Smooth Mat\'ern Gaussian Random Fields: Euler Characteristic, Expected Number and Height Distribution of Critical Points

TL;DR

This paper derives explicit formulas for the Euler characteristic, critical points, and height distribution of smooth Matérn Gaussian random fields on Euclidean space and spheres, aiding in statistical inference and error control.

Contribution

It provides new explicit formulas for topological and critical point characteristics of smooth Gaussian fields with Matérn covariance, applicable to Euclidean and spherical domains.

Findings

Explicit formulas for Euler characteristic of excursion sets

Expected number and height distribution of critical points derived

Results facilitate approximation of excursion probabilities and p-value computations

Abstract

This paper studies Gaussian random fields with Mat\'ern covariance functions with smooth parameter $\nu>2$. 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.