Statistical learning and cross-validation for point processes

TL;DR

This paper introduces a comprehensive statistical learning framework for point processes, utilizing novel concepts like bivariate innovations and point process cross-validation, and demonstrates improved accuracy in intensity estimation tasks.

Contribution

It develops a general supervised learning approach for point processes, combining new discrepancy measures and cross-validation methods, with theoretical analysis and practical improvements.

Findings

Outperforms existing methods in mean squared error for intensity estimation

Provides theoretical properties of bivariate innovations

Applies to parametric, non-parametric, and conditional intensity fitting

Abstract

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.