Multiple testing of local maxima for detection of peaks in random fields

A topological multiple testing scheme is presented for detecting peaks in images under stationary ergodic Gaussian noise, where tests are performed at local maxima of the smoothed observed signals. The procedure generalizes the one-dimensional scheme of Schwartzman, Gavrilov and Adler [Ann. Statist. 39 (2011) 3290-3319] to Euclidean domains of arbitrary dimension. Two methods are developed according to two different ways of computing p-values: (i) using the exact distribution of the height of local maxima, available explicitly when the noise field is isotropic [Extremes 18 (2015) 213- 240; Expected number and height distribution of critical points of smooth isotropic Gaussian random fields (2015) Preprint]; (ii) using an approximation to the overshoot distribution of local maxima above a pre-Threshold, applicable when the exact distribution is unknown, such as when the stationary noise field is nonisotropic [Extremes 18 (2015) 213-240]. The algorithms, combined with the Benjamini-Hochberg procedure for thresholding p-values, provide asymptotic strong control of the False Discovery Rate (FDR) and power consistency, with specific rates, as the search space and signal strength get large. The optimal smoothing bandwidth and optimal pre-Threshold are obtained to achieve maximum power. Simulations show that FDR levels are maintained in nonasymptotic conditions. The methods are illustrated in the analysis of functional magnetic resonance images of the brain.