Statistical Properties of Point Process Learning for Gibbs Processes

Licentiate thesis, 2024

This thesis studies Point Process Learning (PPL), which is a novel statistical learning framework that uses point process cross-validation and point process prediction errors, and includes different hyperparameters. Specifically, statistical properties of PPL are explored, in the context of Gibbs point processes. Paper 1 demonstrates PPL’s advantages over pseudolikelihood, which is a state-of-the-art parameter estimation method and a special case of Takacs- Fiksel estimation (TF), with particular focus on Gibbs hard-core processes. Paper 2 compares PPL to TF, and shows that TF is a special case of PPL, when the cross-validation scheme tends to leave-one-out cross-validation. In addition, Paper 2 shows that for four common Gibbs models, namely Poisson, hard-core, Strauss and Geyer saturation processes, one can choose hyperparameters so that PPL outperforms TF in terms of mean square error.

In Paper 1 and 2, parameter estimation with PPL is done by minimizing loss functions, while Paper 3 explores an alternative approach to PPL, namely estimating equations. Further, statistical properties of the parameter estimator are derived in Paper 3, such as consistency and asymptotic normality for large samples, as well as bias and variance for small samples. It is concluded that the estimating equation approach is not feasible for PPL, whereby the original loss function-based approach is preferred. Moving on, Paper 3 then provides a theoretical foundation for the loss functions through an empirical risk formulation.

To conclude, PPL is shown to be a flexible and robust competitor to state-of-the-art methods for parameter estimation.