Identifying regions of inhomogeneities in spatial processes via an M-RA and mixture priors

Soils have been heralded as a hidden resource that can be leveraged to mitigate and address some of the major global environmental challenges. Specifically, the organic carbon stored in soils, called Soil Organic Carbon (SOC), can, through proper soil management, help offset fuel emissions, increase food productivity, and improve water quality. As collecting data on SOC is costly and time consuming, not much data on SOC is available, although understanding the spatial variability in SOC is of fundamental importance for effective soil management. In this manuscript, we propose a modeling framework that can be used to gain a better understanding of the dependence structure of a spatial process by identifying regions within a spatial domain where the process displays the same spatial correlation range. To achieve this goal, we propose a generalization of the Multi-Resolution Approximation (M-RA) modeling framework of Katzfuss (2017) originally introduced as a strategy to reduce the computational burden encountered when analyzing massive spatial datasets. To allow for the possibility that the correlation of a spatial process might be characterized by a different range in different subregions of a spatial domain, we provide the M-RA basis functions weights with a two-component mixture prior with one of the mixture components a shrinking prior. We call our approach the mixture M-RA. Application of the mixture M-RA model to both stationary and non-stationary data shows that the mixture M-RA model can handle both types of data, can correctly establish the type of spatial dependence structure in the data (e.g. stationary vs not), and can identify regions of local stationarity.