General spatio-temporal factor models for high-dimensional random fields on a lattice

Motivated by the need for analysing large spatio-temporal panel data, we introduce a novel nonparametric methodology for n-dimensional random fields observed across S spatial locations and T time periods. We call it general spatio-temporal factor model (GSTFM). First, we provide the probabilistic and mathematical underpinning needed for the representation of a random field as the sum of two components: the common component (driven by a small number q of latent factors) and the idiosyncratic component (mildly cross-correlated). We show that the two components are identified as n → ∞. Second, we propose an estimator of the common component and derive its statistical guarantees (consistency and rate of convergence) as min(n,S,T ) → ∞. Third, we propose an information criterion to determine the number of factors. Estimation makes use of Fourier analysis in the frequency domain and thus it fully exploits the information on the spatiotemporal covariance structure of the whole panel. Synthetic data examples illustrate the applicability of GSTFM and its advantages over the extant generalized dynamic factor model that ignores the spatial correlations.