Fitting regression models for intensity functions of spatial point processes is of great interest in ecological and epidemiological studies of association between spatially referenced events and geographical or environmental covariates. When Cox or cluster process models are used to accommodate clustering not accounted for by the available covariates, likelihood based inference becomes computationally cumbersome due to the complicated nature of the likelihood function and the associated score function. It is therefore of interest to consider alternative more easily computable estimating functions. We derive the optimal estimating function in a class of first-order estimating functions. The optimal estimating function depends on the solution of a certain Fredholm integral equation which in practise is solved numerically. The derivation of the optimal estimating function has close similarities to the derivation of quasi-likelihood for standard data sets. The approximate solution is further equivalent to a quasi-likelihood score for binary spatial data. We therefore use the term quasi-likelihood for our optimal estimating function approach. We demonstrate in a simulation study and a data example that our quasi-likelihood method for spatial point processes is both statistically and computationally efficient.

中文翻译：

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空间点过程的拟似然性。

在空间参考事件与地理或环境协变量之间的关系的生态学和流行病学研究中，对空间点过程的强度函数拟合回归模型非常感兴趣。当使用Cox或聚类过程模型来容纳可用协变量未解决的聚类时，由于似然函数和相关评分函数的复杂性，基于似然的推理在计算上变得繁琐。因此，感兴趣的是考虑替代性的更容易计算的估计函数。我们在一类一阶估计函数中得出最优估计函数。最佳估计函数取决于一个特定的Fredholm积分方程的解，该积分解在实践中可以通过数值求解。最佳估计函数的推导与标准数据集的拟似然推导有着相似的相似性。近似解还等效于二进制空间数据的准似然分数。因此，我们将术语拟似然性用于我们的最佳估计函数方法。我们在模拟研究和数据示例中证明，我们用于空间点过程的拟似然法在统计和计算上均有效。