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: Firstly, a methodology is created for profile likelihoods for Gaussian spatial models with Matérn family of correlation functions, including anisotropic models. This methodology adopts a novel reparameterization for generation of representative points, and uses GPUs for parallel profile likelihoods computation in software implementation. Then, we show the profile likelihood of the Matérn shape parameter is often quite flat but still identifiable, it can usually rule out very small values. Finally, 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.

中文翻译：

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跨高斯地质统计模型中参数的剖面可能性

由于大方差矩阵的重复分解带来的计算负担，剖面似然很少在地统计模型中使用。考虑到协方差参数的不确定性在地质统计模型中可能会产生非常严重的后果，因为一些协方差参数很难识别，问题非常严重，以至于马特恩相关函数的可微性参数通常被视为固定的。该问题与各向异性空间模型更加复杂，因为需要考虑两个附加参数。在本文中，我们做出了以下贡献：首先，为具有 Matérn 相关函数系列（包括各向异性模型）的高斯空间模型的剖面似然创建了一种方法。该方法采用新颖的重新参数化来生成代表点，并在软件实现中使用 GPU 进行并行轮廓似然计算。然后，我们表明 Matérn 形状参数的轮廓似然通常非常平坦但仍然可识别，它通常可以排除非常小的值。最后，模拟研究和实际数据示例的应用表明，基于轮廓的协方差参数和回归参数的置信区间比传统的标准 Wald 型置信区间具有更好的覆盖范围。