Regression-based covariance functions for nonstationary spatial modeling

In many environmental applications involving spatially-referenced data, limitations on the number and locations of observations motivate the need for practical and efficient models for spatial interpolation, or kriging. A key component of models for continuously-indexed spatial data is the covariance function, which is traditionally assumed to belong to a parametric class of stationary models. However, stationarity is rarely a realistic assumption. Alternative methods which more appropriately model the nonstationarity present in environmental processes often involve high-dimensional parameter spaces, which lead to difficulties in model fitting and interpretability. To overcome this issue, we build on the growing literature of covariate-driven nonstationary spatial modeling. Using process convolution techniques, we propose a Bayesian model for continuously-indexed spatial data based on a flexible parametric covariance regression structure for a convolution-kernel covariance matrix. The resulting model is a parsimonious representation of the kernel process, and we explore properties of the implied model, including a description of the resulting nonstationary covariance function and the interpretational benefits in the kernel parameters. Furthermore, we demonstrate that our model provides a practical compromise between stationary and highly parameterized nonstationary spatial covariance functions that do not perform well in practice. We illustrate our approach through an analysis of annual precipitation data.