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Multiple hypothesis testing with persistent homology

  • * Corresponding author: Mikael Vejdemo-Johansson

    * Corresponding author: Mikael Vejdemo-Johansson 

MVJ would like to acknowledge the support of NVIDIA Corporation with the donation of the Titan X Pascal GPU used for this research. This work was supported by a grant from the Simons Foundation (961833, MVJ)
SM would like to acknowledge partial funding from HFSP RGP005, NSF DMS 17-13012, NSF BCS 1552848, NSF DBI 1661386, NSF IIS 15-46331, NSF DMS 16-13261, as well as high-performance computing partially supported by grant 2016-IDG-1013 from the North Carolina Biotechnology Center
The simulated data used in this paper are available from Figshare, DOI 10.6084/m9.figshare.10262507

Abstract Full Text(HTML) Figure(16) / Table(12) Related Papers Cited by
  • In this paper we propose a computationally efficient multiple hypothesis testing procedure for persistent homology. The computational efficiency of our procedure is based on the observation that one can empirically simulate a null distribution that is universal across many hypothesis testing applications involving persistence homology. Our observation suggests that one can simulate the null distribution efficiently based on a small number of summaries of the collected data and use this null in the same way that p-value tables were used in classical statistics. To illustrate the efficiency and utility of the null distribution we provide procedures for rejecting acyclicity with both control of the Family-Wise Error Rate (FWER) and the False Discovery Rate (FDR). We will argue that the empirical null we propose is very general conditional on a few summaries of the data based on simulations and limit theorems for persistent homology for point processes.

    Mathematics Subject Classification: Primary: 55N31.

    Citation:

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  • 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)

    Table 1.  Rejection rates by estimator and invariant for an axis-aligned rectangular acyclic model. Rates for $ \alpha = 0.01/0.05/0.10 $ are separated with $ / $ in each table cell

    Invariant axis hull unbiased hull
    $\texttt{Linf.0}$ .08/.15/.28 .21/.29/.38 .02/.05/.05
    $\texttt{Linf.1}$ .03/.07/.10 .02/.10/.22 .03/.03/.05
    $\texttt{L1.0}$ .00/.00/.08 .94/.96/.96 .00/.03/.09
    $\texttt{L1.1}$ .06/.06/.24 .10/.26/.36 .08/.08/.10
    $\texttt{L2.0}$ .05/.09/.13 .82/.88/.92 .00/.00/.06
    $\texttt{L2.1}$ .06/.13/.28 .22/.32/.44 .05/.08/.12
     | Show Table
    DownLoad: CSV

    Table 2.  Rejection rates by acyclic model, estimator and invariant for each type of acyclic model. Rates for $ \alpha = 0.01/0.05/0.10 $ are separated with $ / $ in each table cell

    Model Invariant axis unbiased hull
    axis $\texttt{L1.0}$ 0.00/0.00/0.00 0.00/0.00/0.00
    $\texttt{L1.1}$ 0.11/0.21/0.55 0.00/0.00/0.19
    $\texttt{L2.0}$ 0.00/0.00/0.00 0.00/0.00/0.00
    $\texttt{L2.1}$ 0.21/0.31/0.42 0.00/0.08/0.15
    $\texttt{Linf.0}$ 0.00/0.00/0.00 0.00/0.00/0.00
    $\texttt{Linf.1}$ 0.00/0.10/0.10 0.00/0.08/0.08
    ball $\texttt{L1.0}$ 0.00/0.00/0.00 0.00/0.02/0.03
    $\texttt{L1.1}$ 0.00/0.00/0.00 0.00/0.00/0.00
    $\texttt{L2.0}$ 0.00/0.00/0.00 0.00/0.01/0.01
    $\texttt{L2.1}$ 0.00/0.00/0.00 0.00/0.00/0.00
    $\texttt{Linf.0}$ 0.00/0.00/0.00 0.00/0.00/0.01
    $\texttt{Linf.1}$ 0.00/0.02/0.02 0.01/0.01/0.01
    cross $\texttt{L1.0}$ 0.18/0.18/0.18 0.13/0.13/0.21
    $\texttt{L1.1}$ 0.00/0.00/0.00 0.00/0.22/0.32
    $\texttt{L2.0}$ 0.18/0.18/0.18 0.13/0.13/0.21
    $\texttt{L2.1}$ 0.00/0.00/0.00 0.00/0.00/0.32
    $\texttt{Linf.0}$ 0.11/0.11/0.19 0.13/0.13/0.23
    $\texttt{Linf.1}$ 0.00/0.00/0.00 0.00/0.00/0.11
    random.hull $\texttt{L1.0}$ 0.00/0.00/0.00 0.00/0.00/0.00
    $\texttt{L1.1}$ 0.00/0.00/0.00 0.10/0.10/0.10
    $\texttt{L2.0}$ 0.00/0.00/0.00 0.00/0.00/0.00
    $\texttt{L2.1}$ 0.00/0.00/0.00 0.10/0.10/0.10
    $\texttt{Linf.0}$ 0.00/0.00/0.02 0.00/0.00/0.09
    $\texttt{Linf.1}$ 0.00/0.02/0.02 0.01/0.04/0.09
    random.polytope $\texttt{L1.0}$ 0.00/0.00/0.00 0.02/0.02/0.07
    $\texttt{L1.1}$ 0.00/0.02/0.07 0.01/0.03/0.04
    $\texttt{L2.0}$ 0.00/0.00/0.00 0.00/0.02/0.07
    $\texttt{L2.1}$ 0.00/0.02/0.04 0.01/0.03/0.08
    $\texttt{Linf.0}$ 0.00/0.05/0.05 0.04/0.04/0.04
    $\texttt{Linf.1}$ 0.00/0.00/0.05 0.00/0.02/0.13
    simplex $\texttt{L1.0}$ 0.00/0.00/0.00 0.01/0.03/0.09
    $\texttt{L1.1}$ 0.00/0.00/0.00 0.00/0.02/0.04
    $\texttt{L2.0}$ 0.00/0.00/0.00 0.02/0.03/0.05
    $\texttt{L2.1}$ 0.00/0.00/0.00 0.00/0.00/0.02
    $\texttt{Linf.0}$ 0.01/0.02/0.02 0.00/0.03/0.11
    $\texttt{Linf.1}$ 0.00/0.00/0.00 0.00/0.02/0.03
     | Show Table
    DownLoad: CSV

