Efficient Recursive Implementation of Spatial-Temporal Gaussian Process Regression

The current implementation of the spatial-temporal Gaussian process regression has computational complexity O(NM³), where N and M are the number of temporal and spatial data, respectively, and thus can only be applied to data with large N but relatively small M. In this work, we show that by exploring the Kronecker structure in the state-space model realization of the spatial-temporal Gaussian process, we can extend the current implementation with a coordinate transformation and an output transformation (corresponding to data preprocessing), such that the computational complexity is reduced to O(M³ +NM² +NM) and therefore the proposed implementation can be applied to data with large N and moderately large M. Moreover, the proposed implementation can be parallelized and the computational complexity can be further lowered if parallel computing is adopted.