Shadowed sets with higher approximation regions
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METHODOLOGIES AND APPLICATION
Shadowed sets with higher approximation regions M. A. Ibrahim1 · T. O. William-West1 · A. F. D. Kana2 · D. Singh1
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper mainly discusses three points involving shadowed set approximation of a given fuzzy set. Firstly, a principle of uncertainty balance, which guarantees that preservation of uncertainty in the induced shadowed set is studied. Secondly, an alternative formulation for determining the optimum partition thresholds of shadowed sets is suggested. This formulation helps us study principle of uncertainty balance in shadowed sets with higher approximation regions. Thirdly, five-region shadowed set, which effectively deals with the issue of uncertainty balance, is introduced. We provide a closed-form formula for determining its optimum partition thresholds and generalize it to n(≥ 5)-region shadowed sets. Finally, some examples from synthetic and real dataset are provided to demonstrate the feasibility of the suggested methods. Keywords Fuzzy set · Shadowed set · Three-way approximation
1 Introduction Shadowed sets, introduced by Pedrycz (1998), are direct algorithmic construction of fuzzy sets. By relying on Kleene’s three-valued logic (Kleene 1952), they represent fuzzy sets with the aid of three regions (i.e., core, shadow and excluded zones). In fact, shadowed sets stem from the necessity to make crisp decision with fuzzy sets (Nguyen et al. 2000). A key aspect of shadowed sets is the determination of the required pair of thresholds, which balances the uncertainty distorted as a result of transforming a given fuzzy set into three regions (Pedrycz 1998; Pedrycz and Vukovich 2002). There are various methods of inducing shadowed sets (Deng and Yao 2013; Ibrahim and William-West 2019; Pedrycz Communicated by V. Loia.
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T. O. William-West [email protected] M. A. Ibrahim [email protected] A. F. D. Kana [email protected] D. Singh [email protected]
1
Department of Mathematics, Faculty of Physical Sciences, Ahmadu Bello University, Zaria, Nigeria
2
Department of Computer Sciences, Faculty of Physical Sciences, Ahmadu Bello University, Zaria, Nigeria
1998; Tahayori et al. 2013; William-West et al. 2019; Yao et al. 2017; Zhang and Yao 2018; Zhou et al. 2019a, b). These methods anchor on different optimization-based principles. The main goal of shadowed sets is to relocate the fuzziness inherently associated with the original fuzzy set into a shadow region, by determining a pair of symmetric thresholds, (α, 1−α), guided by a principle of uncertainty balance. This principle follows from a general optimization-based principle known as principle of uncertainty and information invariance (Klir 1987). Klir (1987), in general terms, suggested a principle of uncertainty (and its related information) invariance, which states that when making transformations between different mathematical theories characterized by uncertainty, the amount of uncertainty should be preserved under these transformations. I
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