Uncertainty Measurement Of Rough Sets Based On Two-dimensional Direct-product Universe

The double-universes rough set is a basic rough set model for expanding the universe.It relates the single universe rough set theory on the one-dimensional space system to the two-dimensional space system,so it realizes the span of the spatial dimension;however,it requires the Cartesian product structure and related measurement development and system correlation for application.Therefore,this thesis conducts an in-depth research of the doubleuniverses rough set,constructs the rough set based on two-dimensional direct-product universe,and further studies its rough approximation operators and uncertainty measures.Concretely,this thesis includes two parts as follows.1.Based on the Cartesian product,the two-dimensional direct product relation is defined,and then the covering space on two-dimensional direct product universe is established and the related properties of its approximation operators are studied.Finally,the systematic relationships between the upper and lower approximation operators based on the direct product relation and the basic relation(i.e.,the origin relation and the inverse relation)are discussed.2.The rough entropy and knowledge granularity of rough set over two-universes approximate space are used to perform projective decomposition,structural simulation,and granular replacement on two symmetrical single-universe covering spaces,and the direct product integration is implemented in the two-dimensional direct-product universe covering space,and thus the rough entropy and knowledge granularity are determined in the covering space on the two symmetrical single-universes and compositive two-dimensional direct-product universe.For the three sets of double measures,the double-measures sum,supremum and infimum,granulation monotonicity,three-way linear combination are researched respectively to reveal their relevance in three spaces.In addition,the concept of rough entropy is constructed by integrating the rough coefficients,and the in-depth application of rough entropy on the covering space of two-dimensional direct-product universe is explored.Finally,the data simulation and simulation experiment are made to verify the correctness of measure construction and theoretical properties.The research begins from the rough set of two universes and its measurement,and it carries out the essential positioning and simulation promotion,so it realizes the system construction and measurement development of the two-dimensional direct-product universe covering space.The related model construction and uncertainty characterization enrich rough set theory and its applications.