Let us have a population of objects that are subjected to a given event. Each object can be assigned a degree of membership to the population. Assume that for a set of objects, a continuous parameter is measured before and after the given event. We can now form two paired fuzzy samples from the population—Z¹ and Z². We then measure the change ΔZ of Z from Z² to Z¹ and form the fuzzy sample of change Z³. Our aim is to explore if the event has caused statistically significant change ΔZ of Z for that Population. Therefore, we conduct statistical tests for nullity of the central tendencies (mean, median) of change over paired fuzzy samples. We develop two Bootstrap based simulation algorithms to identify the p_value of such tests for the mean of change and for the median of change. Each of the algorithms has eight modifications depending on: (a) whether the synthetic fuzzy samples were generated using ‘quasi-equal-information generation’ (i.e. synthetic fuzzy samples with almost equal amount of information as the original ones) or using ‘equal-size generation’ (i.e. synthetic fuzzy samples with the same size of fuzzy observations as the original ones); (b) whether the approximated sample cumulative distribution functions (CDF) for the synthetic samples generation are empirical (ECDF), or fuzzy empirical (FECDF); (c) whether we perform a one-tail or two-tail test. We demonstrate the consistency of the developed fuzzy Bootstrap nullity tests on two numerical examples where the central tendencies of change are known. We also present a medical case study, where we compare the proposed techniques with an alternative one that utilizes crisp tests. In that case study, we demonstrate the advantages of fuzzy Bootstrap nullity tests in comparison to standard crisp methods over central tendencies. In our discussions, we outline that to declare significance of change, we focus on a whole cluster of tests over fuzzy paired samples as opposed to relying on individual test results.

让我们有一群受给定事件影响的对象。可以为每个对象分配一定程度的成员资格。假设对于一组对象，在给定事件之前和之后测量连续参数。我们现在可以从总体中形成两个成对的模糊样本——Z ¹和Z ²。然后我们测量Z 从Z ²到Z ¹的变化Δ Z并形成变化Z ³的模糊样本。我们的目标是探索该事件是否引起了Z的统计显着变化ΔZ对于那个人口。因此，我们对配对模糊样本的变化的中心趋势（平均值、中位数）的无效性进行统计测试。我们开发了两种基于 Bootstrap 的模拟算法来识别p_值变化的平均值和变化的中位数的此类测试。每种算法都有八种修改，具体取决于：(a) 合成模糊样本是使用“准等信息生成”生成的（即信息量几乎与原始样本相同的合成模糊样本）还是使用“等信息生成”生成的。大小生成'（即与原始模糊观察具有相同大小的合成模糊样本）；(b) 合成样本生成的近似样本累积分布函数 (CDF) 是经验 (ECDF) 还是模糊经验 (FECDF)；(c) 我们是执行单尾测试还是双尾测试。我们在两个已知变化的中心趋势的数值示例上证明了开发的模糊 Bootstrap 无效测试的一致性。我们还提供了一个医学案例研究，我们将所提出的技术与使用清晰测试的替代技术进行比较。在那个案例研究中，我们展示了模糊 Bootstrap 无效性测试与标准清晰方法相比中心趋势的优势。在我们的讨论中，我们概述了为了声明变化的重要性，我们专注于对模糊配对样本的整个测试集群，而不是依赖于单个测试结果。