In this research, the fourth axiom to improve the well-defined examination of similarity measures is studied, where the measures have a symmetric structure. We first provide a theoretic enhancement of three correlation coefficient similarity measures that were proposed by a source paper. Second, we use the same numerical example proposed by the source paper for pattern recognition problems to illustrate that the weighted correlation coefficient similarity measure is dependent on the set of weights. Finally, we demonstrate that the correlation coefficient similarity measure in the intuitionistic fuzzy set environment can address the issue of practical fault diagnosis when solving the turbine engine problems using similarity measures with symmetric characteristics.

中文翻译：

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相似测度第四公理

在本研究中，研究了改进相似性度量的明确定义的第四个公理，其中度量具有对称结构。我们首先提供了由源论文提出的三个相关系数相似性度量的理论增强。其次，我们使用源论文中针对模式识别问题提出的相同数值示例来说明加权相关系数相似性度量取决于权重集。最后，我们证明了直觉模糊集环境中的相关系数相似性度量可以解决使用具有对称特征的相似性度量解决涡轮发动机问题时的实际故障诊断问题。