The abstract notion of rough approximation space is applied to the concrete cases of topological spaces with the particular situation of clopen–topologies generated by partitions, according to the Pawlak approach to rough set theory. In this partition context of a finite universe, typical of complete information systems, the probability space generated by the counting measure is analyzed, with particular regard to a local notion of rough entropy linked to the Shannon approach to these arguments. In the context of partition the notion of entropy as measure of uncertainty is distinguished from the notion of co–entropy as measure of granularity. The above considerations are extended to the case of covering, typical situation of incomplete information systems with the associated similarity relation.
Cattaneo, G., Ciucci, D., Biancucci, D. (2008). Entropy and co-entropy of partitions and coverings with applications to roughness theory. In R. Bello, R. Falcón, W. Pedrycz, J. Kacprzyk (a cura di), Granular computing: at the junction of rough sets and fuzzy sets (pp. 55-77). Springer [10.1007/978-3-540-76973-6_4].
Entropy and co-entropy of partitions and coverings with applications to roughness theory
CATTANEO, GIANPIERO;CIUCCI, DAVIDE ELIO;
2008
Abstract
The abstract notion of rough approximation space is applied to the concrete cases of topological spaces with the particular situation of clopen–topologies generated by partitions, according to the Pawlak approach to rough set theory. In this partition context of a finite universe, typical of complete information systems, the probability space generated by the counting measure is analyzed, with particular regard to a local notion of rough entropy linked to the Shannon approach to these arguments. In the context of partition the notion of entropy as measure of uncertainty is distinguished from the notion of co–entropy as measure of granularity. The above considerations are extended to the case of covering, typical situation of incomplete information systems with the associated similarity relation.
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