  Geostatistical modeling for continuous point-referenced data has been
extensively applied to neuroimaging because it produces efficient and valid
statistical inference. However, diffusion tensor imaging (DTI), a neuroimaging
characterizing the brain structure produces a positive definite (p.d.) matrix
for each voxel. Current geostatistical modeling has not been extended to p.d.
matrices because introducing spatial dependence among positive definite
matrices properly is challenging. In this paper, we use the spatial Wishart
process, a spatial stochastic process (random field) where each p.d.
matrix-variate marginally follows a Wishart distribution, and spatial
dependence between random matrices is induced by latent Gaussian processes.
This process is valid on an uncountable collection of spatial locations and is
almost surely continuous, leading to a reasonable means of modeling spatial
dependence. Motivated by a DTI dataset of cocaine users, we propose a spatial
matrix-variate regression model based on the spatial Wishart process. A
problematic issue is that the spatial Wishart process has no closed-form
density function. Hence, we propose approximation methods to obtain a feasible
working model. A local likelihood approximation method is also applied to
achieve fast computation. The simulation studies and real data analysis
demonstrate that the working model produces reliable inference and improved
performance compared to other methods.
