  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.
