  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.
