A weighted complement-entropy system based on tri-level granular structures

In terms of rough set theory, information-theoretical measures have been introduced to implement uncertainty measurements and system applications, and their robust construction and in-depth development based on hierarchy and granularity become required and valuable. According to the existing complement-entropy system, a weighted complement-entropy system is established by tri-level granular structures of decision table, and its basic properties and systematic equivalency are revealed. Firstly, Bayes' probability formula at micro-bottom induces a mathematical transformation and hierarchical evolution, and three-way weighted complement-entropies are constructed at both meso-middle and macro-top to achieve the hierarchy, systematicness, monotonicity, and algorithm. Secondly, the classical complement-entropy system is hierarchically decomposed to meso-middle and micro-bottom, and the equivalency between both complement-entropy systems is achieved. Finally, relevant measures and properties are effectively verified by table examples and data experiments. This study hierarchically establishes three-way weighted complement-entropies to develop and interpret the traditional complement-entropies, thus facilitating information optimization and uncertainty applications.