We study maximum likelihood estimation for spatial generalized linear mixed models with Gaussian process approximations using a stochastic Newton-Raphson algorithm. We consider two Gaussian Process approximations in this context: spectral Gaussian process approximations and stochastic partial differential equations (SPDE). We refine the stochastic maximum likelihood algorithm and we propose a new stopping criterion for efficient termination to prevent long runs of sampling in the stationary post-convergence phase and a Monte Carlo estimator of fixed effect standard errors. We run a series of simulation comparisons of spatial statistical models alongside the popular Bayesian integrated nested Laplacian approximation method which incorporates SPDE. We show that HSGP provides nominal coverage of fixed and random effect parameters with smooth latent fields but performance degrades for rough fields. SPDE in a stochastic maximum likelihood framework maintains nominal coverage and matches or improves upon the performance of Bayesian integrated nested Laplacian approximation.

翻译：我们研究了使用随机牛顿-拉弗森算法进行高斯过程近似的空间广义线性混合模型的最大似然估计。在此背景下，我们考虑了两种高斯过程近似方法：谱高斯过程近似和随机偏微分方程（SPDE）。我们对随机最大似然算法进行了改进，并提出了一种新的收敛停止准则，以有效终止算法，避免在平稳后收敛阶段进行长时间采样，同时引入了一个蒙特卡洛估计量来计算固定效应标准误差。我们通过一系列模拟比较了空间统计模型与流行的基于SPDE的贝叶斯积分嵌套拉普拉斯近似方法。结果表明，对于光滑潜场，HSGP能够提供名义覆盖率的固定效应和随机效应参数，但对于粗糙场，其性能会下降。在随机最大似然框架下，SPDE能够维持名义覆盖率，并达到或优于贝叶斯积分嵌套拉普拉斯近似方法的性能。