    Table 3.  Rejection rates across all acyclic models. Rates for $ \alpha = 0.01/0.05/0.10 $ are separated with $ / $ in each table cell

    Invariant axis unbiased hull
    $\texttt{L1.0}$ 0.06/0.07/0.08 0.09/0.10/0.15
    $\texttt{L1.1}$ 0.03/0.05/0.07 0.02/0.03/0.06
    $\texttt{L2.0}$ 0.08/0.08/0.10 0.05/0.11/0.14
    $\texttt{L2.1}$ 0.06/0.07/0.09 0.02/0.07/0.07
    $\texttt{Linf.0}$ 0.07/0.10/0.11 0.01/0.03/0.08
    $\texttt{Linf.1}$ 0.00/0.02/0.07 0.00/0.02/0.04
     | Show Table
    DownLoad: CSV

    Table 4.  Rejection rates across all Figure 8 power models, using homological dimension 1. Rates for $ \alpha = 0.01/0.05/0.10 $ are separated with $ / $ in each table cell

    Estimator Invariant Noise 25 50 100 500
    axis L1 0.01 0.80/0.87/0.97 0.80/0.80/0.80 1.00/1.00/1.00 0.00/0.00/0.00
    0.05 0.20/0.20/0.20 0.20/0.27/0.38 0.00/0.00/0.00 0.00/0.00/0.00
    0.1 0.20/0.35/0.40 0.00/0.12/0.20 0.00/0.00/0.00 0.00/0.00/0.00
    L2 0.01 0.82/0.99/1.00 0.80/0.80/0.80 1.00/1.00/1.00 1.00/1.00/1.00
    0.05 0.20/0.20/0.21 0.60/0.60/0.60 0.41/0.53/0.71 0.80/0.80/0.80
    0.1 0.20/0.35/0.40 0.02/0.19/0.25 0.00/0.00/0.00 0.00/0.00/0.00
    Linf 0.01 0.26/0.60/0.62 0.80/0.80/0.80 1.00/1.00/1.00 1.00/1.00/1.00
    0.05 0.02/0.02/0.02 0.43/0.59/0.60 0.49/0.98/1.00 1.00/1.00/1.00
    0.1 0.40/0.40/0.40 0.00/0.27/0.60 0.00/0.16/0.37 0.40/0.41/0.48
    unbiased.hull L1 0.01 0.62/0.79/0.80 0.60/0.74/0.80 0.83/0.99/1.00 0.00/0.00/0.00
    0.05 0.20/0.20/0.22 0.40/0.46/0.59 0.00/0.05/0.19 0.00/0.00/0.00
    0.1 0.20/0.23/0.35 0.00/0.12/0.20 0.00/0.00/0.00 0.00/0.00/0.00
    L2 0.01 0.80/0.80/0.91 0.80/0.80/0.80 1.00/1.00/1.00 1.00/1.00/1.00
    0.05 0.20/0.20/0.30 0.60/0.60/0.60 0.63/0.78/0.80 1.00/1.00/1.00
    0.1 0.20/0.26/0.38 0.40/0.40/0.40 0.00/0.00/0.10 0.00/0.00/0.00
    Linf 0.01 0.40/0.47/0.58 0.80/0.80/0.80 1.00/1.00/1.00 1.00/1.00/1.00
    0.05 0.20/0.23/0.35 0.60/0.60/0.64 1.00/1.00/1.00 1.00/1.00/1.00
    0.1 0.06/0.21/0.22 0.24/0.52/0.66 0.40/0.56/0.63 0.42/0.60/0.66
     | Show Table
    DownLoad: CSV

