Objective tools for characterizing materials at the atomic level are often difficult to develop because of the size or structure of the data. Atom probe tomography (APT) is a measurement tool that maps the location and type of atoms in materials in three‐dimensions (3D), producing data sets with potentially billions of observations. In this work, we present a set of spatial statistics methods developed to test the null hypotheses of no global spatial association; no local spatial association; and no local spatial cross‐correlation and apply these for the first time to APT data. The empirical and modeled covariogram and Moran's I can be used to study the global structure of a spatially referenced atomic element. The local indicator of spatial association (LISA) identifies volumes where high levels of values (hot spots) or low levels of values (cold spots) of elemental clustering exist. The local indicator of spatial cross‐correlation (LISC) reports where simultaneously high levels or low levels of two atomic elements occur. For each test statistic at each location, an associated p‐value is produced that can be used to weigh the evidence in favor of spatial clustering. The size of APT data sets presents some challenges, so the effect of weight functions and neighborhood selection on the computation and significance of the test statistics are discussed, and the issue of multiple statistical testing is also considered. These methods are illustrated using an APT data set with atomic percentages reported in voxels binned to 1 nm³.

中文翻译：

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客观识别局部空间结构以进行材料表征

由于数据的大小或结构，通常难以开发用于在原子级表征材料的客观工具。原子探针层析成像（APT）是一种测量工具，可按三维（3D）映射材料中原子的位置和类型，从而生成具有数十亿次观测值的数据集。在这项工作中，我们提出了一套空间统计方法，以检验没有全局空间关联的零假设。没有局部空间联系；并且没有局部空间互相关，并将其首次应用于APT数据。经验和建模的协变量图和Moran's I可用于研究空间参考原子元素的整体结构。空间关联的本地指示符（LISA）标识元素聚类的高值（热点）或低值（冷点）存在的体积。空间互相关的局部指标（LISC）报告同时发生两个原子元素的高位或低位的情况。对于每个位置的每个测试统计信息，产生了p值，可以用来权衡证据以利于空间聚类。APT数据集​​的大小提出了一些挑战，因此讨论了权重函数和邻域选择对检验统计量的计算和意义的影响，并考虑了多重统计检验的问题。这些方法使用APT数据集​​进行了说明，其中原子百分比报告在装在1 nm ^3的体素中。