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.

我们提出了一类新颖的时空区域数据动态因子模型。这类新颖的模型假设时空过程可以由一些根据动态线性模型随时间演变的潜在因素来表示。由于潜在因子向量的维数通常远小于子区域的数量，因此我们提出的模型类可以实现大幅降维。在每个时间点，观测向量通过因子载荷矩阵与潜在因子向量线性相关。该矩阵的每一列可以被视为将一个潜在因子与观测向量相关联的因子载荷的向量化图。因此，为了考虑空间依赖性，我们假设因子载荷矩阵的每一列都遵循内在条件自回归 (ICAR) 过程。因此，我们将我们的模型类别称为动态 ICAR 时空因子模型 (DIFM)。我们开发了吉布斯采样器来探索后验分布。此外，我们通过预测密度的拉普拉斯都会估计器开发模型选择。我们提出两个案例研究。第一个案例研究针对模拟数据，表明我们的 DIFM 是可识别的，并且我们提出的推理程序在恢复底层数据生成过程方面效果良好。最后，第二个案例研究展示了我们的 DIFM 框架的实用性和灵活性，适用于 2015 年至 2021 年美国药物过量流行病。