Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Smooth Matérn Gaussian Random Fields: Euler Characteristic, Expected Number and Height Distribution of Critical Points

Dan Cheng

TL;DR

For smooth Gaussian random fields with Mat\'ern covariance function with smooth parameter $\nu>2, the explicit formulae for the expected Euler characteristic of the excursion set, the expected number and height distribution of critical points are derived.

Abstract

This paper studies Gaussian random fields with Matérn covariance functions with smooth parameter $ν>2$. Two cases of parameter spaces, the Euclidean space and $N$-dimensional sphere, are considered. For such smooth Gaussian fields, we have derived the explicit formulae for the expected Euler characteristic of the excursion set, the expected number and height distribution of critical points. The results are valuable for approximating the excursion probability in family-wise error control and for computing p-values in peak inference.

Smooth Matérn Gaussian Random Fields: Euler Characteristic, Expected Number and Height Distribution of Critical Points

TL;DR

For smooth Gaussian random fields with Mat\'ern covariance function with smooth parameter $\nu>2, the explicit formulae for the expected Euler characteristic of the excursion set, the expected number and height distribution of critical points are derived.

Abstract

This paper studies Gaussian random fields with Matérn covariance functions with smooth parameter . Two cases of parameter spaces, the Euclidean space and -dimensional sphere, are considered. For such smooth Gaussian fields, we have derived the explicit formulae for the expected Euler characteristic of the excursion set, the expected number and height distribution of critical points. The results are valuable for approximating the excursion probability in family-wise error control and for computing p-values in peak inference.
Paper Structure (5 sections, 7 theorems, 39 equations)

This paper contains 5 sections, 7 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Smooth Matérn Gaussian Random Fields on Euclidean Space
  3. The expected Euler characteristic of excursion set
  4. Expected number and height distribution of critical points
  5. Smooth Matérn Gaussian Random Fields on Spheres

Key Result

Proposition 2.1

Let $\rho(d) = \mathcal{M}(\sqrt{d})$, and let $\rho'=\rho'(0)$ and $\rho"=\rho"(0)$. Suppose $\nu>2$. Then In particular,

Theorems & Definitions (14)

  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.5
  • proof
  • Proposition 3.1
  • proof
  • ...and 4 more