Abstract
Rules induced from rough sets are described by an approach using possible coverings in an incomplete data set with similarity of values. In a complete data set only one covering is derived, whereas lots of possible coverings are derived in an incomplete data set. This seems to present some difficulties due to computational complexity, but it is not. No difficulty comes from the lattice structure that the family of possible coverings has. Four approximations that rough sets are consisted of are derived by using only two coverings: the minimum and maximum possible ones. Four types of rules can be induced from the approximations. So, we can derive the rules without worrying about computational complexity coming from the number of values with incomplete information.
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Notes
- 1.
Unless confused, \(R_{a_{i}}\) is used.
- 2.
\(\delta \) is used in place of \(\delta _{a_{i}}\) if no confusion.
- 3.
Therefore, \(R_{a_{i}}^{\delta }\) becomes a tolerance relation [5].
- 4.
C(o) or \(C(o)_{a_{i}}\) is used in place of \(C(o)_{a_{i}}^{\delta }\) if no confusion.
- 5.
\(\sqsubseteq \) is defined as \(\mathcal{E} \sqsubseteq \mathcal{E'}\) if \(\forall E \in \mathcal{E} \exists E' \in \mathcal{E'} \wedge E \subseteq E'\).
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Nakata, M., Saito, N., Sakai, H., Fujiwara, T. (2022). Rules Induced Using Possible Coverings from Incomplete Data Sets. In: Kahraman, C., Cebi, S., Cevik Onar, S., Oztaysi, B., Tolga, A.C., Sari, I.U. (eds) Intelligent and Fuzzy Techniques for Emerging Conditions and Digital Transformation. INFUS 2021. Lecture Notes in Networks and Systems, vol 307. Springer, Cham. https://doi.org/10.1007/978-3-030-85626-7_103
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