Krigings over space and time based on latent low-dimensional structures

We propose a new nonparametric approach to represent the linear dependence structure of a spatio-temporal process in terms of latent common factors. Though it is formally similar to the existing reduced rank approximation methods, the fundamental difference is that the low-dimensional structure is completely unknown in our setting, which is learned from the data collected irregularly over space but regularly over time. Furthermore, a graph Laplacian is incorporated in the learning in order to take the advantage of the continuity over space, and a new aggregation method via randomly partitioning space is introduced to improve the efficiency. We do not impose any stationarity conditions over space either, as the learning is facilitated by the stationarity in time. Krigings over space and time are carried out based on the learned low-dimensional structure, which is scalable to the cases when the data are taken over a large number of locations and/or over a long time period. Asymptotic properties of the proposed methods are established. Illustration with both simulated and real data sets is also reported.