    Table 5.  Rejection rates by acyclic model, estimator and invariant for each type of acyclic model using the multiplicative invariants $ \pi $ and $ \ell $. Rates for $ \alpha = 0.01/0.05/0.10 $ are separated with $ / $ in each table cell

    Model Invariant axis hull unbiased hull
    axis $\texttt{ell.L1.1}$ 0.00/0.00/0.00 0.00/0.00/0.00 0.00/0.00/0.00
    $\texttt{ell.L2.1}$ 0.21/0.33/0.38 0.13/0.13/0.28 0.11/0.23/0.32
    $\texttt{ell.Linf.1}$ 0.00/0.00/0.00 0.13/0.13/0.13 0.00/0.07/0.07
    $\texttt{pi.L1.1}$ 0.11/0.23/0.56 0.26/0.34/0.42 0.11/0.26/0.41
    $\texttt{pi.L2.1}$ 0.21/0.23/0.43 0.26/0.34/0.34 0.11/0.26/0.41
    $\texttt{pi.Linf.1}$ 0.00/0.00/0.00 0.13/0.13/0.13 0.07/0.07/0.07
    ball $\texttt{ell.L1.1}$ 0.00/0.00/0.00 0.00/0.00/0.00 0.00/0.00/0.00
    $\texttt{ell.L2.1}$ 0.02/0.06/0.07 0.03/0.08/0.11 0.01/0.05/0.06
    $\texttt{ell.Linf.1}$ 0.00/0.00/0.08 0.02/0.02/0.09 0.00/0.03/0.04
    $\texttt{pi.L1.1}$ 0.01/0.03/0.06 0.00/0.02/0.07 0.00/0.09/0.18
    $\texttt{pi.L2.1}$ 0.01/0.03/0.05 0.00/0.01/0.07 0.00/0.05/0.13
    $\texttt{pi.Linf.1}$ 0.00/0.02/0.09 0.00/0.02/0.07 0.01/0.06/0.08
    cross $\texttt{ell.L1.1}$ 0.00/0.00/0.00 0.00/0.00/0.00 0.00/0.00/0.00
    $\texttt{ell.L2.1}$ 0.00/0.00/0.00 0.00/0.00/0.00 0.00/0.00/0.00
    $\texttt{ell.Linf.1}$ 0.00/0.00/0.00 0.00/0.00/0.00 0.00/0.00/0.00
    $\texttt{pi.L1.1}$ 0.18/0.31/0.31 0.12/0.33/0.33 0.11/0.22/0.31
    $\texttt{pi.L2.1}$ 0.18/0.18/0.31 0.12/0.24/0.36 0.11/0.22/0.22
    $\texttt{pi.Linf.1}$ 0.00/0.00/0.00 0.00/0.00/0.12 0.00/0.00/0.21
    random.hull $\texttt{ell.L1.1}$ 0.00/0.00/0.00 0.00/0.00/0.00 0.00/0.00/0.00
    $\texttt{ell.L2.1}$ 0.00/0.02/0.11 0.02/0.11/0.11 0.03/0.03/0.06
    $\texttt{ell.Linf.1}$ 0.00/0.10/0.15 0.00/0.01/0.01 0.00/0.03/0.24
    $\texttt{pi.L1.1}$ 0.16/0.16/0.16 0.18/0.19/0.20 0.14/0.14/0.14
    $\texttt{pi.L2.1}$ 0.16/0.16/0.16 0.18/0.19/0.20 0.14/0.14/0.14
    $\texttt{pi.Linf.1}$ 0.11/0.19/0.20 0.04/0.10/0.21 0.09/0.15/0.20
    random.polytope $\texttt{ell.L1.1}$ 0.00/0.00/0.00 0.00/0.00/0.00 0.00/0.00/0.00
    $\texttt{ell.L2.1}$ 0.03/0.09/0.15 0.02/0.05/0.12 0.01/0.01/0.08
    $\texttt{ell.Linf.1}$ 0.00/0.00/0.04 0.00/0.00/0.06 0.00/0.00/0.02
    $\texttt{pi.L1.1}$ 0.01/0.09/0.13 0.07/0.09/0.11 0.02/0.06/0.11
    $\texttt{pi.L2.1}$ 0.01/0.09/0.15 0.04/0.06/0.11 0.01/0.07/0.11
    $\texttt{pi.Linf.1}$ 0.00/0.02/0.02 0.00/0.00/0.08 0.00/0.06/0.11
    simplex $\texttt{ell.L1.1}$ 0.00/0.00/0.00 0.00/0.00/0.00 0.00/0.00/0.00
    $\texttt{ell.L2.1}$ 0.01/0.01/0.07 0.01/0.03/0.09 0.05/0.07/0.14
    $\texttt{ell.Linf.1}$ 0.01/0.01/0.05 0.02/0.02/0.02 0.00/0.04/0.09
    $\texttt{pi.L1.1}$ 0.02/0.03/0.05 0.01/0.02/0.06 0.00/0.05/0.07
    $\texttt{pi.L2.1}$ 0.01/0.02/0.07 0.01/0.02/0.07 0.00/0.05/0.07
    $\texttt{pi.Linf.1}$ 0.01/0.03/0.03 0.00/0.00/0.02 0.01/0.05/0.05
     | Show Table
    DownLoad: CSV

