# Efficient Spatio-Temporal Gaussian Regression via Kalman Filtering

arXiv: https://arxiv.org/abs/1705.01485 · HTML: https://mlsanity.com/paper/1705.01485/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 May 2017.

## Built on by (most cited first, 5 of 5)

- [Deep networks for system identification: a Survey](https://mlsanity.com/paper/2301.12832/) (2023)
- [Computation-Aware Kalman Filtering and Smoothing](https://mlsanity.com/paper/2405.08971/) (2024)
- [The Rank-Reduced Kalman Filter: Approximate Dynamical-Low-Rank Filtering In High Dimensions](https://mlsanity.com/paper/2306.07774/) (2023)
- [Spatio-Temporal Gaussian Process for Building Terrain-Incorporating Wind Power Curves](https://mlsanity.com/paper/2607.00051/) (2026)
- [Effect Decomposition of Functional-Output Computer Experiments via Orthogonal Additive Gaussian Processes](https://mlsanity.com/paper/2506.12701/) (2025)

## Abstract (arXiv)

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 tk represents a sufficient statistic to compute the minimum variance estimate of the process at any t ≥ tk 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.