This article studies the use of asymmetric loss functions for the optimal prediction of positive-valued spatial processes. We focus on the family of power-divergence loss functions with properties such as continuity, convexity, connections to well known divergence measures, and the ability to control the asymmetry and behaviour of the loss function via a power parameter. The properties of power-divergence loss functions, optimal power-divergence (OPD) spatial predictors, and related measures of uncertainty quantification are studied. In addition, we examine in general the notion of asymmetry in loss functions defined for positive-valued spatial processes and define an asymmetry measure, which we apply to the family of power-divergence loss functions and other common loss functions. The paper concludes with a simulation study comparing the optimal power-divergence predictor to predictors derived from other common loss functions. Finally, we illustrate OPD spatial prediction on a dataset of zinc measurements in the soil of a floodplain of the Meuse River, Netherlands.

中文翻译：

---

正值空间过程的最优预测：不对称功率发散损失

本文研究了使用不对称损失函数对正值空间过程进行最优预测。我们专注于幂散度损失函数系列，其属性包括连续性、凸性、与众所周知的散度度量的连接，以及通过幂参数控制损失函数的不对称性和行为的能力。研究了功率散度损失函数、最优功率散度（OPD）空间预测器的性质以及不确定性量化的相关度量。此外，我们一般性地研究了为正值空间过程定义的损失函数中的不对称概念，并定义了不对称度量，我们将其应用于功率散度损失函数和其他常见损失函数系列。本文最后进行了一项模拟研究，将最佳功率散度预测器与从其他常见损失函数导出的预测器进行了比较。最后，我们说明了对荷兰默兹河漫滩土壤中锌测量数据集的 OPD 空间预测。