Multiple hypothesis testing with persistent homology

Figure 1.  Circle point clouds for a range of noise levels ($ \sigma^2 $) and point cloud sizes (25, 50,100 and 500 points)

Figure 2.  Some examples of point clouds and their axis-aligned or convex hull estimated null model point clouds. The first two rows are the cross-polytope acyclic model, third row is the concentric circles model with outer radius 2, fourth row is the figure 8 model with left radius 1.5 and the last row is a Thomas process with scale 0.05.
The left-most column is the observed acyclic or cyclic model, and the four columns to the right are sample point clouds drawn from the estimated null model (using estimators, from the top, axis, hull, axis, hull, axis

Figure 3.  ECDFs of each topological invariant of a sample of 100 standardized simulated point clouds for each of 100 different randomly selected acyclic model setups

Figure 4.  In the left-most column, what we would expect accurate and unbiased estimation and null models to look like. In the middle, the undilated convex hull results. To the right, the results from dilating the convex hull estimation as described. Best results among the convex hull approaches are achieved with the $ L_\infty $ persistence norm methods and the dilated unbiased convex hull estimation method, while $ L_1 $ and $ L_2 $ perform worse in the unbiased convex hull approach, and all the biased hull methods perform worse again

Figure 5.  QQ-plot for p-values split up by acyclic model type and by estimator type. The fact that these QQ-curves almost all lie over the diagonal line means that in almost all cases, we under-produce false positives

Figure 6.  QQ-plots for the FWER procedure on collections of acyclic point clouds against a uniform distribution

Figure 7.  ECDF-plots for the FWER procedure on collections of acyclic point clouds with a single power estimation point cloud, split by type of power estimation

Figure 8.  QQ-plots for the single-hypothesis case of all the composite invariants we study here, split up by acyclic model type and null model estimator

Figure 9.  QQ-plots for FWER method for all composite invariants we study here, split up by type ($ L_1/L_2/L_\infty $ or $ \pi $ or $ \ell $) and by null model estimator

Figure 10.  ECDF of the Concentric Circle models across different sample sizes, estimators (axis and unbiased convex hull) and additive topological invariants

Figure 11.  ECDF of the Figure 8 models across different sample sizes, estimators (axis and unbiased convex hull) and additive topological invariants

Figure 12.  ECDF of the Sphere models across different sample sizes, estimators (axis and unbiased convex hull) and additive topological invariants

Figure 13.  ECDF of the Thomas models across different sample sizes, estimators (axis and unbiased convex hull) and additive topological invariants

Figure 14.  Concentric circles point clouds for a range of noise levels ($ \sigma^2 $), point cloud sizes (25, 50,100 and 500 points), and size imbalances ($ r $)

Figure 15.  Figure 8 point clouds for a range of noise levels ($ \sigma^2 $), point cloud sizes (25, 50,100 and 500 points), and size imbalances ($ r $)

Figure 16.  Thomas process point clouds for a range of noise levels ($ \sigma^2 $) and point cloud sizes (25, 50,100 and 500 points)