Modeling data with non-stationary covariance structure is important to represent heterogeneity in geophysical and other environmental spatial processes. In this work, we investigate a multistage approach to modeling non-stationary covariances that is efficient for large data sets. First, we use likelihood estimation in local, moving windows to infer spatially varying covariance parameters. These surfaces of covariance parameters can then be encoded into a global covariance model specifying the second-order structure for the complete spatial domain. The resulting global model allows for efficient simulation and prediction. We investigate the non-stationary spatial autoregressive (SAR) model related to Gaussian Markov random field (GMRF) methods, which is amenable to plug in local estimates and practical for large data sets. In addition we use a simulation study to establish the accuracy of local Mat\'ern parameter estimation as a reliable technique when replicate fields are available and small local windows are exploited to reduce computation. This multistage modeling approach is implemented on a non-stationary climate model output data set with the goal of emulating the variation in the numerical model ensemble using a Gaussian process.

中文翻译：

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使用局部似然估计和 Matérn 到 SAR 转换对空间数据进行建模

具有非平稳协方差结构的建模数据对于表示地球物理和其他环境空间过程中的异质性很重要。在这项工作中，我们研究了一种对大型数据集有效的非平稳协方差建模的多阶段方法。首先，我们在局部移动窗口中使用似然估计来推断空间变化的协方差参数。然后可以将这些协方差参数表面编码为全局协方差模型，指定完整空间域的二阶结构。由此产生的全局模型允许有效的模拟和预测。我们研究了与高斯马尔可夫随机场 (GMRF) 方法相关的非平稳空间自回归 (SAR) 模型，该模型适合插入局部估计值，适用于大型数据集。此外，我们使用模拟研究来建立局部 Mat\'ern 参数估计的准确性，作为一种可靠的技术，当重复场可用并且利用小的局部窗口来减少计算时。这种多阶段建模方法是在非平稳气候模型输出数据集上实施的，目的是使用高斯过程模拟数值模型集合的变化。