Statistical learning and cross-validation for point processes

• 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.

Subject headings

NATURVETENSKAP  -- Data- och informationsvetenskap (hsv//swe)

NATURAL SCIENCES  -- Computer and Information Sciences (hsv//eng)

NATURVETENSKAP  -- Matematik -- Sannolikhetsteori och statistik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Probability Theory and Statistics (hsv//eng)

Keyword

Bivariate innovation

Cross-Validation

Generalised random samples

Kernel intensity estimation

Loss function

Monte-Carlo cross-validation

Multinomial k-fold cross-validation

Papangelou conditional intensity function

Prediction

Subsampling

Test function

Thinning

Publication and Content Type

vet (subject category)

rap (subject category)