An approximation to peak detection power using Gaussian random field theory

TL;DR

This paper develops and validates formulas for approximating peak detection power in Gaussian random fields, extending existing methods to more general mean functions and demonstrating their effectiveness on simulated and real fMRI data.

Contribution

It extends explicit formulas for expected local maxima to rotationally symmetric mean functions and validates their accuracy in practical scenarios.

Findings

Formulas accurately approximate peak detection power in various asymptotic regimes.

Adjusted formulas improve accuracy when the expected number of maxima exceeds 1.

Application to fMRI data demonstrates practical utility in neuroimaging analysis.

Abstract

We study power approximation formulas for peak detection using Gaussian random field theory. The approximation, based on the expected number of local maxima above the threshold $u$, $\mathbb{E}[M_u]$, is proved to work well under three asymptotic scenarios: small domain, large threshold, and sharp signal. An adjusted version of $\mathbb{E}[M_u]$ is also proposed to improve accuracy when the expected number of local maxima $\mathbb{E}[M_{-\infty}]$ exceeds 1. Cheng and Schwartzman (2018) developed explicit formulas for $\mathbb{E}[M_u]$ of smooth isotropic Gaussian random fields with zero mean. In this paper, these formulas are extended to allow for rotational symmetric mean functions, so that they are suitable for power calculations. We also apply our formulas to 2D and 3D simulated datasets, and the 3D data is induced by a group analysis of fMRI data from the Human Connectome Project to measure performance in a realistic setting.