    Table 6.  Rejection rates across all types of acyclic model, for each estimator and invariant using the multiplicative invariants $ \pi $ and $ \ell $. Rates for $ \alpha = 0.01/0.05/0.10 $ are separated with $ / $ in each table cell

    Invariant axis hull unbiased hull
    $\texttt{ell.L1.1}$ 0.00/0.00/0.00 0.00/0.00/0.00 0.00/0.00/0.00
    $\texttt{ell.L2.1}$ 0.04/0.09/0.13 0.04/0.07/0.12 0.04/0.07/0.11
    $\texttt{ell.Linf.1}$ 0.00/0.02/0.05 0.03/0.03/0.05 0.00/0.03/0.08
    $\texttt{pi.L1.1}$ 0.08/0.14/0.21 0.11/0.17/0.20 0.06/0.14/0.20
    $\texttt{pi.L2.1}$ 0.10/0.12/0.20 0.10/0.14/0.19 0.06/0.13/0.18
    $\texttt{pi.Linf.1}$ 0.02/0.04/0.06 0.03/0.04/0.10 0.03/0.07/0.12
     | Show Table
    DownLoad: CSV

    Table 7.  Rejection rates for the FWER procedure as applied to randomly selected collections of acyclic point clouds. We have excluded the cross-polytope acyclic models due to their surprisingly high rejection rates already in the single-hypothesis case. Rates for $ \alpha = 0.01/0.05/0.10 $ are separated with $ / $ in each table cell

    Invariant axis unbiased hull
    ell.L1.1 0.00/0.00/0.00 0.00/0.00/0.00
    ell.L2.1 0.00/0.05/0.12 0.03/0.12/0.17
    ell.Linf.1 0.05/0.09/0.15 0.08/0.12/0.16
    L1.0 0.02/0.04/0.04 0.00/0.05/0.12
    L1.1 0.03/0.05/0.07 0.00/0.02/0.03
    L2.0 0.02/0.04/0.06 0.02/0.05/0.08
    L2.1 0.01/0.02/0.07 0.01/0.01/0.02
    Linf.0 0.04/0.10/0.11 0.03/0.09/0.14
    Linf.1 0.01/0.02/0.04 0.01/0.03/0.06
    pi.L1.1 0.02/0.07/0.16 0.05/0.07/0.11
    pi.L2.1 0.02/0.07/0.14 0.05/0.06/0.16
    pi.Linf.1 0.01/0.04/0.09 0.01/0.06/0.12
     | Show Table
    DownLoad: CSV

    Table 8.  Rejection rates for the FWER procedure as applied to randomly selected collections of acyclic point clouds, with a single power estimation point cloud included. We have excluded the cross-polytope acyclic models due to their surprisingly high rejection rates already in the single-hypothesis case

