Modeling Complex Spatial Dependencies: Low-Rank Spatially Varying Cross-Covariances With Application to Soil Nutrient Data

Advances in geo-spatial technologies have created data-rich environments which provide extraordinary opportunities to understand the complexity of large and spatially indexed data in ecology and the natural sciences. Our current application concerns analysis of soil nutrients data collected at La Selva Biological Station, Costa Rica, where inferential interest lies in capturing the spatially varying relationships among the nutrients. The objective here is to interpolate not just the nutrients across space, but also associations among the nutrients that are posited to vary spatially. This requires spatially varying cross-covariance models. Fully process-based specifications using matrix-variate processes are theoretically attractive but computationally prohibitive. Here we develop fully process-based low-rank but non-degenerate spatially varying cross-covariance processes that can effectively yield interpolate cross-covariances at arbitrary locations. We show how a particular low-rank process, the predictive process, which has been widely used to model large geostatistical datasets, can be effectively deployed to model non-degenerate cross-covariance processes. We produce substantive inferential tools such as maps of nonstationary cross-covariances that constitute the premise of further mechanistic modeling and have hitherto not been easily available for environmental scientists and ecologists.