🤖 AI Summary
Existing spatial functional regression models neglect spatial dependence, while functional kriging—though capable of predicting unobserved location curves—lacks parametric modeling and Bayesian inference capabilities. To address this, we propose the first hierarchical Bayesian function-on-function regression framework that jointly accounts for intra-curve, inter-curve, and spatial dependencies. We introduce a novel synchronized band scoring method to identify statistically significant regions of influence for the regression functions. Our approach integrates basis expansion for dimension reduction, an extension of functional kriging that avoids specifying variogram functions, and a tailored MCMC algorithm to mitigate slow mixing induced by high-dimensional parameters. Experiments on areal and point-referenced data demonstrate that our method is computationally efficient, yields accurate parameter estimates, and achieves significantly higher predictive accuracy than state-of-the-art alternatives.
📝 Abstract
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.