The log Gaussian Cox process (LGCP) is a frequently applied method for modeling point pattern data. The normalization constant of the LGCP likelihood involves an integral over a latent field. That integral is computationally expensive, making it troublesome to perform inference with standard methods. The so‐called stochastic partial differential equation–integrated nested Laplace approximation (SPDE‐INLA) framework enables fast approximate inference for a range of hierarchical models, where a key component is to approximate the latent field by a triangulated mesh. Recent research has made it possible to fit LGCP models with this framework using an approximate integration method to compute the integral. We carefully describe several alternative variants of that approximate integration method and derive an analytical formula for the integral in question, which actually is exact under the triangular mesh assumption used by SPDE‐INLA. We compare the different integration strategies through a comprehensive simulation study and find that the analytical formula is often more accurate, but not always. Among the approximate integration methods, we recommend a simple extension to a method implemented in an R‐package for fitting LGCP models.

中文翻译：

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研究对数高斯Cox过程可能性中的归一化常数的基于网格的近似方法

对数高斯考克斯过程（LGCP）是一种常用的建模点模式数据的方法。LGCP可能性的归一化常数涉及一个潜场上的积分。该积分的计算量很大，因此很难用标准方法进行推理。所谓的随机偏微分方程集成的嵌套拉普拉斯近似（SPDE-INLA）框架可用于一系列层次模型的快速近似推论，其中关键的组成部分是通过三角网格近似潜在场。最近的研究已使使用近似积分方法计算积分的LGCP模型适合此框架。我们仔细描述了该近似积分方法的几种替代形式，并导出了所讨论积分的解析公式，该公式实际上在SPDE-INLA使用的三角形网格假设下是精确的。我们通过全面的仿真研究比较了不同的集成策略，发现分析公式通常更准确，但并非总是如此。在近似的集成方法中，我们建议对R包中实现的适合LGCP模型的方法进行简单扩展。