Distribution of the Height of Local Maxima of Gaussian Random Fields

Abstract

Let $\{f(t):t\in T\}$ be a smooth Gaussian random field over a parameter space $T$, where $T$ may be a subset of Euclidean space or, more generally, a Riemannian manifold. For any local maximum of $f(t)$ located at $t_0$ in the interior of $T$, we provide general formulae and asymptotic approximations for both the tail distribution of the height of a local maximum $\mathbb{P}\{f(t_0)>u|t_0\text{is a local maximum of}f(t)\}$ and the overshoot distribution of a local maximum $\mathbb{P}\{f(t_0)>u+v|t_0\text{is a local maximum of}f(t)\text{and}f(t_0)>v\}$. Assuming further that $f$ is isotropic, we apply techniques from random matrix theory related to the Gaussian orthogonal ensemble to compute such conditional probabilities explicitly when $T$ is Euclidean or a sphere of arbitrary dimension. Such calculations are motivated by the statistical problem of detecting peaks in the presence of smooth Gaussian noise.

40 pages, 2 figures

References in corpus (3)

• Strictly and non-strictly positive definite functions on spheres
• Multiple testing of local maxima for detection of peaks in 1D
• Excursion probability of Gaussian random fields on sphere

Cited by in corpus (3)

• Excursion probability of Gaussian random fields on sphere
• Multiple Testing of Local Extrema for Detection of Change Points
• A Note on Global Suprema of Band-Limited Spherical Random Functions