  Spatial data are often derived from multiple sources (e.g. satellites,
in-situ sensors, survey samples) with different supports, but associated with
the same properties of a spatial phenomenon of interest. It is common for
predictors to also be measured on different spatial supports than the response
variables. Although there is no standard way to work with spatial data with
different supports, a prevalent approach used by practitioners has been to use
downscaling or interpolation to project all the variables of analysis towards a
common support, and then using standard spatial models. The main disadvantage
with this approach is that simple interpolation can introduce biases and, more
importantly, the uncertainty associated with the change of support is not taken
into account in parameter estimation. In this article, we propose a Bayesian
spatial latent Gaussian model that can handle data with different rectilinear
supports in both the response variable and predictors. Our approach allows to
handle changes of support more naturally according to the properties of the
spatial stochastic process being used, and to take into account the uncertainty
from the change of support in parameter estimation and prediction. We use
spatial stochastic processes as linear combinations of basis functions where
Gaussian Markov random fields define the weights. Our hierarchical modelling
approach can be described by the following steps: (i) define a latent model
where response variables and predictors are considered as latent stochastic
processes with continuous support, (ii) link the continuous-index set
stochastic processes with its projection to the support of the observed data,
(iii) link the projected process with the observed data. We show the
applicability of our approach by simulation studies and modelling land
suitability for improved grassland in Rhondda Cynon Taf, a county borough in
Wales.
