  Gaussian random field (GRF) models are widely used in spatial statistics to
capture spatially correlated error. We investigate the results of replacing
Gaussian processes with Laplace moving averages (LMAs) in spatial generalized
linear mixed models (SGLMMs). We demonstrate that LMAs offer improved
predictive power when the data exhibits localized spikes in the response.
SGLMMs with LMAs are shown to maintain analogous parameter inference and
similar computing to Gaussian SGLMMs. We propose a novel discrete space LMA
model for irregular lattices and construct conjugate samplers for LMAs with
georeferenced and areal support. We provide a Bayesian analysis of SGLMMs with
LMAs and GRFs over multiple data support and response types.
