A Unified Spatiotemporal Framework for Modeling Censored and Missing Areal Responses

Abstract: We propose a new Bayesian approach for spatiotemporal areal data with censored and missing observations. The method introduces a flexible random effect that combines the spatial dependence structures of the Simultaneous Autoregressive (SAR) and Directed Acyclic Graph Autoregressive (DAGAR) models with a temporal autoregressive component. We demonstrate that this formulation extends both spatial models into a unified spatiotemporal framework, expressing them as Gaussian Markov random fields in their innovation form. The resulting model captures spatial, temporal, and joint spatiotemporal correlations in an interpretable way. Simulation studies show that the proposed model outperforms common ad hoc imputation strategies, such as replacing censored values with the limit of detection (LOD) or imputing missing data by the sample mean. We further apply the method to carbon monoxide (CO) concentration data from Beijing's air quality network, comparing the proposed DAGAR-AR model with the traditional Conditional Autoregressive (CAR) approach. The results indicate that while the CAR model achieves slightly better predictive performance, the DAGAR-AR specification offers clearer interpretability and a more coherent representation of the spatiotemporal dependence structure.