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Linguistic rough sets

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Abstract

We introduce linguistic rough set (LRS) by integrating linguistic quantifiers in the rough set framework. The proposed LRS is inspired by the ways in which humans process imprecise information. It operates directly with the linguistic summaries and caters to imprecision implicit in the real world with partial knowledge. The measures of LRS are developed and its properties are investigated in detail. An approach is proposed for approximation of fuzzy concepts with the proposed LRS. This approach is applied in a real world case-study on the credit scoring analysis problem.

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Abbreviations

\(U\) :

Universe of objects

\(x_{i}\) :

Object or alternative

\(C\) :

Set of attributes/criteria

f :

Function

V :

Domain of values

\({\text{IS}}\) :

Information system

\({\text{IND}}\) :

Indiscernibility relation

\(U/B\) :

Partition of \(U\) with respect to attribute set \(B\)

\(\left[ {x_{i} } \right]\) :

Equivalence class generated with \({\text{IND}}\)

\(\underline{B} \left( D \right)\) :

Lower approximation of set \(D\) with respect to attribute set \(B\)

\(\overline{B}\left( D \right)\) :

Upper approximation of \(D\) with respect to attribute set \(B\)

\(\widetilde{R}\left( {x_{i} ,x_{j} } \right)\) :

Fuzzy similarity relation between \(x_{i}\) and \(x_{j}\)

\(\mu_{{C_{j} }} \left( {x_{i} } \right)\) :

Fuzzy membership of \(x_{i}\) in \(C_{j}\)

\(\mu_{{\underline{{\tilde{R}}} \left( Y \right)}} \left( {C_{j} } \right)\) :

Lower approximation of membership of fuzzy concept \(C_{j}\) in fuzzy concept \(Y\)

\(\mu_{{\overline{{\tilde{R}}} \left( Y \right)}} \left( {C_{j} } \right)\) :

Upper approximation of membership of \(C_{j}\) in \(Y\)

\({\mathcal{F}}\left( U \right)\) :

Power set of all fuzzy subsets defined on U

\({\text{supp}} V\) :

Support of a fuzzy set \(V\)

\({\text{card}} V\) :

Cardinality of a fuzzy set \(V\)

\(S\) :

Linguistic label set

\({\mathcal{D}}\left( {W,V} \right)\) :

Degree of inclusion of fuzzy concept \(V\) in \(W\)

\(\lambda\) :

Degree of certainty of approximations

\(\underline{C}_{\lambda } \left( Y \right)\) :

Lower approximation of \(Y\) in terms of a set \(C\) of fuzzy concepts

\(\overline{C}_{\lambda } \left( Y \right)\) :

Upper approximation of \(Y\) in terms of set \(C\) of fuzzy concepts

\(T_{C} \left( Y \right)\) :

Quality of approximation of fuzzy concept \(Y\)

\(\kappa\) :

Crispness coefficient

\(\alpha_{\lambda } \left( Y \right)\) :

Accuracy of approximation of \(Y\) with degree of certainty \(\lambda\)

References

  1. Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning, Part I. Inf Sci 8:199–249

    Article  MathSciNet  MATH  Google Scholar 

  2. Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning, Part II. Inf Sci 8:301–357

    Article  MathSciNet  MATH  Google Scholar 

  3. Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning, Part III. Inf Sci 9:43–80

    Article  MathSciNet  MATH  Google Scholar 

  4. Zadeh LA (1999) From computing with numbers to computing with words from manipulation of measurements to manipulation of perceptions. IEEE Trans Circuits Syst 45(1):105–119

    Article  MathSciNet  MATH  Google Scholar 

  5. Zadeh LA (2005) Toward a generalized theory of uncertainty (GTU)—an outline. Inf Sci 172(1–2):1–40

    Article  MathSciNet  MATH  Google Scholar 

  6. Zadeh LA (2006) Generalized theory of uncertainty (GTU)—principal concepts and ideas. Comput Stat Data Anal 51(1):15–46

    Article  MathSciNet  MATH  Google Scholar 

  7. Zadeh LA (2002) Toward a perception-based theory of probabilistic reasoning with imprecise probabilities. J Statistical Plan Inference 105(1):233–264

    Article  MathSciNet  MATH  Google Scholar 

  8. Zadeh LA (2000) Outline of a computational theory of perceptions based on computing with words. In: Sinha NK, Gupta MM (eds) Soft computing and intelligent systems: theory and applications. Academic Press, London, pp 3–22

    Google Scholar 

  9. Zadeh LA (2004) A note on web intelligence, world knowledge and fuzzy logic. Data Knowl Eng 50(3):291–304

    Article  Google Scholar 

  10. Zadeh LA (2001) A new direction in AI: toward a computational theory of perceptions. AI Mag 22(1):73–84

    MATH  Google Scholar 

  11. Zadeh LA (1983) A computational approach to fuzzy quantifiers in natural languages. Comput Math Appl 9(1):149–184

    Article  MathSciNet  MATH  Google Scholar 

  12. Zadeh LA (1997) Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic. Fuzzy Sets Syst 90(2):111–127

    Article  MathSciNet  MATH  Google Scholar 

  13. Yager RR (1991) On linguistic summaries of data. In: Piatetsky-Shapiro G, Frawley B (eds) Knowledge discovery in databases. MIT Press, Cambridge, pp 347–363

    Google Scholar 

  14. Yager RR (1982) A new approach to the summarization of data. Inf Sci 28:69–86

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Dubois D, Prade H (1990) Rough fuzzy sets and fuzzy rough sets. Int J Gen Syst 17:191–209

