Abstract: Modeling high-dimensional functional responses utilizing multi-dimensional functional covariates is complicated by spatial and/or temporal dependence in the observations in addition to high-dimensional predictors. To utilize such rich sources of information we develop multi-dimensional spatial functional models that employ low-rank basis function expansions to facilitate model implementation. These models are developed within a hierarchical Bayesian framework that accounts for several sources of uncertainty, including the error that arises from truncating the infinite-dimensional basis function expansions, error in the observations, and uncertainty in the parameters. We illustrate the predictive ability of such a model through a simulation study and an application that considers spatial models of soil electrical conductivity depth profiles using spatially dependent near-infrared spectral images of electrical conductivity covariates.

Key words and phrases: Basis functions, diffuse reflectance spectroscopy, Karhunen-Loève, matrix normal, penetrometer, principal components, soil electrical conductivity.