Expected Number and Height Distribution of Critical Points of Smooth Isotropic Gaussian Random Fields

On the number of excursion sets of planar Gaussian fields

The Nazarov–Sodin constant describes the average number of nodal set components of smooth Gaussian fields on large scales. We generalise this to a functional describing the corresponding number of level set components for arbitrary levels. Using results from Morse theory, we express this functional as an integral over the level densities of different types of critical points, and as a result deduce the absolute continuity of the functional as the level varies. We further give upper and lower bounds showing that the functional is at least bimodal for certain isotropic fields, including the important special case of the random plane wave.

Peak p-values and false discovery rate inference in neuroimaging

Peaks are a mainstay of neuroimage analysis for reporting localization results. The current peak detection procedure in SPM12 requires a pre-threshold for approximating p-values and a false discovery rate (FDR) nominal level for inference. However, the pre-threshold is an undesirable feature, while the FDR level is meaningless if the null hypothesis is not properly defined. This article provides: 1) a peak height distribution for smooth Gaussian error fields, which does not require a screening pre-threshold; 2) a signal-plus-noise model where FDR of peaks can be controlled and properly interpreted. Matlab code for calculation of p-values using the exact peak height distribution is available as an SPM extension.

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 [31] 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 [9, 10]; (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 non-isotropic [9]. 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 non-asymptotic conditions. The methods are illustrated in the analysis of functional magnetic resonance images of the brain.