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Estimation of Expected Euler Characteristic Curves of Nonstationary Smooth Gaussian Random Fields

Fabian Telschow, Armin Schwartzman, Dan Cheng, Pratyush Pranav

Abstract

The expected Euler characteristic (EEC) curve of excursion sets of a Gaussian random field is used to approximate the distribution of its supremum for high thresholds. Viewed as a function of the excursion threshold, the EEC is expressed by the Gaussian kinematic formula (GKF) as a linear function of the Lipschitz-Killing curvatures (LKCs) of the field, which solely depend on the domain and covariance function of the field. So far its use for non-stationary Gaussian fields over non-trivial domains has been limited because in this case the LKCs are difficult to estimate. In this paper, consistent estimators of the LKCs are proposed as linear projections of "pinned" observed Euler characteristic curves and a linear parametric estimator of the EEC curve is obtained, which is more efficient than its nonparametric counterpart for repeated observations. A multiplier bootstrap modification reduces the variance of the estimator, and allows estimation of LKCs and EEC of the limiting field of non-Gaussian fields satisfying a functional CLT. The proposed methods are evaluated using simulations of 2D fields and illustrated in thresholding of 3D fMRI brain activation maps and cosmological simulations on the 2-sphere.

Estimation of Expected Euler Characteristic Curves of Nonstationary Smooth Gaussian Random Fields

Abstract

The expected Euler characteristic (EEC) curve of excursion sets of a Gaussian random field is used to approximate the distribution of its supremum for high thresholds. Viewed as a function of the excursion threshold, the EEC is expressed by the Gaussian kinematic formula (GKF) as a linear function of the Lipschitz-Killing curvatures (LKCs) of the field, which solely depend on the domain and covariance function of the field. So far its use for non-stationary Gaussian fields over non-trivial domains has been limited because in this case the LKCs are difficult to estimate. In this paper, consistent estimators of the LKCs are proposed as linear projections of "pinned" observed Euler characteristic curves and a linear parametric estimator of the EEC curve is obtained, which is more efficient than its nonparametric counterpart for repeated observations. A multiplier bootstrap modification reduces the variance of the estimator, and allows estimation of LKCs and EEC of the limiting field of non-Gaussian fields satisfying a functional CLT. The proposed methods are evaluated using simulations of 2D fields and illustrated in thresholding of 3D fMRI brain activation maps and cosmological simulations on the 2-sphere.

Paper Structure

This paper contains 37 sections, 10 theorems, 65 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The expected Euler characteristic curve
  3. Previous work on estimation of LKCs
  4. Our proposed Hermite projection estimator
  5. Simulations and data applications
  6. Estimation of the LKCs for Gaussian fields
  7. LKCs as projections of the EC curve
  8. Estimation of the LKCs from a single observation
  9. Properties of the Hermite projection estimator
  10. Estimation of the LKCs from repeated observations
  11. LKC estimation from non-Gaussian fields
  12. Bootstrapped Hermite projection estimator
  13. Properties of the bootstrapped Hermite projection estimator
  14. Estimation of the EEC and Applications
  15. Plug-in estimation of the EEC
  16. ...and 22 more sections

Key Result

Theorem 1

Assume $f$ is Gaussian and satisfies (G1)-(G4). Since the paths of $f$ are almost surely Morse functions, we only have finitely many critical values, which we order and denote with $u_0<...<u_M$. Thus, almost surely with random $a_m=\chi_f(u_{m})\in\mathbb{Z}$ and hence

Figures (11)

  • Figure 1: (left) Single realization of the isotropic Gaussian random field \ref{['eq:isotropic-2D']}. (middle) EC curves (gray) for $N=10$ realizations of the field and their average (blue). (right) Corresponding smoothed EC curves (gray) and their pointwise average (blue). Dashed blue lines are pointwise 95% confidence bands for the true EEC curve (red).
  • Figure 2: Isotropic Gaussian field ($\nu = 5$ and $L=50$): comparison of mean and standard deviation of different LKC estimators. Black lines represent the theoretical LKCs.
  • Figure 3: Isotropic non-Gaussian field ($\nu = 5$ and $L=50$): comparison of mean and standard deviation of different LKC estimators. Black lines represent the theoretical LKCs.
  • Figure 4: Isotropic Gaussian field: Covariance functions of the sample average EC curve (left) and the HPE of the ${\rm EEC}$ (middle) from $1000$ samples. Right panel shows their standard deviation functions in red and blue, respectively.
  • Figure 5: Isotropic Gaussian field: pointwise coverage of the EEC curve under respectively "true" variance of the EC curves (top row) and estimated variance (bottom row) for different sample sizes. The dashed lines represent the target confidence level 95%.
  • ...and 6 more figures

Theorems & Definitions (17)

  • Remark
  • Theorem 1
  • Remark
  • Theorem 2
  • Corollary 1
  • Corollary 2
  • Definition 1: Gaussian Multiplier Field
  • Theorem 3
  • Theorem 4
  • Remark
  • ...and 7 more