH  Google Scholar 

  46. Wang G, Ma X, Yu H (2015) Monotonic uncertainty measures for attribute reduction in probabilistic rough set model. Int J Approx Reason 59:41–67

    Article  MathSciNet  MATH  Google Scholar 

  47. Wang R, Wang XZ, Kwong S, Xu C (2017) Incorporating diversity and informativeness in multiple-instance active learning. IEEEE Trans Fuzzy Syst 25(6):1460–1475

    Article  Google Scholar 

  48. Wang XZ, Aamir R, Fu AM (2015) Fuzziness based sample categorization for classifier performance improvement. J Intell Fuzzy Syst 29(3):1185–1196

    Article  MathSciNet  Google Scholar 

  49. Wang XZ, He YL, Wang DD (2014) Non-naive Bayesian classifiers for classification problems with continuous attributes. IEEE Trans Cybern 44(1):21–39

    Article  Google Scholar 

  50. Wang XZ, Wang R, Feng HM, Wang H (2014) A new approach to classifier fusion based on upper integral. IEEE Trans Cybern 44(5):620–635

    Article  MathSciNet  Google Scholar 

  51. Wang XZ, Wang R, Xu C (2018) Discovering the relationship between generalization and uncertainty by incorporating complexity of classification. IEEE Trans Cybern 48(2):703–715

    Article  MathSciNet  Google Scholar 

  52. Wang XZ, Xing HJ, Li Y, Hua Q, Dong CR, Pedrycz W (2015) A study on relationship between generalization abilities and fuzziness of base classifiers in ensemble learning. IEEE Trans Fuzzy Syst 23(5):1638–1654

    Article  Google Scholar 

  53. Wong SKM, Ziarko W, Ye RL (1986) Comparison of rough-set and statistical methods in inductive learning. Int J Man Mach Stud 25(1):53–72

    Article  MATH  Google Scholar 

  54. Xie G, Zhang J, Lai K, Yu L (2008) Variable precision rough set for group decision-making: an application. Int J Approx Reason 49:331–343

    Article  MATH  Google Scholar 

  55. Xu W, Guo Y (2016) Generalized multigranulation double-quantitative decision-theoretic rough set. Knowl Based Syst 105:190–205

    Article  Google Scholar 

  56. Yao Y (1992) A decision theoretic framework for approximating concepts. Int J Man Mach Stud 37(6):793–809

    Article  Google Scholar 

  57. Yao Y (2003) Probabilistic approaches to rough sets. Expert Syst 20:287–297

    Article  Google Scholar 

  58. Yao Y (2011) The superiority of three-way decisions in probabilistic rough set models. Inf Sci 181:1080–1096

    Article  MathSciNet  MATH  Google Scholar 

  59. Yao Y (2010) Three-way decisions with probabilistic rough sets. Inf Sci 180:341–353

    Article  MathSciNet  Google Scholar 

  60. Yao Y, Deng X (2014) Quantitative rough sets based on subsethood measures. Inf Sci 267:306–322

    Article  MathSciNet  MATH  Google Scholar 

  61. Yao Y, Lin TY (1996) Generalization of rough sets using modal logic. Intell Autom Soft Comput 2(2):103–119

    Article  Google Scholar 

  62. Yao Y, Wong S.K, Lingras P (1990) A decision-theoretic rough set model. In: Proceedings of international symposium on methodlogies for intelligent systems, vol 5, pp 17-25

  63. Yu H, Liu Z, Wang G (2014) An automatic method to determine the number of clusters using decision-theoretic rough set. Int J Approx Reason 55(1):101–115

    Article  MathSciNet  MATH  Google Scholar 

  64. Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353

    Article  MATH  Google Scholar 

  65. Zhang J, Li T, Ruan D (2012) Neighborhood rough sets for dynamic data mining. Int J Intell Syst 27(4):317–342

    Article  Google Scholar 

  66. Zhang J, Li T, Ruan D (2012) Rough sets based matrix approaches with dynamic attribute variation in set-valued information systems. Int J Approx Reason 53(4):620–635

    Article  MathSciNet  MATH  Google Scholar 

  67. Zhang Q, Xie Q, Wang G (2018) A novel three-way decision model with decision-theoretic rough sets using utility theory. Knowl Based Syst 159:321–335

    Article  Google Scholar 

  68. Zhang Q, Yang S, Wang G (2017) Measuring uncertainty of probabilistic rough set model from its three regions. IEEE Trans Syst Man Cybern Syst 47(12):3299–3309

    Article  Google Scholar 

  69. Zhang Q, Zhang Q, Wang G (2016) The uncertainty of probabilistic rough sets in multi-granulation spaces. Int J Approx Reason 77:38–54

    Article  MathSciNet  MATH  Google Scholar 

  70. Zhang X, Miao D (2015) An expanded double-quantitative model regarding probabilities and grades and its hierarchical double-quantitative attribute reduction. Inf Sci 299:312–336

    Article  MathSciNet  MATH  Google Scholar 

  71. Zhang X, Miao D (2016) Double-quantitative fusion of accuracy and importance: Systematic measure mining, benign integration construction, hierarchcal attribute reducion. Knowl Based Syst 91:219–240

    Article  Google Scholar 

  72. Zhang X, Miao D (2014) Quantitative information architecture, granular computing and rough set models in the double-quantitative approximation space of precision and grade. Inf Sci 268:147–168

    Article  MathSciNet  MATH  Google Scholar 

  73. Zhang X, Miao D (2013) Two basic double-quantitative rough set models for precision and graded and their investigation using granular computing. Int J Approx Reason 54:1130–1148

    Article  MATH  Google Scholar 

  74. Zhang X, Mo Z, Xiong F, Cheng W (2012) Comparative study of variable precision rough set model and graded rough set model. Int J Approx Reason 53(1):104–116

    Article  MathSciNet  MATH  Google Scholar 

  75. Zhou B (2014) Multi-class decision-theoretic rough sets. Int J Approx Reason 55:211–224

    Article  MathSciNet  MATH  Google Scholar 

  76. Ziarko W (2005) The investigation of the Bayesian rough set model. Int J Approx Reason 40(1):81–91

    MathSciNet  MATH  Google Scholar 

  77. Ziarko W (1993) Variable precision rough set model. J Comput Syst Sci 46(1):39–59

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work is supported by Natural Science Foundation of China (Nos. 11671109, No. 11671111, 61472463, 61772002), Science and Technology Research Program of Chongqing Municipal Education Commission (No. KJ1709221), and Wentao Li is supported by the China Scholarship Council under Grant No. 201606120161.

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Authors and Affiliations

  1. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China

    Wentao Li & Xiaoping Xue

  2. Department of Electrical and Computer Engineering, University of Alberta, Edmonton, T6R 2V4, Canada

    Wentao Li & Witold Pedrycz

  3. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, People’s Republic of China

    Weihua Xu

  4. Faculty of Information Technology, Macau University of Science and Technology, Taipa, Macau

    Bingjiao Fan

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  1. Wentao Li
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  2. Witold Pedrycz
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  3. Xiaoping Xue
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  5. Bingjiao Fan
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Correspondence to Xiaoping Xue.

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Li, W., Pedrycz, W., Xue, X. et al. Fuzziness and incremental information of disjoint regions in double-quantitative decision-theoretic rough set model. Int. J. Mach. Learn. & Cyber. 10, 2669–2690 (2019). https://doi.org/10.1007/s13042-018-0893-7

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