  In modern spatial statistics, the structure of data that is collected has
become more heterogeneous. Depending on the type of spatial data, different
modeling strategies for spatial data are used. For example, a kriging approach
for geostatistical data; a Gaussian Markov random field model for lattice data;
or a log Gaussian Cox process for point-pattern data. Despite these different
modeling choices, the nature of underlying scientific data-generating (latent)
processes is often the same, which can be represented by some continuous
spatial surfaces. In this paper, we introduce a unifying framework for
process-based multivariate spatial fusion models. The framework can jointly
analyze all three aforementioned types of spatial data (or any combinations
thereof). Moreover, the framework accommodates different conditional
distributions for geostatistical and lattice data. We show that some
established approaches, such as linear models of coregionalization, can be
viewed as special cases of our proposed framework. We offer flexible and
scalable implementations in R using Stan and INLA. Simulation studies confirm
that the predictive performance of latent processes improves as we move from
univariate spatial models to multivariate spatial fusion models. The introduced
framework is illustrated using a cross-sectional study linked with a national
cohort dataset in Switzerland, we examine differences in underlying spatial
risk patterns between respiratory disease and lung cancer.
