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Simultaneous Confidence Bands for Functional Data Using the Gaussian Kinematic Formula

Fabian J. E. Telschow, Armin Schwartzman

Abstract

This article constructs simultaneous confidence bands (SCBs) for functional parameters using the Gaussian Kinematic formula of $t$-processes (tGKF). Although the tGKF relies on Gaussianity, we show that a central limit theorem (CLT) for the parameter of interest is enough to obtain asymptotically precise covering rates even for non-Gaussian processes. As a proof of concept we study the functional signal-plus-noise model and derive a CLT for an estimator of the Lipschitz-Killing curvatures, the only data dependent quantities in the tGKF SCBs. Extensions to discrete sampling with additive observation noise are discussed using scale space ideas from regression analysis. Here we provide sufficient conditions on the processes and kernels to obtain convergence of the functional scale space surface. The theoretical work is accompanied by a simulation study comparing different methods to construct SCBs for the population mean. We show that the tGKF works well even for small sample sizes and only a Rademacher multiplier-$t$ bootstrap performs similarily well. For larger sample sizes the tGKF often outperforms the bootstrap methods and is computational faster. We apply the method to diffusion tensor imaging (DTI) fibers using a scale space approach for the difference of population means. R code is available in our Rpackage SCBfda.

Simultaneous Confidence Bands for Functional Data Using the Gaussian Kinematic Formula

Abstract

This article constructs simultaneous confidence bands (SCBs) for functional parameters using the Gaussian Kinematic formula of -processes (tGKF). Although the tGKF relies on Gaussianity, we show that a central limit theorem (CLT) for the parameter of interest is enough to obtain asymptotically precise covering rates even for non-Gaussian processes. As a proof of concept we study the functional signal-plus-noise model and derive a CLT for an estimator of the Lipschitz-Killing curvatures, the only data dependent quantities in the tGKF SCBs. Extensions to discrete sampling with additive observation noise are discussed using scale space ideas from regression analysis. Here we provide sufficient conditions on the processes and kernels to obtain convergence of the functional scale space surface. The theoretical work is accompanied by a simulation study comparing different methods to construct SCBs for the population mean. We show that the tGKF works well even for small sample sizes and only a Rademacher multiplier- bootstrap performs similarily well. For larger sample sizes the tGKF often outperforms the bootstrap methods and is computational faster. We apply the method to diffusion tensor imaging (DTI) fibers using a scale space approach for the difference of population means. R code is available in our Rpackage SCBfda.

Paper Structure

This paper contains 40 sections, 17 theorems, 95 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Organisation of the Article
  3. Simultaneous Confidence Bands
  4. SCBs for Functional Parameters
  5. Functional Signal-Plus-Noise Model
  6. Estimation of the Quantile
  7. Estimation of the Quantile Using the tGKF
  8. The Gaussian Kinematic Formula for t-processes
  9. The tGKF-estimator of $q_{\alpha,N}$
  10. Estimation of the Quantile Using the Bootstrap
  11. Non-parametric bootstrap-$t$
  12. Multiplier-$t$ Bootstrap
  13. Asymptotic Covering Rates
  14. Asymptotic SCBs for Functional Parameters
  15. Asymptotic SCBs for the Signal-Plus-Noise Model
  16. ...and 25 more sections

Key Result

Proposition 1

Any $(\mathcal{L} ^p,\delta)-$Lipschitz process over a compact set $\mathcal{S}$ has finite $p$-th $\mathcal{C}(\mathcal{S} )$-moment.

Figures (9)

  • Figure 1: Left: Samples of two Gaussian process with observation noise and different population mean function and SCBs from tGKF for the smoothed data. Right: Example of dependence of the covering rate of SCBs of the smoothed mean function on the sample size, constructed using various methods.
  • Figure 2: Simulation result for smooth Gaussian processes. Top row: samples from the signal-plus-noise models. Middle row: samples from the error processes. Bottom row: simulated covering rates. The solid black line is the targeted level of the SCBs and the dashed black line are twice the standard error for a Bernoulli random variable with $p=0.95$.
  • Figure 3: Simulation result for smooth non Gaussian processes (Model A). Left: samples from the error processes. Right: simulated covering rates. The solid black line is the targeted level of the SCBs and the dashed black line are twice the standard error for a Bernoulli random variable with $p=0.95$.
  • Figure 4: Simulation result for smooth non Gaussian processes (Model B). Left: samples from the error processes. Right: simulated covering rates. The solid black line is the targeted level of the SCBs and the dashed black line are twice the standard error for a Bernoulli random variable with $p=0.95$.
  • Figure 5: Simulation results for Gaussian processes (Model A) with observation noise. Top row: samples from the error processes. Bottom row: simulated covering rates. The solid black line is the targeted level of the SCBs and the dashed black line are twice the standard error for a Bernoulli random variable with $p=0.95$.
  • ...and 4 more figures

Theorems & Definitions (40)

  • Definition 1
  • Remark 1
  • Definition 2
  • Proposition 1
  • Remark 2
  • Remark 3
  • Theorem \oldthetheorem
  • Remark 4
  • Remark 5
  • Theorem \oldthetheorem
  • ...and 30 more