Inference for asymptotically Gaussian random fields

In this thesis we consider two outstanding problems in the statistical analysis of random field data. The first relates to inference for the mean function, the second to inference for the covariance function.We propose a modification to the Gaussian random field (GRF) theory for computing the p-value of a supra-threshold test when there are departures from normality in the data. Specifically, we derive a saddlepoint approximation to the expected Euler characteristic (EC) of the excursion set at a level u > 0 of a normalized sum of n i.i.d. stationary random fields. We show that as u and n grow simultaneously, the 'tilted' approximation is asymptotically more accurate than a simple Gaussian assumption. We illustrate the improvement using data from astrophysics, cognitive psychology ('bubbles') and brain mapping. In the 'bubbles' setting we perform a simulation to ascertain the accuracy of the tilted expected EC as a p-value approximation. We also present a geometric interpretation for the formula, which suggests an asymptotic generalization of Jonathan Taylor's Gaussian Kinematic Formula (GKF) [86].We then treat the problem of inference for the principal component (PC) or eigenimages of the covariance function of GRF data. Approximate p-values for thresholding tests of PC random fields have proved elusive due to a lack of useful results about the (finite-dimensional) distribution(s) of such processes. We take a Bayesian approach, discretize the problem, and formulate a parsimonious hierarchical model for the covariance structure of the process. The model is based on that of Smith & Kohn [75] but includes several modifications for improved stability, computational efficiency and sampling speed. It also involves a sparsity parameter modeled a priori as a Markov random field. We devise a simulation study to calibrate the model with generic GRF data and also test it using real data from neuroimaging and remote sensing applications