Theoretical Analysis of Heteroscedastic Gaussian Processes with Posterior Distributions

TL;DR

This paper develops a theoretical framework for heteroscedastic Gaussian processes, deriving exact posterior distributions and applying these findings to improve chance-constrained control in systems with unknown disturbances.

Contribution

It provides the first exact analytical expressions for posterior means, variances, and distributions of heteroscedastic Gaussian processes, enhancing their application in control systems.

Findings

Derived exact posterior distributions for HGPs.

Applied theoretical results to chance-constrained control.

Demonstrated improved disturbance handling in control systems.

Abstract

This study introduces a novel theoretical framework for analyzing heteroscedastic Gaussian processes (HGPs) that identify unknown systems in a data-driven manner. Although HGPs effectively address the heteroscedasticity of noise in complex training datasets, calculating the exact posterior distributions of the HGPs is challenging, as these distributions are no longer multivariate normal. This study derives the exact means, variances, and cumulative distributions of the posterior distributions. Furthermore, the derived theoretical findings are applied to a chance-constrained tracking controller. After an HGP identifies an unknown disturbance in a plant system, the controller can handle chance constraints regarding the system despite the presence of the disturbance.