In this paper we present the methodology for detecting outliers and testing the goodness-of-fit of random sets using topological data analysis. We construct the filtration from level sets of the signed distance function and consider various summary functions of the persistence diagram derived from the obtained persistence homology. The outliers are detected using functional depths for the summary functions. Global envelope tests using the summary statistics as test statistics were used to construct the goodness-of-fit test. The procedures were justified by a simulation study using germ-grain random set models.

翻译：本文提出了利用拓扑数据分析检测随机集异常值及检验其拟合优度的方法。我们通过符号距离函数的水平集构造过滤过程，并考虑从所得持续同源性中提取的持续图的各种汇总函数。采用汇总函数的函数深度检测异常值，同时以汇总统计量作为检验统计量构建全局包络检验以进行拟合优度检验。通过使用种子-颗粒随机集模型的仿真研究验证了该方法的有效性。