  We aim at analyzing geostatistical and areal data observed over irregularly
shaped spatial domains and having a distribution within the exponential family.
We propose a generalized additive model that allows to account for
spatially-varying covariate information. The model is fitted by maximizing a
penalized log-likelihood function, with a roughness penalty term that involves
a differential quantity of the spatial field, computed over the domain of
interest. Efficient estimation of the spatial field is achieved resorting to
the finite element method, which provides a basis for piecewise polynomial
surfaces. The proposed model is illustrated by an application to the study of
criminality in the city of Portland, Oregon, USA.
