Spatial Process Gradients and Their Use in Sensitivity Analysis for Environmental Processes

Abstract

This paper develops methodology for local sensitivity analysis based on directional derivatives associated with spatial processes. Formal gradient analysis for spatial processes was elaborated in previous papers, focusing on distribution theory for directional derivatives associated with a response variable assumed to follow a Gaussian process model. In the current work, these ideas are extended to additionally accommodate a continuous covariate whose directional derivatives are also of interest and to relate the behavior of the directional derivatives of the response surface to those of the covariate surface. It is of interest to assess whether, in some sense, the gradients of the response follow those of the explanatory variable. The joint Gaussian structure of all variables, including the directional derivatives, allows for explicit distribution theory and, hence, kriging across the spatial region using multivariate normal theory. Working within a Bayesian hierarchical modeling framework, posterior samples enable all gradient analysis to occur post model fitting. As a proof of concept, we show how our methodology can be applied to a standard geostatistical modeling setting using a simulation example. For a real data illustration, we work with point pattern data, deferring our gradient analysis to the intensity surface, adopting a log-Gaussian Cox process model. In particular, we relate elevation data to point patterns associated with several tree species in Duke Forest.