An Efficient Implementation for Spatial-Temporal Gaussian Process Regression and Its Applications

TL;DR

This paper introduces a computationally efficient method for spatial-temporal Gaussian process regression by exploiting Kronecker structure, enabling applications to larger datasets with improved prediction accuracy.

Contribution

It presents a novel implementation that reduces computational complexity from d7(NM^3) to d7(M^3+NM^2) by leveraging Kronecker structure, expanding applicability to larger data.

Findings

Reduced computational complexity enables larger datasets processing.

Improved prediction performance on weather and temperature data.

Effective kernel design enhances model accuracy.

Abstract

Spatial-temporal Gaussian process regression is a popular method for spatial-temporal data modeling. Its state-of-art implementation is based on the state-space model realization of the spatial-temporal Gaussian process and its corresponding Kalman filter and smoother, and has computational complexity $\mathcal{O}(NM^3)$, where $N$ and $M$ are the number of time instants and spatial input locations, respectively, and thus can only be applied to data with large $N$ but relatively small $M$. In this paper, our primary goal is to show that by exploring the Kronecker structure of the state-space model realization of the spatial-temporal Gaussian process, it is possible to further reduce the computational complexity to $\mathcal{O}(M^3+NM^2)$ and thus the proposed implementation can be applied to data with large $N$ and moderately large $M$. The proposed implementation is illustrated over applications in weather data prediction and spatially-distributed system identification. Our secondary goal is to design a kernel for both the Colorado precipitation data and the GHCN temperature data, such that while having more efficient implementation, better prediction performance can also be achieved than the state-of-art result.