Numerical Gaussian process Kalman filtering

TL;DR

This paper introduces a novel numerical Gaussian process Kalman filter that enables recursive filtering of infinite-dimensional systems without spatial discretization, learning noise parameters online, demonstrated on an advection equation.

Contribution

It embeds numerical Gaussian processes into Kalman filtering, allowing for filtering of infinite-dimensional systems with automatic noise parameter learning.

Findings

Successfully applied to the advection equation simulation.

Eliminates the need for spatial discretization.

Automatically learns process and measurement noise levels.

Abstract

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.