Motivated by the rapidly increasing relevance of virtual material design in the domain of materials science, it has become essential to assess whether topological properties of stochastic models for a spatial tessellation are in accordance with a given dataset. Recently, tools from topological data analysis such as the persistence diagram have allowed to reach profound insights in a variety of application contexts. In this work, we establish the asymptotic normality of a variety of test statistics derived from a tessellation-adapted refinement of the persistence diagram. Since in applications, it is common to work with tessellation data subject to interactions, we establish our main results for Voronoi and Laguerre tessellations whose generators form a Gibbs point process. We elucidate how these conceptual results can be used to derive goodness of fit tests, and then investigate their power in a simulation study. Finally, we apply our testing methodology to a tessellation describing real foam data.
翻译:材料科学领域虚拟材料设计的相关性迅速提高,因此,必须评估空间星系切换模型的地形特性是否符合特定数据集。最近,来自地形数据分析的工具,如持久性图等,在各种应用环境中可以深入地洞察到各种应用背景。在这项工作中,我们确定了从对持久性图进行套接式调整后得出的各种测试统计数据的无症状常态性。由于在应用中,与可互动的星系数据合作是常见的,因此,我们为Voronoi和Laguerre熔融得出了我们的主要结果,而Voronoi和Laguerre 熔融形成一个吉布点过程。我们阐述了这些概念结果如何用于获得适当测试的良性,然后在模拟研究中调查它们的力量。最后,我们将我们的测试方法应用于描述真实泡沫数据的星系。