Abstract Three-way approximations of fuzzy sets are an important scheme of granular computing, by abstracting a fuzzy set to its discrete three option-alternatives which adhere to human cognitive behaviors and reduce the computational burden. The key point of such three-way approximations of fuzzy sets is how to choose a suitable design leading to their realization. Undesired three-way approximations might be produced if the selected mechanism is unsuitable to data distribution. In this study, the principles for constructing three-way approximations of fuzzy sets are summarized. The following taxonomy of these principles is provided, namely (i) uncertainty balance-based principle, (ii) prototype-based principle, and (iii) model-based invoking the tradeoff between classification error and the number of data that have to be classified. A number of detailed optimization models are discussed in detail. To evaluate the performance of different construction principles, a general unsupervised learning framework based on three-way approximations of fuzzy sets is exhibited. Some synthetic data sets along with sixteen data sets from UCI repository are involved for experiments. Friedman testing followed by Holm-Bonferroni testing are exploited to test the performance significance of the proposed criteria, which can provide insights and deliver guidance when choosing a principle for constructing three-way approximations of fuzzy sets in the real-world scenarios. The research methods in this paper can also be extended to supervised and semi-supervised learning areas.

中文翻译：

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模糊集三向逼近的构建原则：基于无监督学习的比较评价

摘要 模糊集的三向逼近是粒计算的一个重要方案，它通过将模糊集抽象为其离散的三个选项，这些选项遵循人类的认知行为并减少计算负担。模糊集的这种三向近似的关键在于如何选择合适的设计来实现它们。如果所选机制不适合数据分布，则可能会产生不希望的三向近似。在这项研究中，总结了构建模糊集三向逼近的原则。提供了这些原则的以下分类法，即（i）基于不确定性平衡的原则，（ii）基于原型的原则，以及（iii）基于模型的调用分类错误和必须分类的数据数量之间的权衡. 详细讨论了许多详细的优化模型。为了评估不同构造原理的性能，展示了一种基于模糊集三向近似的通用无监督学习框架。实验涉及一些合成数据集以及来自 UCI 存储库的 16 个数据集。利用弗里德曼测试和 Holm-Bonferroni 测试来测试所提出标准的性能重要性，这可以在选择在现实世界场景中构建模糊集三向近似的原则时提供见解和指导。本文的研究方法也可以扩展到监督和半监督学习领域。展示了一种基于模糊集三向近似的通用无监督学习框架。实验涉及一些合成数据集以及来自 UCI 存储库的 16 个数据集。利用弗里德曼测试和 Holm-Bonferroni 测试来测试所提出标准的性能重要性，这可以在选择在现实世界场景中构建模糊集三向近似的原则时提供见解和指导。本文的研究方法也可以扩展到监督和半监督学习领域。展示了一种基于模糊集三向近似的通用无监督学习框架。实验涉及一些合成数据集以及来自 UCI 存储库的 16 个数据集。利用弗里德曼测试和 Holm-Bonferroni 测试来测试所提出标准的性能重要性，这可以在选择在现实世界场景中构建模糊集三向近似的原则时提供见解和指导。本文的研究方法也可以扩展到监督和半监督学习领域。利用弗里德曼测试和 Holm-Bonferroni 测试来测试所提出标准的性能重要性，这可以在选择在现实世界场景中构建模糊集三向近似的原则时提供见解和指导。本文的研究方法也可以扩展到监督和半监督学习领域。利用弗里德曼测试和 Holm-Bonferroni 测试来测试所提出标准的性能重要性，这可以在选择在现实世界场景中构建模糊集三向近似的原则时提供见解和指导。本文的研究方法也可以扩展到监督和半监督学习领域。