    Model type Invariant axis unbiased hull
    concentric ell.L1.1 0.00/0.00/0.00 0.00/0.00/0.00
    ell.L2.1 0.05/0.15/0.21 0.09/0.14/0.22
    ell.Linf.1 0.36/0.43/0.50 0.31/0.41/0.47
    L1.0 0.02/0.04/0.06 0.03/0.05/0.08
    L1.1 0.23/0.24/0.29 0.17/0.29/0.31
    L2.0 0.02/0.09/0.10 0.02/0.04/0.07
    L2.1 0.44/0.49/0.51 0.44/0.48/0.52
    Linf.0 0.29/0.33/0.38 0.29/0.34/0.35
    Linf.1 0.47/0.52/0.57 0.44/0.47/0.54
    pi.L1.1 0.18/0.25/0.30 0.18/0.31/0.37
    pi.L2.1 0.30/0.34/0.41 0.27/0.37/0.42
    pi.Linf.1 0.42/0.48/0.50 0.40/0.46/0.51
    fig 8 ell.L1.1 0.00/0.00/0.00 0.00/0.00/0.00
    ell.L2.1 0.03/0.10/0.14 0.04/0.10/0.15
    ell.Linf.1 0.32/0.34/0.37 0.28/0.36/0.38
    L1.0 0.03/0.05/0.06 0.01/0.01/0.01
    L1.1 0.42/0.45/0.49 0.38/0.42/0.46
    L2.0 0.03/0.06/0.07 0.01/0.03/0.04
    L2.1 0.54/0.62/0.66 0.55/0.60/0.64
    Linf.0 0.04/0.16/0.23 0.14/0.22/0.27
    Linf.1 0.59/0.64/0.66 0.60/0.65/0.65
    pi.L1.1 0.36/0.44/0.53 0.38/0.47/0.56
    pi.L2.1 0.39/0.51/0.59 0.46/0.54/0.63
    pi.Linf.1 0.53/0.58/0.60 0.54/0.58/0.64
    sphere ell.L1.1 0.00/0.00/0.00 0.00/0.00/0.00
    ell.L2.1 0.06/0.14/0.18 0.04/0.16/0.27
    ell.Linf.1 0.19/0.29/0.35 0.22/0.31/0.34
    L1.0 0.00/0.01/0.02 0.06/0.09/0.11
    L1.1 0.16/0.19/0.24 0.13/0.15/0.20
    L2.0 0.01/0.04/0.05 0.03/0.05/0.12
    L2.1 0.27/0.32/0.36 0.26/0.29/0.30
    Linf.0 0.08/0.16/0.19 0.15/0.18/0.20
    Linf.1 0.28/0.32/0.35 0.27/0.34/0.36
    pi.L1.1 0.16/0.26/0.33 0.18/0.28/0.35
    pi.L2.1 0.28/0.34/0.40 0.25/0.36/0.42
    pi.Linf.1 0.30/0.36/0.42 0.29/0.40/0.46
    thomas ell.L1.1 0.00/0.00/0.00 0.00/0.00/0.00
    ell.L2.1 0.00/0.04/0.10 0.04/0.08/0.14
    ell.Linf.1 0.01/0.05/0.14 0.05/0.14/0.17
    L1.0 0.18/0.21/0.21 0.25/0.29/0.33
    L1.1 0.07/0.13/0.16 0.07/0.07/0.10
    L2.0 0.34/0.35/0.38 0.45/0.47/0.48
    L2.1 0.07/0.11/0.14 0.08/0.09/0.11
    Linf.0 0.40/0.47/0.51 0.47/0.51/0.57
    Linf.1 0.07/0.09/0.12 0.05/0.10/0.13
    pi.L1.1 0.04/0.08/0.17 0.03/0.11/0.17
    pi.L2.1 0.04/0.10/0.19 0.03/0.09/0.18
    pi.Linf.1 0.07/0.14/0.21 0.07/0.16/0.22
     | Show Table
    DownLoad: CSV

    Table 9.  Rejection rates for the FWER procedure as applied to randomly selected collections of acyclic point clouds, with a single planar power estimation point cloud of a specific noise level included. We have excluded the cross-polytope acyclic models due to their surprisingly high rejection rates already in the single-hypothesis case. Rates for $ \alpha = 0.01/0.05/0.10 $ are separated with $ / $ in each table cell

