Granular rough sets and granular shadowed sets: Three-way approximations in Pawlak approximation spaces
A Pawlak approximation space is a pair of a ground set/space and a quotient set/space of the ground set induced by an equivalence relation on the ground set. The quotient space is a simple granulation of the ground space such that an equivalence class is a granule of objects in the ground space and,...
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| Veröffentlicht in: | International journal of approximate reasoning Jg. 142; S. 231 - 247 |
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| Hauptverfasser: | , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Elsevier Inc
01.03.2022
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| Schlagworte: | |
| ISSN: | 0888-613X, 1873-4731 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | A Pawlak approximation space is a pair of a ground set/space and a quotient set/space of the ground set induced by an equivalence relation on the ground set. The quotient space is a simple granulation of the ground space such that an equivalence class is a granule of objects in the ground space and, at the same time, a single granular object in the quotient space. The new two-space view leads to more insights into and a deeper understanding of rough set theory. In this paper, we revisit results from rough sets from the two-space perspective and introduce the notions of granular rough sets and probabilistic granular rough sets in the quotient space, as three-way approximations of sets in the ground space. We propose a concept of granular shadowed sets in the quotient space, as three-way approximations of fuzzy sets in the ground space. We formulate a cost-sensitive method to construct a granular shadowed set from a fuzzy set. We show that, when the costs satisfy some conditions, the three granular approximations become the same for the special case where a fuzzy set is in fact a set. |
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| ISSN: | 0888-613X 1873-4731 |
| DOI: | 10.1016/j.ijar.2021.11.012 |