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：在地统计模型中，由于大型方差矩阵的重复分解带来的计算负担，剖面似然函数很少被使用。考虑到协方差参数的不确定性在地统计模型中可能产生重大影响，因为某些协方差参数辨识度较低，这一问题严重到通常将Matern相关函数的可微参数视为固定值。对于各向异性空间模型，由于需要额外考虑两个参数，问题更为复杂。本文做出以下贡献：1. 建立了一种适用于高斯空间模型（包括各向异性模型）中Matern相关函数族参数的剖面似然方法。该方法采用新的参数化策略生成代表性点，并在软件实现中利用GPU进行并行剖面似然计算。2. 我们证明Matern形状参数的剖面似然通常较为平坦但仍可辨识，通常可排除极小值。3. 模拟研究与实际数据应用表明，基于剖面似然的协方差参数和回归参数置信区间在覆盖程度上优于传统标准Wald型置信区间。