The similarity between two fuzzy sets (FSs) is an important concept in fuzzy logic. As the research interest on general type-2 (GT2) FSs has increased recently, many similarity measures for them have also been proposed. This paper gives a comprehensive overview of existing similarity measures for GT2 FSs, points out their limitations, and, by using an intuitive geometric explanation, proposes a Jaccard similarity measure for GT2 FSs that is an extension of the popular Jaccard similarity measure for type-1 and interval type-2 FSs. The fundamental difference between the proposed Jaccard similarity measure for GT2 FSs and all existing similarity measures is that the Jaccard similarity measure considers the overall geometries of two GT2 FSs and does not depend on a specific representation of the GT2 FSs, whereas all existing similarity measures for GT2 FSs depend either on the vertical slice representation or the $\alpha$-plane representation. We show that the Jaccard similarity measure for GT2 FSs satisfies four properties of a similarity measure and demonstrate its reasonableness using two examples.

中文翻译：

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封闭一般类型 2 模糊集的相似性测度：概述、比较和几何方法

两个模糊集（FS）之间的相似性是模糊逻辑中的一个重要概念。随着最近对一般 2 型 (GT2) FS 的研究兴趣增加，也提出了许多针对它们的相似性度量。本文全面概述了现有的 GT2 FS 相似性度量，指出了它们的局限性，并通过使用直观的几何解释，提出了 GT2 FS 的 Jaccard 相似性度量，它是流行的 1 类 Jaccard 相似性度量的扩展和间隔类型 2 FS。为 GT2 FS 提出的 Jaccard 相似性度量与所有现有相似性度量之间的根本区别在于，Jaccard 相似性度量考虑了两个 GT2 FS 的整体几何形状，而不依赖于 GT2 FS 的特定表示，而 GT2 FS 的所有现有相似性度量取决于垂直切片表示或 $\alpha$ 平面表示。我们表明 GT2 FS 的 Jaccard 相似性度量满足相似性度量的四个属性，并使用两个示例证明其合理性。