Fuzzy set theory has shown potential for reducing uncertainty as a result of data sparsity and also provides advantages for quantifying gradational changes like those of pollutant concentrations through fuzzy clustering based approaches. The ability to lower the sampling frequency and perform laboratory analyses on fewer samples, yet still produce an adequate pollutant distribution map, would reduce the initial cost of new remediation projects. To assess the ability of fuzzy modeling to make spatial predictions using fewer sample points, its predictive ability was compared with the ordinary kriging (OK) and inverse distance weighting (IDW) methods under increasingly sparse data conditions. This research used a Takagi–Sugeno (TS) fuzzy modelling approach with fuzzy c-means (FCM) clustering to make spatial predictions of the lead concentrations in soil. The performance of the TS model was very dependent on the number of outliers in the respective validation set. For modeling under sparse data conditions, the TS fuzzy modeling approach using FCM clustering and constant width Gaussian shaped membership functions did not show any advantages over IDW and OK for the type of data tested. Therefore, it was not possible to speculate on a possible reduction in sampling frequency for delineating the extent of contamination for new remediation projects.

中文翻译：

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稀疏分布地理空间数据空间预测的 Takagi-Sugeno 模糊建模研究

由于数据稀疏性，模糊集理论已显示出降低不确定性的潜力，并且还为通过基于模糊聚类的方法量化污染物浓度等渐变变化提供了优势。降低采样频率并在更少的样品上进行实验室分析的能力，但仍能产生足够的污染物分布图，将减少新的修复项目的初始成本。为了评估模糊建模使用较少样本点进行空间预测的能力，在数据日益稀疏的情况下，将其预测能力与普通克里金法 (OK) 和反距离加权 (IDW) 方法进行了比较。这项研究使用了Takagi–Sugeno（TS）模糊建模方法和模糊c均值（FCM）聚类来对土壤中的铅浓度进行空间预测。TS 模型的性能非常依赖于各自验证集中的异常值数量。对于稀疏数据条件下的建模，使用FCM聚类和等宽高斯形状隶属函数的TS模糊建模方法在测试数据类型方面没有表现出优于IDW和OK的优势。因此，无法推测可能会降低采样频率以描述新修复项目的污染程度。对于测试的数据类型，使用 FCM 聚类和恒定宽度高斯形状隶属函数的 TS 模糊建模方法没有显示出优于 IDW 和 OK 的任何优势。因此，无法推测可能会降低采样频率以描述新修复项目的污染程度。对于测试的数据类型，使用 FCM 聚类和恒定宽度高斯形状隶属函数的 TS 模糊建模方法没有显示出优于 IDW 和 OK 的任何优势。因此，无法推测可能会降低采样频率以描述新修复项目的污染程度。