The Past Does Matter: Correlation of Subsequent States in Trajectory Predictions of Gaussian Process Models

TL;DR

This paper addresses the challenge of trajectory distribution computation in Gaussian Process models of dynamical systems, highlighting the flaws of previous independence assumptions and proposing a new piecewise linear approximation to improve accuracy.

Contribution

It identifies the incorrect independence assumption in prior uncertainty propagation methods and introduces a novel piecewise linear approximation for Gaussian Processes in continuous models.

Findings

Previous independence assumptions lead to inaccuracies in trajectory predictions.

The proposed piecewise linear approximation better captures the correlation of subsequent states.

The method reduces computational costs compared to sampling-based approaches.

Abstract

Computing the distribution of trajectories from a Gaussian Process model of a dynamical system is an important challenge in utilizing such models. Motivated by the computational cost of sampling-based approaches, we consider approximations of the model's output and trajectory distribution. We show that previous work on uncertainty propagation, focussed on discrete state-space models, incorrectly included an independence assumption between subsequent states of the predicted trajectories. Expanding these ideas to continuous ordinary differential equation models, we illustrate the implications of this assumption and propose a novel piecewise linear approximation of Gaussian Processes to mitigate them.