Prediction of Spatial Point Processes: Regularized Method with Out-of-Sample Guarantees

Spatial point processes can be found in a range of applications from astronomy and biology to ecology and criminology. These processes can be characterized by a nonnegative intensity function λpxq which predicts the number of events that occur across space parameterized byxPX [8, 4]. A standard approach to estimate the intensity function of a process is to use nonparametric kernel density-based methods [6, 7]. These smoothing techniques require, however, careful tuning of kernel bandwidth parameters and are, more importantly, subject to selection biases. That is, in regions where no events have been observed, the intensity is inferred to be zero and no measure is readily available for a user to assess the uncertainty of such predictions. More advanced methods infer the intensity by assuming a parameterized model of the data-generating process, such as inhomogeneous Poisson point process models.