Efficiency assessment of approximated spatial predictions for large datasets

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.