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Complicated spatial structures (CSS) are common in biological data (e.g. fibrin clots, fibroblasts), but are difficult to quantitatively analyze without losing important information. Topological data analysis (TDA) provides a way for biologists to better understand, visualize, and interpret such data. TDA is a statistical framework for extracting topological information from data and using it to estimate properties of the underlying structures. It has potential to dramatically improve the analysis of biological data by retrieving and quantifying crucial information that is missed in ad-hoc methods by specifically targeting shape-related features.

We present a framework for hypothesis testing of CSS using persistent homology. The randomness in the data (due to measurement error or topological noise) is transferred to randomness in the topological summaries, which provides an infrastructure for inference. These tests allow for statistical comparisons between CSS. We present several possible test statistics using persistence diagrams and carryout a simulation study to investigate the suitableness of the proposed test statistics.