Due to the numerous applications that utilize different types of fuzzy membership functions (MFs), it may sometimes be difficult to choose an appropriate MF for a particular application. In this paper, we establish preliminary guidelines to direct this selection by proposing a three-stage method. In the “forward” stage, different MFs, such as crisp MFs, type-1 (T1) fuzzy MFs, and type-2 (T2) fuzzy MFs, are generated from multidimensional data sets. Next, in the “reverse” stage, data is generated back from these MFs by considering different bin sizes. In doing so, various data sets may be generated for different applications which require fuzzy data. Finally, for the “similarity analysis” stage, we propose an iterative algorithm that makes use of the results of Wilcoxon signed rank (WSR) and Wilcoxon rank sum (WRS) tests to compare the original data and the generated data. From the results of these tests, recommendations concerning the suitability of MFs for a specific application may be suggested by observing the accuracy of representation and the requirements of the application. With this analysis, the objective is to gain insight on when T2 fuzzy sets may be considered to outperform T1 fuzzy sets, and vice versa. Several examples are provided using synthetic and real data to validate the iterative algorithm for data sets in various dimensions.

中文翻译：

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多维 1 型和 2 型模糊隶属函数生成的数据分析


由于许多应用程序使用不同类型的模糊隶属函数 (MF)，有时可能很难为特定应用程序选择合适的 MF。在本文中，我们通过提出三阶段方法建立了指导这种选择的初步指南。在“前向”阶段，从多维数据集生成不同的 MF，例如清晰 MF、1 型（T1）模糊 MF 和 2 型（T2）模糊 MF。接下来，在“反向”阶段，通过考虑不同的 bin 大小，从这些 MF 生成数据。这样做时，可以为需要模糊数据的不同应用生成各种数据集。最后，对于“相似性分析”阶段，我们提出了一种迭代算法，利用 Wilcoxon 符号秩（WSR）和 Wilcoxon 秩和（WRS）测试的结果来比较原始数据和生成的数据。根据这些测试的结果，可以通过观察表示的准确性和应用的要求来提出有关 MF 对于特定应用的适用性的建议。通过此分析，目标是了解何时可以认为 T2 模糊集优于 T1 模糊集，反之亦然。提供了使用合成数据和真实数据的几个示例，以验证各种维度的数据集的迭代算法。