基于严凸函数的知识粒度与相对粒度

摘要

首次将严凸函数引入知识粒度研究中，提出基于严凸函数的知识粒度理论框架。根据该理论框架，给出一系列知识粒度度量函数，证明现有多种常见的知识粒度度量是该理论框架的特殊情形或变种。给出基于严凸函数的相对粒度定义，虽然对任意严凸函数导出的相对粒度不满足单调性，但对一些特殊严凸函数导出的相对粒度证明其单调性，并给出等号成立的条件。证明现有条件信息熵都是文中提出的严凸函数相对粒度的特殊情形，揭示它们的知识粒度本质。针对一致决策表，证明相对粒度与正区域不变等价，从而得到一致决策表代数约简的相对粒度判定方法。数值算例验证文中结论的正确性。%The strictly convex function is introduced into the research of knowledge granularity for the first time. Based on the strictly convex function, a theory framework for constructing knowledge granularity is proposed. A series of knowledge granularity measuring functions is derived under this framework. It is proved that the existing knowledge granularity measuring functions are the special cases or variations of knowledge granularity measures which are derived by strictly convex functions. The definition of the relative knowledge granularity based on strictly convex function is given. Its monotonicity is proved for some special strictly convex functions and the corresponding equality conditions are provided, although it does not hold for general strictly convex functions. It is proved that the existing two conditional information entropies are the special forms of the proposed relative knowledge granularity. Their knowledge granularity essence is revealed. For a consistent decision table, it is proved that the relative knowledge granularity is equivalent to positive region for each other. Therefore, the attribute reduction judgment method of algebraic reduction is presented by the relative granularity in consistent decision table. The correctness of the proposed conclusions is showed by a numerical example.