    Article  MATH  Google Scholar 

  17. Radzikowska AM, Kerre EE (2002) A comparative study of fuzzy rough sets. Fuzzy Sets Syst 126(2):137–155

    Article  MathSciNet  MATH  Google Scholar 

  18. Jensen R, Shen Q (2004) Semantics-preserving dimensionality reduction: rough and fuzzy-rough based approaches. IEEE Trans Knowl Data Eng 16(12):1457–1471

    Article  Google Scholar 

  19. Beauboef T, Petry FE, Arora G (1998) Information-theoretic measures of uncertainty for rough sets and rough relational databases. Inf Sci 109:185–195

    Article  Google Scholar 

  20. Bodjanova S (1997) Approximation of fuzzy concepts in decision making. Fuzzy Sets Syst 85(1):23–29

    Article  MathSciNet  MATH  Google Scholar 

  21. Chunsheng F, Fending C, Qiang W, Shouchao W, Ju W (2010) Application of rough sets theory to sources analysis of atmospheric particulates, bioinformatics and biomedical engineering (iCBBE). In: 2010 4th International Conference, 18–20 June, pp 1–5

  22. Shi C (2009) The application of rough sets and fuzzy sets in the supply chain knowledge sharing risk early-warning. Fuzzy systems and knowledge discovery. FSKD ‘09. In: Sixth International Conference, vol 6, 14–16 Aug, pp 441–445

  23. Bordogna G, Fedrizzi M, Passi G (1997) A linguistic modeling of consensus in group decision making based on OWA operator. IEEE Trans Syst Man Cybernetics Part A Syst Hum 27:126–132

    Article  Google Scholar 

  24. Levrat E, Voisin A, Bombardier S, Bremont J (1997) Subjective evaluation of car seat comfort with fuzzy set techniques. Int J Intell Syst 12:891–913

    Article  Google Scholar 

  25. Zeshui Xu (2007) A method for multiple attribute decision making with incomplete weight information in linguistic setting. Knowl Based Syst 20(8):719–725

    Article  Google Scholar 

  26. Zhang WX, Leung Y (1996) Theory of inclusion degrees and its applications to uncertainty inferences. Soft computing in intelligent systems and information processing, IEEE, New York, pp 496–501

  27. Xu ZB, Liang JY, Dang CY, Chin KS (2002) Inclusion degree: a perspective on measures for rough set data analysis. Inf Sci 141:227–236

    Article  MathSciNet  MATH  Google Scholar 

  28. Liu Dun, Li Tianrui, Ruan Da (2011) Probabilistic model criteria with decision-theoretic rough sets. Inf Sci 181(17):3709–3722

    Article  MathSciNet  Google Scholar 

  29. Dombi J (1982) Basic concepts for a theory of evaluation: the aggregative operator. Eur J Oper Res 10(3):282–293

    Article  MathSciNet  MATH  Google Scholar 

  30. Lawry J, Tang Y (2010) Granular knowledge representation and inference using labels and label expressions. Fuzzy Syst IEEE Trans 18(3):500–514

    Article  Google Scholar 

  31. Zadeh LA (1996) Fuzzy logic = computing with words. IEEE Trans Fuzzy Syst 4:103–111

    Article  Google Scholar 

  32. Dubois D, Prade H (1987) Rough fuzzy sets and fuzzy rough sets. Fuzzy Sets Syst 23:3–18

    Article  MathSciNet  MATH  Google Scholar 

  33. Wang X, Tsang ECC, Zhao S, Chen D, Yeung DS (2007) Learning fuzzy rules from fuzzy samples based on rough set techniques. Inf Sci 177:4493–4514

    Article  MathSciNet  MATH  Google Scholar 

  34. Wang Xizhao, Zhai Junhai, Lu Shuxia (2008) Induction of multiple fuzzy decision trees based on rough set technique. Inf Sci 178(16):3188–3202

    Article  MathSciNet  MATH  Google Scholar 

  35. Lin TY (1997) Granular computing. Announcement of the BISC Special Interest Group on Granular Computing

  36. Qian YH, Liang JY, Yao YY, Dang CY (2010) Incomplete mutigranulation rough set. IEEE Trans Syst Man Cybern Part A 20:420–430

    Article  Google Scholar 

  37. Liu Xin, Qian Yuhua, Liang Jiye (2014) A rule-extraction framework under multigranulation rough sets. Int J Mach Learn Cybernet 5(2):319–326

    Article  Google Scholar 

  38. Lin Guoping, Liang Jiye, Qian Yuhua (2014) Topological approach to multigranulation rough sets. Int J Mach Learn Cybernet 5(2):233–243

    Article  Google Scholar 

  39. Fang Lian-Hua, Li Ke-Dian, Li Jin-Jin (2013) Properties of two types of covering-based rough sets. Int J Mach Learn Cybernet 4(6):685–691

    Article  Google Scholar 

  40. Kang Xiangping, Li Deyu (2013) Dependency space, closure system and rough set theory. Int J Mach Learn Cybernet 4(6):595–599

    Article  Google Scholar 

  41. Pawlak Z (1991) Rough sets. Theoretical aspects of reasoning about data, system theory, knowledge engineering and problem solving, vol 9, Kluwer, Dordrecht

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Indian Institute of Technology, New Delhi, India

    Manish Agarwal

  2. Department of Information Engineering and Computer Science, University of Trento, Trento, Italy

    Themis Palpanas

Authors
  1. Manish Agarwal
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  2. Themis Palpanas
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Correspondence to Manish Agarwal.

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Agarwal, M., Palpanas, T. Linguistic rough sets. Int. J. Mach. Learn. & Cyber. 7, 953–966 (2016). https://doi.org/10.1007/s13042-014-0297-2

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