Extending hypothesis testing with persistent homology to three or more groups

Abstract

We extend the work of Robinson and Turner to use hypothesis testing with persistence homology to test for measurable differences in shape between point clouds from three or more groups. Using samples of point clouds from three distinct groups, we conduct a large-scale simulation study to validate our proposed extension. We consider various combinations of groups, samples sizes and measurement errors in the simulation study, providing for each combination the percentage of p-values below an alpha-level of 0.05. Additionally, we apply our method to a Cardiotocography data set and find statistically significant evidence of measurable differences in shape between normal, suspect and pathologic health status groups.2010 Mathematics Subject Classification. 55N35, 62H15.

Hypothesis Testing for Shapes using Vectorized Persistence Diagrams

“…Most hypothesis test methods for persistent homology have been studied based on permutation tests (Chen et al, 2015;Robinson and Turner, 2017;Cericola et al, 2018;Berry et al, 2020). It is difficult to define probability distributions on the space of persistence diagrams because it is infinite in dimension and has complicated geometry (Robinson and Turner, 2017).…”

“…Algorithm 1 presents the permutation test procedure of Robinson and Turner (2017). Cericola et al (2018) extend the two-sample test scheme to multiple label group testing using the one-way analysis of variance (ANOVA) procedure and Vejdemo-Johansson and Mukherjee (2018) propose procedures to control a multiple testing problem.…”

“…permutation tests using a pairwise distance of persistence diagrams (Robinson and Turner, 2017;Cericola et al, 2018) and functional summaries (Chen et al, 2015;Berry et al, 2020), multiple hypothesis testing procedures using uniform point cloud data (Vejdemo-Johansson and Mukherjee, 2018), tests using mean persistence landscapes (Bubenik, 2015) and kernel hypothesis tests in RKHS (Kusano, 2019).…”

Multiple testing with persistent homology

“…An alternative approach was considered in a series of papers where instead of considering a persistence diagram as a summary a probability density was used as a topological summary, an approach called distance to measure [9,10,11] In the context of hypothesis testing [4] proposed using goodness of fit statistics -Kolmogorov-Smirnov, χ 2 or Mann-Whitney -to test compare empirical distributions from two samples of persistence diagrams. The ideas most closely related to the procedures we develop in this paper was to define hypothesis testing procedures directly on persistence diagrams using permutation testing and barcode distances [8,22]. In this paper we will extend two sample single hypothesis testing and ANOVA procedures to the multiple hypothesis test setting.…”

“…Hypothesis testing in the based on topological summaries of data has been an area of Topological Data Analysis (TDA) that has seen growth recently as both applied and mathematical statistics have been developed using TDA. Almost of all the current literature on hypothesis testing in TDA has focused on two sample tests [22] or extensions to analysis of variance (ANOVA) settings [8] where differences across more than two conditions are considered. Neither of these papers take into account multiple testing because the number of hypotheses tested is small, for example one in two sample tests.…”