The size and structure of spatial molecular and atomic clustering can significantly impact material properties and is therefore important to accurately quantify. Ripley's K-function (K(r)), a measure of spatial correlation, can be used to perform such quantification when the material system of interest can be represented as a marked point pattern. This work demonstrates how machine learning models based on K(r)-derived metrics can accurately estimate cluster size and intra-cluster density in simulated three dimensional (3D) point patterns containing spherical clusters of varying size; over 90% of model estimates for cluster size and intra-cluster density fall within 11% and 18% error of the true values, respectively. These K(r)-based size and density estimates are then applied to an experimental APT reconstruction to characterize MgZn clusters in a 7000 series aluminum alloy. We find that the estimates are more accurate, consistent, and robust to user interaction than estimates from the popular maximum separation algorithm. Using K(r) and machine learning to measure clustering is an accurate and repeatable way to quantify this important material attribute.

中文翻译：

---

使用 Ripley's K 函数和机器学习进行原子探针断层扫描的三维聚类分析

空间分子和原子簇的大小和结构可以显着影响材料特性，因此准确量化很重要。当感兴趣的材料系统可以表示为标记点模式时，Ripley 的 K 函数 (K(r)) 是一种空间相关性的度量，可用于执行此类量化。这项工作展示了基于 K(r) 衍生度量的机器学习模型如何在包含不同大小球形簇的模拟三维 (3D) 点模式中准确估计簇大小和簇内密度；超过 90% 的集群大小和集群内密度的模型估计值分别落在真实值的 11% 和 18% 以内。然后将这些基于 K(r) 的尺寸和密度估计应用于实验性 APT 重建，以表征 7000 系列铝合金中的 MgZn 簇。我们发现，与流行的最大分离算法的估计相比，这些估计对用户交互更加准确、一致和稳健。使用 K(r) 和机器学习来测量聚类是量化这一重要材料属性的准确且可重复的方法。