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.

中文翻译：

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基于三级粒度结构的加权补熵系统

在粗糙集理论方面，已经引入了信息论措施来实现不确定性测量和系统应用，并且它们基于层次和粒度的稳健构建和深入开发变得必要和有价值。根据现有的补熵系统，通过决策表的三级粒度结构建立加权补熵系统，揭示其基本性质和系统等价性。首先，在微观底层贝叶斯概率公式引发数学变换和层次演化，在中层和宏观顶层构建三向加权补熵，实现层级性、系统性、单调性和算法。第二，将经典的补体-熵系统分层分解为中-中、微-底，实现了两个补体-熵系统之间的等价性。最后，通过表例和数据实验，有效地验证了相关的措施和性质。本研究分层建立三向加权互补熵来开发和解释传统的互补熵，从而促进信息优化和不确定性应用。