  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.
