Model averaging, as an appealing ensemble technique, strategically integrates all valuable information from candidate models to construct fast and accurate prediction. Despite of having been widely practiced in many fields such as cross-sectional data, censored data and longitudinal data, its application to spatial data characterized by inherent spatial heterogeneity remains surprisingly limited. To mitigate risk of model misspecification and enhance the flexibility of prediction, we propose a combined estimator constructed by computing the weighted average of estimators derived from a set of spatially varying coefficient candidate models. Herein, the model weights are determined via a Mallows-type criterion, which dynamically calibrates the relative importance of individual candidate models in the ensemble. Theoretically, we establish desirable asymptotic properties under two practical scenarios. First, in the case where all candidate models are misspecified, the proposed model averaging estimator attains asymptotic optimality in the sense that it minimizes the squared error loss function asymptotically. Second, when the candidate model set encompasses at least one quasi-correct model, the weights assigned by the Mallows-type criterion asymptotically concentrate on the quasi-correct models, and the resulting model averaging estimator converges in probability to the true conditional mean. Both simulation studies and a real-world empirical example demonstrate that the proposed method generally outperforms alternative comparative approaches in terms of predictive accuracy and robustness.

翻译：模型平均作为一种极具吸引力的集成技术，通过策略性地整合候选模型中的所有有价值信息，以构建快速且准确的预测。尽管该方法已在横截面数据、删失数据和纵向数据等诸多领域得到广泛应用，但其在具有固有空间异质性特征的空间数据中的应用却出人意料地有限。为降低模型设定错误的风险并增强预测的灵活性，我们提出一种组合估计量，该估计量通过计算一组空间变化系数候选模型所得估计量的加权平均而构建。其中，模型权重通过Mallows型准则确定，该准则动态校准了集成中各个候选模型的相对重要性。在理论上，我们在两种实际场景下建立了理想的渐近性质。首先，在所有候选模型均设定错误的情况下，所提出的模型平均估计量在渐近意义上达到最优性，即它渐近地最小化平方误差损失函数。其次，当候选模型集合包含至少一个拟正确模型时，Mallows型准则分配的权重渐近集中于拟正确模型，且所得的模型平均估计量依概率收敛于真实条件均值。模拟研究和实际实证案例均表明，所提方法在预测精度和鲁棒性方面通常优于其他对比方法。