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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- \(U\) :
-
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 \(\lambda\)
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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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DOI: https://doi.org/10.1007/s13042-014-0297-2