  This paper presents the first general (supervised) statistical learning
framework for point processes in general spaces. Our approach is based on the
combination of two new concepts, which we define in the paper: i) bivariate
innovations, which are measures of discrepancy/prediction-accuracy between two
point processes, and ii) point process cross-validation (CV), which we here
define through point process thinning. The general idea is to carry out the
fitting by predicting CV-generated validation sets using the corresponding
training sets; the prediction error, which we minimise, is measured by means of
bivariate innovations. Having established various theoretical properties of our
bivariate innovations, we study in detail the case where the CV procedure is
obtained through independent thinning and we apply our statistical learning
methodology to three typical spatial statistical settings, namely parametric
intensity estimation, non-parametric intensity estimation and Papangelou
conditional intensity fitting. Aside from deriving theoretical properties
related to these cases, in each of them we numerically show that our
statistical learning approach outperforms the state of the art in terms of mean
(integrated) squared error.