    Noise Invariant axis unbiased hull
    0.01 ell.L1.1 0.00/0.00/0.00 0.00/0.00/0.00
    ell.L2.1 0.07/0.17/0.19 0.06/0.14/0.19
    ell.Linf.1 0.43/0.51/0.53 0.41/0.44/0.51
    L1.0 0.00/0.02/0.04 0.03/0.06/0.10
    L1.1 0.47/0.54/0.55 0.41/0.50/0.54
    L2.0 0.01/0.02/0.03 0.03/0.04/0.08
    L2.1 0.76/0.78/0.79 0.73/0.75/0.78
    Linf.0 0.27/0.31/0.33 0.22/0.28/0.31
    Linf.1 0.73/0.78/0.79 0.70/0.74/0.74
    pi.L1.1 0.43/0.53/0.62 0.44/0.55/0.62
    pi.L2.1 0.63/0.67/0.74 0.66/0.69/0.74
    pi.Linf.1 0.59/0.65/0.71 0.64/0.68/0.69
    0.05 ell.L1.1 0.00/0.00/0.00 0.00/0.00/0.00
    ell.L2.1 0.05/0.11/0.15 0.05/0.11/0.16
    ell.Linf.1 0.32/0.42/0.49 0.33/0.45/0.48
    L1.0 0.00/0.04/0.05 0.03/0.09/0.09
    L1.1 0.20/0.31/0.34 0.17/0.27/0.32
    L2.0 0.01/0.05/0.06 0.02/0.05/0.08
    L2.1 0.47/0.59/0.63 0.53/0.60/0.65
    Linf.0 0.21/0.26/0.30 0.26/0.30/0.34
    Linf.1 0.50/0.60/0.64 0.52/0.57/0.62
    pi.L1.1 0.14/0.26/0.36 0.15/0.30/0.35
    pi.L2.1 0.26/0.44/0.50 0.26/0.43/0.48
    pi.Linf.1 0.43/0.49/0.61 0.40/0.50/0.60
    0.1 ell.L1.1 0.00/0.00/0.00 0.00/0.00/0.00
    ell.L2.1 0.06/0.08/0.18 0.06/0.11/0.20
    ell.Linf.1 0.24/0.32/0.35 0.23/0.31/0.40
    L1.0 0.01/0.04/0.04 0.05/0.10/0.11
    L1.1 0.18/0.20/0.28 0.16/0.22/0.27
    L2.0 0.02/0.05/0.05 0.03/0.05/0.08
    L2.1 0.32/0.35/0.37 0.27/0.37/0.41
    Linf.0 0.23/0.28/0.32 0.22/0.30/0.39
    Linf.1 0.35/0.40/0.40 0.35/0.43/0.47
    pi.L1.1 0.14/0.21/0.25 0.15/0.22/0.25
    pi.L2.1 0.21/0.27/0.30 0.18/0.27/0.34
    pi.Linf.1 0.35/0.45/0.49 0.36/0.45/0.51
     | Show Table
    DownLoad: CSV

    Table 10.  Single hypothesis rejection rates by acyclic model, estimator and composite invariant for each type of acyclic model. Rates for $ \alpha = 0.01/0.05/0.10 $ are separated with $ / $ in each table cell

