Semi-monolayer cover rough set: Concept, property and granular algorithm

In this paper, we define four pairs of SMC rough approximation operators, namely GC, GU, DE and DA, and then discuss their properties and relations. It has been proven that (1) GU is the equivalent granular-form approximation operator of DE. (2) DE and DA are two pairs of cross dual approximation operators. That is to say that upper DE and lower DA, lower DE and upper DA are two pairs of dual approximation operators. The equivalent relation and cross dual relation serve as a bridge connecting granular-form approximation operators with the dot-form ones. (3) A DE or DA approximation set can be rebuilt by the corresponding GC approximation set and an SMC-Mapping. The equivalent granular form of DE and DA gives rise to some advanced algorithms that improve the efficiency of calculating DE and DA approximation sets. Experimental results on a UCI data set show the granular algorithm achieves better performance than the basic dot algorithm, and the speedup of efficiency tends to be exponential with the size of universe.