Recently, Cronie et al. (2024) introduced the notion of cross-validation for point processes and a new statistical methodology called Point Process Learning (PPL). In PPL one splits a point process/pattern into a training and a validation set, and then predicts the latter from the former through a parametrised Papangelou conditional intensity. The model parameters are estimated by minimizing a point process prediction error; this notion was introduced as the second building block of PPL. It was shown that PPL outperforms the state-of-the-art in both kernel intensity estimation and estimation of the parameters of the Gibbs hard-core process. In the latter case, the state-of-the-art was represented by pseudolikelihood estimation. In this paper we study PPL in relation to Takacs-Fiksel estimation, of which pseudolikelihood is a special case. We show that Takacs-Fiksel estimation is a special case of PPL in the sense that PPL with a specific loss function asymptotically reduces to Takacs-Fiksel estimation if we let the cross-validation regime tend to leave-one-out cross-validation. Moreover, PPL involves a certain type of hyperparameter given by a weight function which ensures that the prediction errors have expectation zero if and only if we have the correct parametrisation. We show that the weight function takes an explicit but intractable form for general Gibbs models. Consequently, we propose different approaches to estimate the weight function in practice. In order to assess how the general PPL setup performs in relation to its special case Takacs-Fiksel estimation, we conduct a simulation study where we find that for common Gibbs models we can find loss functions and hyperparameters so that PPL typically outperforms Takacs-Fiksel estimation significantly in terms of mean square error. Here, the hyperparameters are the cross-validation parameters and the weight function estimate.

翻译：最近，Cronie等人（2024）引入了点过程的交叉验证概念以及一种称为点过程学习（PPL）的新统计方法。在PPL中，将一个点过程/模式划分为训练集和验证集，然后通过参数化的Papangelou条件强度从前者预测后者。模型参数通过最小化点过程预测误差来估计；这一概念被引入作为PPL的第二个构建模块。研究表明，PPL在核强度估计和Gibbs硬核过程参数估计方面均优于现有最先进方法。在后一种情况下，现有最先进方法以伪似然估计为代表。本文研究了PPL与Takacs-Fiksel估计之间的关系，其中伪似然估计是后者的一个特例。我们证明，Takacs-Fiksel估计是PPL的一个特例，其意义在于：若采用特定损失函数，且让交叉验证机制趋于留一交叉验证，则PPL在渐近意义上可简化为Takacs-Fiksel估计。此外，PPL涉及一种由权重函数给出的超参数，该函数确保当且仅当参数化正确时，预测误差的期望为零。我们证明，对于一般Gibbs模型，该权重函数具有显式但难以处理的形式。因此，我们提出了在实践中估计权重函数的不同方法。为了评估通用PPL框架相对于其特例Takacs-Fiksel估计的性能，我们进行了一项模拟研究，发现对于常见的Gibbs模型，我们可以找到损失函数和超参数，使得PPL在均方误差方面通常显著优于Takacs-Fiksel估计。这里的超参数包括交叉验证参数和权重函数估计值。