Modeling Multivariate Spatial-Temporal Data with Latent Low-Dimensional Dynamics

High-dimensional multivariate spatial-temporal data arise frequently in a wide range of applications; however, there are relatively few statistical methods that can simultaneously deal with spatial, temporal and variable-wise dependencies in large data sets. In this paper, we propose a new approach to utilize the correlations in variable, space and time to achieve dimension reduction and to facilitate spatial/temporal predictions in the high-dimensional settings. The multivariate spatial-temporal process is represented as a linear transformation of a lower-dimensional latent factor process. The spatial dependence structure of the factor process is further represented non-parametrically in terms of latent empirical orthogonal functions. The low-dimensional structure is completely unknown in our setting and is learned entirely from data collected irregularly over space but regularly over time. We propose innovative estimation and prediction methods based on the latent low-rank structures. Asymptotic properties of the estimators and predictors are established. Extensive experiments on synthetic and real data sets show that, while the dimensions are reduced significantly, the spatial, temporal and variable-wise covariance structures are largely preserved. The efficacy of our method is further confirmed by the prediction performances on both synthetic and real data sets.