  In this work we study the non-parametric reconstruction of spatio-temporal
dynamical Gaussian processes (GPs) via GP regression from sparse and noisy
data. GPs have been mainly applied to spatial regression where they represent
one of the most powerful estimation approaches also thanks to their universal
representing properties. Their extension to dynamical processes has been
instead elusive so far since classical implementations lead to unscalable
algorithms. We then propose a novel procedure to address this problem by
coupling GP regression and Kalman filtering. In particular, assuming space/time
separability of the covariance (kernel) of the process and rational time
spectrum, we build a finite-dimensional discrete-time state-space process
representation amenable of Kalman filtering. With sampling over a finite set of
fixed spatial locations, our major finding is that the Kalman filter state at
instant $t_k$ represents a sufficient statistic to compute the minimum variance
estimate of the process at any $t \geq t_k$ over the entire spatial domain.
This result can be interpreted as a novel Kalman representer theorem for
dynamical GPs. We then extend the study to situations where the set of spatial
input locations can vary over time. The proposed algorithms are finally tested
on both synthetic and real field data, also providing comparisons with standard
GP and truncated GP regression techniques.
