Infinite-Dimensional Kalman Filtering Approach to Spatio-Temporal Gaussian Process Regression

We show how spatio-temporal Gaussian process (GP) regression problems (or the equivalent Kriging problems) can be formulated as infinite-dimensional Kalman filtering and Rauch-Tung-Striebel (RTS) smoothing problems, and present a procedure for converting spatio-temporal covariance functions into infinite-dimensional stochastic differential equations (SDEs). The resulting infinite-dimensional SDEs belong to the class of stochastic pseudo-differential equations and can be numerically treated using the methods developed for deterministic counterparts of the equations. The scaling of the computational cost in the proposed approach is linear in the number of time steps as opposed to the cubic scaling of the direct GP regression solution. We also show how separable covariance functions lead to a finite-dimensional Kalman filtering and RTS smoothing problem, present analytical and numerical examples, and discuss numerical methods for computing the solutions.