    Model Invariant axis unbiased hull
    axis $\texttt{ell}$ 0.02/0.05/0.11 0.04/0.08/0.13
    $\texttt{ell.L1}$ 0.02/0.05/0.09 0.03/0.03/0.10
    $\texttt{ell.L2}$ 0.03/0.04/0.08 0.03/0.09/0.10
    $\texttt{ell.Linf}$ 0.01/0.04/0.06 0.01/0.07/0.07
    $\texttt{L}$ 0.08/0.14/0.21 0.03/0.04/0.16
    $\texttt{L1}$ 0.02/0.08/0.12 0.03/0.03/0.10
    $\texttt{L2}$ 0.05/0.11/0.18 0.01/0.05/0.12
    $\texttt{Linf}$ 0.06/0.13/0.19 0.01/0.06/0.14
    $\texttt{pi}$ 0.02/0.05/0.11 0.04/0.08/0.13
    $\texttt{pi.L1}$ 0.02/0.05/0.09 0.03/0.03/0.10
    $\texttt{pi.L2}$ 0.03/0.04/0.08 0.03/0.09/0.10
    $\texttt{pi.Linf}$ 0.01/0.04/0.06 0.01/0.07/0.07
    ball $\texttt{ell}$ 0.01/0.03/0.04 0.00/0.00/0.03
    $\texttt{ell.L1}$ 0.01/0.01/0.01 0.00/0.00/0.01
    $\texttt{ell.L2}$ 0.01/0.01/0.04 0.00/0.00/0.01
    $\texttt{ell.Linf}$ 0.00/0.03/0.05 0.02/0.02/0.03
    $\texttt{L}$ 0.01/0.03/0.03 0.00/0.01/0.03
    $\texttt{L1}$ 0.01/0.01/0.01 0.01/0.01/0.05
    $\texttt{L2}$ 0.01/0.01/0.03 0.01/0.01/0.02
    $\texttt{Linf}$ 0.00/0.02/0.03 0.00/0.02/0.03
    $\texttt{pi}$ 0.01/0.03/0.04 0.00/0.00/0.03
    $\texttt{pi.L1}$ 0.01/0.01/0.01 0.00/0.00/0.01
    $\texttt{pi.L2}$ 0.01/0.01/0.04 0.00/0.00/0.01
    $\texttt{pi.Linf}$ 0.00/0.03/0.05 0.02/0.02/0.03
    cross $\texttt{ell}$ 0.00/0.00/0.03 0.00/0.03/0.05
    $\texttt{ell.L1}$ 0.00/0.01/0.03 0.02/0.07/0.08
    $\texttt{ell.L2}$ 0.00/0.01/0.01 0.01/0.02/0.04
    $\texttt{ell.Linf}$ 0.00/0.00/0.01 0.00/0.01/0.04
    $\texttt{L}$ 0.26/0.32/0.36 0.12/0.17/0.25
    $\texttt{L1}$ 0.14/0.14/0.15 0.14/0.16/0.18
    $\texttt{L2}$ 0.15/0.16/0.16 0.10/0.15/0.19
    $\texttt{Linf}$ 0.25/0.30/0.35 0.02/0.08/0.16
    $\texttt{pi}$ 0.00/0.00/0.03 0.00/0.03/0.05
    $\texttt{pi.L1}$ 0.00/0.01/0.03 0.02/0.07/0.08
    $\texttt{pi.L2}$ 0.00/0.01/0.01 0.01/0.02/0.04
    $\texttt{pi.Linf}$ 0.00/0.00/0.01 0.00/0.01/0.04
    random.hull $\texttt{ell}$ 0.00/0.00/0.01 0.02/0.04/0.04
    $\texttt{ell.L1}$ 0.00/0.00/0.02 0.02/0.03/0.05
    $\texttt{ell.L2}$ 0.00/0.00/0.01 0.03/0.03/0.05
    $\texttt{ell.Linf}$ 0.00/0.01/0.02 0.01/0.02/0.03
    $\texttt{L}$ 0.00/0.00/0.00 0.01/0.03/0.04
    $\texttt{L1}$ 0.00/0.00/0.00 0.02/0.03/0.04
    $\texttt{L2}$ 0.00/0.00/0.00 0.02/0.03/0.05
    $\texttt{Linf}$ 0.00/0.00/0.01 0.00/0.02/0.03
    $\texttt{pi}$ 0.00/0.00/0.01 0.02/0.04/0.04
    $\texttt{pi.L1}$ 0.00/0.00/0.02 0.02/0.03/0.05
    $\texttt{pi.L2}$ 0.00/0.00/0.01 0.03/0.03/0.05
    $\texttt{pi.Linf}$ 0.00/0.01/0.02 0.01/0.02/0.03
    random.polytope $\texttt{ell}$ 0.00/0.01/0.03 0.00/0.03/0.07
    $\texttt{ell.L1}$ 0.00/0.04/0.06 0.01/0.05/0.08
    $\texttt{ell.L2}$ 0.00/0.02/0.05 0.01/0.04/0.11
    $\texttt{ell.Linf}$ 0.01/0.01/0.03 0.00/0.01/0.04
    $\texttt{L}$ 0.00/0.00/0.02 0.04/0.06/0.09
    $\texttt{L1}$ 0.00/0.02/0.05 0.01/0.05/0.08
    $\texttt{L2}$ 0.00/0.02/0.03 0.00/0.04/0.06
    $\texttt{Linf}$ 0.01/0.02/0.02 0.03/0.06/0.08
    $\texttt{pi}$ 0.00/0.01/0.03 0.00/0.03/0.07
    $\texttt{pi.L1}$ 0.00/0.04/0.06 0.01/0.05/0.08
    $\texttt{pi.L2}$ 0.00/0.02/0.05 0.01/0.04/0.11
    $\texttt{pi.Linf}$ 0.01/0.01/0.03 0.00/0.01/0.04
    simplex $\texttt{ell}$ 0.00/0.00/0.01 0.00/0.02/0.08
    $\texttt{ell.L1}$ 0.00/0.01/0.01 0.00/0.08/0.12
    $\texttt{ell.L2}$ 0.00/0.00/0.01 0.00/0.01/0.05
    $\texttt{ell.Linf}$ 0.00/0.00/0.00 0.00/0.03/0.03
    $\texttt{L}$ 0.00/0.01/0.02 0.05/0.08/0.15
    $\texttt{L1}$ 0.00/0.01/0.01 0.01/0.10/0.14
    $\texttt{L2}$ 0.00/0.00/0.00 0.03/0.05/0.07
    $\texttt{Linf}$ 0.00/0.01/0.01 0.02/0.05/0.07
    $\texttt{pi}$ 0.00/0.00/0.01 0.00/0.02/0.08
    $\texttt{pi.L1}$ 0.00/0.01/0.01 0.00/0.08/0.12
    $\texttt{pi.L2}$ 0.00/0.00/0.01 0.00/0.01/0.05
    $\texttt{pi.Linf}$ 0.00/0.00/0.00 0.00/0.03/0.03
     | Show Table
    DownLoad: CSV

    Table 11.  FWER corrected rejection rates by acyclic model, estimator and composite invariant for each type of acyclic model. Rates for $ \alpha = 0.01/0.05/0.10 $ are separated with $ / $ in each table cell

