Spatial random field models based on Lévy indicator convolutions

Abstract

Process convolutions yield random fields with flexible marginal distributions and dependence beyond Gaussianity, but statistical inference is often hampered by a lack of closed-form marginal distributions, and simulation-based inference may be prohibitively computer-intensive. We here remedy such issues through a class of process convolutions based on smoothing a (d+1)-dimensional Lévy basis with an indicator function kernel to construct a d-dimensional convolution process. Indicator kernels ensure univariate distributions in the Lévy basis family, which provides a sound basis for interpretation, parametric modeling and statistical estimation. We propose a class of isotropic stationary convolution processes constructed through hypograph indicator sets defined as the space between the curve (s,H(s)) of a spherical probability density function H and the plane (s,0). If H is radially decreasing, the covariance is expressed through the univariate distribution function of H. The bivariate joint tail behavior in such convolution processes is studied in detail. Simulation and modeling extensions beyond isotropic stationary spatial models are discussed, including latent process models. For statistical inference of parametric models, we develop pairwise likelihood techniques and illustrate these on spatially indexed weed counts in the Bjertop data set, and on daily wind speed maxima observed over 30 stations in the Netherlands.