🤖 AI Summary
This study develops a statistical inference framework for topological features derived from persistent homology and applies it to goodness-of-fit testing for spatial point patterns. By introducing stabilization techniques—previously unexplored in persistent homology analysis—the authors establish a central limit theorem for bounded functionals of persistence diagrams generated by germ–grain random set models, particularly those exhibiting exponential decay of correlations, as the observation window expands. Building on this asymptotic theory, they construct effective goodness-of-fit tests by combining rectangular partitions of persistence diagrams with functional summaries such as accumulated persistence functions (APFs) and support functions of lift zonoids. The proposed methodology accurately distinguishes between clustered and repulsive spatial structures and successfully detects spatial interaction patterns in breast histology images, demonstrating its practical utility.
📝 Abstract
This paper establishes a central limit theorem (CLT) for functionals of $M$-bounded persistence diagrams arising from germ-grain random set models. Building on stabilisation methods for marked point processes, we show that, under certain conditions, these topological summaries exhibit asymptotic normality as the observation window increases, particularly for models with exponential decay of correlations. These results are applied in goodness-of-fit tests designed to detect spatial interactions such as clustering or repulsion. Using test statistics derived from rectangular partitions of persistence diagrams and functional summaries (e.g., the APF or the support function of the lift zonoid), the study distinguishes between different models. Finally, the methodology is applied to histological images of breast tissue.