Due to the well-known computational showstopper of the exact Maximum Likelihood Estimation (MLE) for large geospatial observations, a variety of approximation methods have been proposed in the literature, which usually require tuning certain inputs. For example, the recently developed Tile Low-Rank approximation (TLR) method involves many tuning parameters, including numerical accuracy. To properly choose the tuning parameters, it is crucial to adopt a meaningful criterion for the assessment of the prediction efficiency with different inputs. Unfortunately, the most commonly-used Mean Square Prediction Error (MSPE) criterion cannot directly assess the loss of efficiency when the spatial covariance model is approximated. Though the Kullback–Leibler Divergence criterion can provide the information loss of the approximated model, it cannot give more detailed information that one may be interested in, e.g., the accuracy of the computed MSE. In this paper, we present three other criteria, the Mean Loss of Efficiency (MLOE), Mean Misspecification of the Mean Square Error (MMOM), and Root mean square MOM (RMOM), and show numerically that, in comparison with the common MSPE criterion and the Kullback–Leibler Divergence criterion, our criteria are more informative, and thus more adequate to assess the loss of the prediction efficiency by using the approximated or misspecified covariance models. Hence, our suggested criteria are more useful for the determination of tuning parameters for sophisticated approximation methods of spatial model fitting. To illustrate this, we investigate the trade-off between the execution time, estimation accuracy, and prediction efficiency for the TLR method with extensive simulation studies and suggest proper settings of the TLR tuning parameters. We then apply the TLR method to a large spatial dataset of soil moisture in the area of the Mississippi River basin, and compare the TLR with the Gaussian predictive process and the composite likelihood method, showing that our suggested criteria can successfully be used to choose the tuning parameters that can keep the estimation or the prediction accuracy in applications.

由于用于大型地理空间观测的精确最大似然估计 (MLE) 具有众所周知的计算优势，因此文献中提出了各种近似方法，这些方法通常需要调整某些输入。例如，最近开发的 Tile Low-Rank approximation (TLR) 方法涉及许多调整参数，包括数值精度。为了正确选择调整参数，采用有意义的标准来评估不同输入的预测效率至关重要。不幸的是，当空间协方差模型被近似时，最常用的均方预测误差 (MSPE) 标准不能直接评估效率的损失。虽然 Kullback-Leibler Divergence 准则可以提供近似模型的信息损失，它不能提供人们可能感兴趣的更详细的信息，例如，计算出的 MSE 的准确性。在本文中，我们提出了其他三个标准，即平均效率损失（MLOE），均方误差的均值错误指定（MMOM）和均方根MOM（RMOM），并与普通的MSPE进行了数值比较标准和 Kullback-Leibler 散度标准，我们的标准提供更多信息，因此更适合通过使用近似或错误指定的协方差模型来评估预测效率的损失。因此，我们建议的标准对于确定空间模型拟合的复杂近似方法的调整参数更有用。为了说明这一点，我们研究了执行时间、估计精度、TLR方法的预测效率和预测效率，并进行了广泛的仿真研究，并提出了TLR调整参数的正确设置。然后我们将 TLR 方法应用于密西西比河流域地区土壤水分的大型空间数据集，并将 TLR 与高斯预测过程和复合似然法进行比较，表明我们建议的标准可以成功地用于选择可以在应用程序中保持估计或预测准确性的调整参数。