Spatial functional data arise in many settings, such as particulate matter curves observed at monitoring stations and age population curves at each areal unit. Most existing functional regression models have limited applicability because they do not consider spatial correlations. Although functional kriging methods can predict the curves at unobserved spatial locations, they are based on variogram fittings rather than constructing hierarchical statistical models. In this manuscript, we propose a Bayesian framework for spatial function-on-function regression that can carry out parameter estimations and predictions. However, the proposed model has computational and inferential challenges because the model needs to account for within and between-curve dependencies. Furthermore, high-dimensional and spatially correlated parameters can lead to the slow mixing of Markov chain Monte Carlo algorithms. To address these issues, we first utilize a basis transformation approach to simplify the covariance and apply projection methods for dimension reduction. We also develop a simultaneous band score for the proposed model to detect the significant region in the regression function. We apply our method to both areal and point-level spatial functional data, showing the proposed method is computationally efficient and provides accurate estimations and predictions.

翻译：空间函数型数据出现在多种场景中，例如在监测站观测到的颗粒物浓度曲线以及各区域单元的人口年龄分布曲线。大多数现有的函数型回归模型由于未考虑空间相关性，其适用性受到限制。尽管函数型克里金方法可以预测未观测空间位置处的曲线，但这些方法基于变异函数拟合而非构建分层统计模型。本文提出了一种用于空间函数对函数回归的贝叶斯框架，该框架能够进行参数估计与预测。然而，所提出的模型面临计算与推断上的挑战，因为模型需要同时考虑曲线内部及曲线之间的依赖性。此外，高维且空间相关的参数可能导致马尔可夫链蒙特卡洛算法收敛缓慢。为解决这些问题，我们首先利用基变换方法简化协方差结构，并采用投影方法进行降维。我们还为所提出的模型开发了同步带评分方法，以检测回归函数中的显著区域。我们将所提方法应用于区域级和点级空间函数型数据，结果表明该方法计算高效，并能提供准确的估计与预测。