Dynamic ICAR Spatiotemporal Factor Models

We propose a novel class of dynamic factor models for spatiotemporal areal data. This novel class of models assumes that the spatiotemporal process may be represented by some few latent factors that evolve through time according to dynamic linear models. As the dimension of the vector of latent factors is typically much smaller than the number of subregions, our proposed class of models may achieve substantial dimension reduction. At each time point, the vector of observations is linearly related to the vector of latent factors through a matrix of factor loadings. Each column of this matrix may be seen as a vectorized map of factor loadings relating one latent factor to the vector of observations. Thus, to account for spatial dependence, we assume that each column of the matrix of factor loadings follows an intrinsic conditional autoregressive (ICAR) process. Hence, we call our class of models the Dynamic ICAR Spatiotemporal Factor Models (DIFM). We develop a Gibbs sampler for exploration of the posterior distribution. In addition, we develop model selection through a Laplace-Metropolis estimator of the predictive density. We present two case studies. The first case study, which is for simulated data, demonstrates that our DIFMs are identifiable and that our proposed inferential procedure works well at recovering the underlying data generating process. Finally, the second case study demonstrates the utility and flexibility of our DIFM framework with an application to the drug overdose epidemic in the United States from 2015 to 2021.