Detecting Sparse Signals in Random Fields, With an Application to Brain Mapping

Abstract

Brain mapping data have been modeled as Gaussian random fields, and local increases in mean are detected by local maxima of a random field of test statistics derived from these data. Accurate p values for local maxima are readily available for isotropic data based on the expected Euler characteristic of the excursion set of the test statistic random field. In this article we give a simple method for dealing with nonisotropic data. Our approach has connections to the model of Sampson and Guttorp for nonisotropy in which there exists an unknown mapping of the support of the data to a space in which the random fields are isotropic. Heuristic justification for our approach comes from the Nash embedding theorem. Formally, we show that our method gives consistent unbiased estimators for the true p values based on new results of Taylor and Adler for random fields on manifolds, which replace the Euclidean metric by the variogram. The results are used to detect gender differences in the cortical thickness of the brain and to detect regions of the brain involved in sentence comprehension measured by functional magnetic resonance imaging.