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Two-sample tests for relevant differences in persistence diagrams

Johannes Krebs, Daniel Rademacher

Abstract

We study two-sample tests for relevant differences in persistence diagrams obtained from $L^p$-$m$-approximable data $(\mathcal{X}_t)_t$ and $(\mathcal{Y}_t)_t$. To this end, we compare variance estimates w.r.t.\ the Wasserstein metrics on the space of persistence diagrams. In detail, we consider two test procedures. The first compares the Fr{é}chet variances of the two samples based on estimators for the Fr{é}chet mean of the observed persistence diagrams $PD(\mathcal{X}_i)$ ($1\le i\le m$), resp., $PD(\mathcal{Y}_j)$ ($1\le j\le n$) of a given feature dimension. We use classical functional central limit theorems to establish consistency of the testing procedure. The second procedure relies on a comparison of the so-called independent copy variances of the respective samples. Technically, this leads to functional central limit theorems for U-statistics built on $L^p$-$m$-approximable sample data.

Two-sample tests for relevant differences in persistence diagrams

Abstract

We study two-sample tests for relevant differences in persistence diagrams obtained from --approximable data and . To this end, we compare variance estimates w.r.t.\ the Wasserstein metrics on the space of persistence diagrams. In detail, we consider two test procedures. The first compares the Fr{é}chet variances of the two samples based on estimators for the Fr{é}chet mean of the observed persistence diagrams (), resp., () of a given feature dimension. We use classical functional central limit theorems to establish consistency of the testing procedure. The second procedure relies on a comparison of the so-called independent copy variances of the respective samples. Technically, this leads to functional central limit theorems for U-statistics built on --approximable sample data.
Paper Structure (7 sections, 13 theorems, 119 equations)

This paper contains 7 sections, 13 theorems, 119 equations.

Table of Contents

  1. Tests for persistence diagrams
  2. A test statistic for relevant differences of Fréchet variances
  3. A test statistic for relevant differences of inco-variances
  4. Mathematical background: a two-parameter FCLT for U-statistics
  5. Mathematical details
  6. Two sample test for Fréchet variances
  7. Two sample tests for inco-variances

Key Result

Theorem 1.2

Let the regularity conditions of Assumption A:FrechetVariances (r) be satisfied for some $r>4$ and suppose there is a constant $\tau \in (0,1)$ such that $\lim_{m,n \to \infty}m/(m+n) = \tau$. Moreover, let $X^*_t = W_r^2(\operatorname{PD}(\mathcal{X}_t), \widehat{\mu}_\mathcal{X})$, resp., $Y^*_t = where $B$ is a standard Brownian motion, $\xi = 2\sqrt{\frac{\Gamma_\mathcal{X}}{\tau} + \frac{\Gam

Theorems & Definitions (20)

  • Theorem 1.2
  • Theorem 1.3
  • proof : Proof of Theorem \ref{['T:FrechetVarTest']}
  • Theorem 1.4
  • Theorem 1.5
  • Remark 2.2
  • Theorem 2.3: FCLT for $U_n(h)$
  • proof : Proof of Theorem \ref{['Thrm:FCLTforUStatistic']}
  • Theorem 2.4: Two-parameter Donsker type FCLT
  • Theorem 2.5: Moment condition for $U_n(h_2)$
  • ...and 10 more