Profile likelihoods are rarely used in geostatistical models due to the computational burden imposed by repeated decompositions of large variance matrices. Accounting for uncertainty in covariance parameters can be highly consequential in geostatistical models as some covariance parameters are poorly identified, the problem is severe enough that the differentiability parameter of the Matern correlation function is typically treated as fixed. The problem is compounded with anisotropic spatial models as there are two additional parameters to consider. In this paper, we make the following contributions: 1, A methodology is created for profile likelihoods for Gaussian spatial models with Mat\'ern family of correlation functions, including anisotropic models. This methodology adopts a novel reparametrization for generation of representative points, and uses GPUs for parallel profile likelihoods computation in software implementation. 2, We show the profile likelihood of the Mat\'ern shape parameter is often quite flat but still identifiable, it can usually rule out very small values. 3, Simulation studies and applications on real data examples show that profile-based confidence intervals of covariance parameters and regression parameters have superior coverage to the traditional standard Wald type confidence intervals.

翻译：由于对大型方差矩阵进行多次分解带来的巨大计算负担，轮廓似然函数在地统计模型中很少被使用。在地统计模型中，考虑协方差参数的不确定性可能极为重要，因为部分协方差参数难以识别，这一问题严重到马特隆相关函数的可微参数通常被视为固定值。对于各向异性空间模型，由于需要额外考虑两个参数，问题变得更加复杂。本文做出以下贡献：1. 提出了一种针对具有马特隆相关函数族（包括各向异性模型）的高斯空间模型的轮廓似然函数方法。该方法采用了新的参数化策略生成代表点，并在软件实现中利用GPU进行并行轮廓似然计算。2. 我们证明马特隆形状参数的轮廓似然函数通常相当平坦但仍可识别，且通常能排除极小的参数值。3. 模拟研究及实际数据案例应用表明，基于轮廓的协方差参数和回归参数置信区间在覆盖性能上优于传统的标准Wald型置信区间。