  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.
