Numerical Gaussian process Kalman filtering
This work addresses a domain-specific problem for researchers in spatiotemporal modeling and filtering, offering an incremental improvement by combining existing numerical Gaussian processes with Kalman filtering.
The paper tackles the problem of performing Kalman filtering on infinite-dimensional systems by embedding numerical Gaussian processes into recursive Kalman filter equations, enabling filtering without spatial discretization and manual noise tuning, as demonstrated in a simulation study of the advection equation.
In this manuscript we introduce numerical Gaussian process Kalman filtering (GPKF). Numerical Gaussian processes have recently been developed to simulate spatiotemporal models. The contribution of this paper is to embed numerical Gaussian processes into the recursive Kalman filter equations. This embedding enables us to do Kalman filtering on infinite-dimensional systems using Gaussian processes. This is possible because i) we are obtaining a linear model from numerical Gaussian processes, and ii) the states of this model are by definition Gaussian distributed random variables. Convenient properties of the numerical GPKF are that no spatial discretization of the model is necessary, and manual setting up of the Kalman filter, that is fine-tuning the process and measurement noise levels by hand is not required, as they are learned online from the data stream. We showcase the capability of the numerical GPKF in a simulation study of the advection equation.