An approximation to peak detection power using Gaussian random field theory

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.