Functional kriging approaches have been developed to predict the curves at unobserved spatial locations. However, most existing approaches are based on variogram fittings rather than constructing hierarchical statistical models. Therefore, it is challenging to analyze the relationships between functional variables, and uncertainty quantification of the model is not trivial. In this manuscript, we propose a Bayesian framework for spatial function-on-function regression. However, inference for 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 the methods to simulated and real datasets, including data on particulate matter in Japan and mobility data in South Korea. The proposed method is computationally efficient and provides accurate estimations and predictions.

翻译：函数型克里金方法已被开发用于预测未观测空间位置上的曲线。然而，现有方法大多基于变异函数拟合而非构建分层统计模型，因此难以分析函数型变量间的关系，且模型的不确定性量化存在非平凡性。本文提出一种空间函数对函数回归的贝叶斯框架。但该模型的推断面临计算与推断双重挑战，因其需同时考虑曲线内部及曲线间的依赖关系。此外，高维且空间相关的参数会导致马尔可夫链蒙特卡洛算法的混合缓慢。为解决这些问题，我们首先采用基变换方法简化协方差结构，并利用投影方法实现降维。同时，我们为该模型开发了一种同步带状评分方法，用于检测回归函数中的显著区域。我们将所提方法应用于模拟数据集及真实数据集，包括日本颗粒物数据与韩国移动性数据。结果表明，所提方法计算高效且能提供准确的估计与预测。