In this paper, we harness a result in point process theory, specifically the expectation of the weighted $K$-function, where the weighting is done by the true first-order intensity function. This theoretical result can be employed as an estimation method to derive parameter estimates for a particular model assumed for the data. The underlying motivation is to avoid the difficulties associated with dealing with complex likelihoods in point process models and their maximization. The exploited result makes our method theoretically applicable to any model specification. In this paper, we restrict our study to Poisson models, whose likelihood represents the base for many more complex point process models. In this context, our proposed method can estimate the vector of local parameters that correspond to the points within the analyzed point pattern without introducing any additional complexity compared to the global estimation. We illustrate the method through simulation studies for both purely spatial and spatio-temporal point processes and show complex scenarios based on the Poisson model through the analysis of two real datasets concerning environmental problems.

翻译：本文利用点过程理论中的一个结果，即加权$K$函数的期望，其中加权由真实的一阶强度函数完成。该理论结果可作为估计方法，用于推导数据假设的特定模型参数估计值。其根本动机在于避免处理点过程模型中复杂似然函数及其最大化所带来的困难。所利用的结果使得我们的方法在理论上适用于任何模型设定。本文将研究范围限定在泊松模型，其似然函数代表了许多更复杂点过程模型的基础。在此背景下，我们提出的方法能够估计分析点模式内各点对应的局部参数向量，且不会引入比全局估计更复杂的计算过程。我们通过纯空间和时空点过程的模拟研究展示了该方法的有效性，并基于泊松模型，通过对两个涉及环境问题的真实数据集的分析，展示了复杂场景下的应用。