    Invariant axis unbiased hull
    ell 0.06/0.06/0.09 0.03/0.08/0.11
    ell.L1 0.05/0.06/0.07 0.03/0.08/0.10
    ell.L2 0.04/0.07/0.09 0.04/0.08/0.09
    ell.Linf 0.01/0.04/0.07 0.00/0.03/0.06
    L 0.05/0.10/0.12 0.05/0.08/0.10
    L1 0.05/0.06/0.06 0.04/0.07/0.12
    L2 0.04/0.07/0.09 0.04/0.08/0.10
    Linf 0.02/0.06/0.11 0.02/0.04/0.06
    pi 0.06/0.06/0.09 0.03/0.08/0.11
    pi.L1 0.05/0.06/0.07 0.03/0.08/0.10
    pi.L2 0.04/0.07/0.09 0.04/0.08/0.09
    pi.Linf 0.01/0.04/0.07 0.00/0.03/0.06
     | Show Table
    DownLoad: CSV

    Table 12.  Rejection rates for the FWER procedure with composite invariants as applied to randomly selected collections of acyclic point clouds, with a single power estimation point cloud included. We have excluded the cross-polytope acyclic models due to their surprisingly high rejection rates already in the single-hypothesis case. Rates for $ \alpha = 0.01/0.05/0.10 $ are separated with $ / $ in each table cell

    Model Invariant axis unbiased hull
    concentric ell 0.48/0.53/0.60 0.42/0.55/0.57
    ell.L1 0.18/0.29/0.33 0.13/0.22/0.26
    ell.L2 0.44/0.49/0.54 0.43/0.50/0.55
    ell.Linf 0.45/0.51/0.53 0.45/0.51/0.54
    L 0.48/0.55/0.58 0.47/0.61/0.61
    L1 0.17/0.28/0.32 0.13/0.22/0.26
    L2 0.44/0.50/0.54 0.42/0.49/0.55
    Linf 0.46/0.50/0.54 0.49/0.59/0.61
    pi 0.48/0.53/0.60 0.42/0.55/0.57
    pi.L1 0.18/0.29/0.33 0.13/0.22/0.26
    pi.L2 0.44/0.49/0.54 0.43/0.50/0.55
    pi.Linf 0.45/0.51/0.53 0.45/0.51/0.54
    fig8 ell 0.63/0.66/0.68 0.63/0.65/0.69
    ell.L1 0.36/0.42/0.44 0.35/0.41/0.45
    ell.L2 0.54/0.60/0.65 0.57/0.64/0.66
    ell.Linf 0.57/0.59/0.62 0.55/0.61/0.68
    L 0.60/0.65/0.67 0.62/0.66/0.68
    L1 0.36/0.41/0.43 0.34/0.40/0.45
    L2 0.53/0.59/0.64 0.57/0.62/0.65
    Linf 0.54/0.58/0.62 0.53/0.61/0.63
    pi 0.63/0.66/0.68 0.63/0.65/0.69
    pi.L1 0.36/0.42/0.44 0.35/0.41/0.45
    pi.L2 0.54/0.60/0.65 0.57/0.64/0.66
    pi.Linf 0.57/0.59/0.62 0.55/0.61/0.68
    sphere ell 0.26/0.32/0.35 0.26/0.28/0.31
    ell.L1 0.13/0.14/0.19 0.08/0.11/0.15
    ell.L2 0.21/0.24/0.30 0.23/0.25/0.29
    ell.Linf 0.25/0.32/0.36 0.21/0.25/0.37
    L 0.24/0.32/0.38 0.30/0.39/0.40
    L1 0.12/0.14/0.18 0.08/0.11/0.16
    L2 0.20/0.25/0.29 0.23/0.25/0.30
    Linf 0.23/0.31/0.40 0.27/0.35/0.39
    pi 0.26/0.32/0.35 0.26/0.28/0.31
    pi.L1 0.13/0.14/0.19 0.08/0.11/0.15
    pi.L2 0.21/0.24/0.30 0.23/0.25/0.29
    pi.Linf 0.25/0.32/0.36 0.21/0.25/0.37
    thomas ell 0.10/0.17/0.20 0.10/0.16/0.18
    ell.L1 0.04/0.08/0.16 0.04/0.08/0.11
    ell.L2 0.07/0.10/0.13 0.09/0.11/0.11
    ell.Linf 0.12/0.16/0.20 0.13/0.17/0.21
    L 0.56/0.63/0.67 0.60/0.64/0.66
    L1 0.22/0.24/0.30 0.21/0.27/0.32
    L2 0.34/0.40/0.42 0.36/0.38/0.42
    Linf 0.47/0.53/0.55 0.53/0.59/0.62
    pi 0.10/0.17/0.20 0.10/0.16/0.18
    pi.L1 0.04/0.08/0.16 0.04/0.08/0.11
    pi.L2 0.07/0.10/0.13 0.09/0.11/0.11
    pi.Linf 0.12/0.16/0.20 0.13/0.17/0.21
     | Show Table
    DownLoad: CSV
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