Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Theoretical Analysis of Heteroscedastic Gaussian Processes with Posterior Distributions

Yuji Ito

TL;DR

A novel theoretical framework for analyzing heteroscedastic Gaussian processes (HGPs) that identify unknown systems in a data-driven manner is introduced and the derived theoretical findings are applied to a chance-constrained tracking controller.

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.

Theoretical Analysis of Heteroscedastic Gaussian Processes with Posterior Distributions

TL;DR

A novel theoretical framework for analyzing heteroscedastic Gaussian processes (HGPs) that identify unknown systems in a data-driven manner is introduced and the derived theoretical findings are applied to a chance-constrained tracking controller.

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.
Paper Structure (15 sections, 6 theorems, 51 equations, 1 figure, 1 table)

This paper contains 15 sections, 6 theorems, 51 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Proposed method with HGPR
  4. Theoretical analyses of HGPR
  5. Practical implementation for HGPR
  6. Control application with demonstration
  7. Chance-constrained tracking control using HGPR
  8. Simulation setting
  9. Simulation results
  10. Conclusion
  11. Proof of Theorem \ref{['thm:GP_fm']}
  12. Proof of Theorem \ref{['thm:GP_prob_fm']}
  13. Proof of Proposition \ref{['thm:SNIS_PDF_ratio']}
  14. Proof of Theorem \ref{['thm:PDF_lnfv1']}
  15. Proof of Theorem \ref{['thm:L1_control']}

Key Result

Theorem 1

Suppose Assumption ass_GPs. For any ${x} \in {\mathcal{X}}$, the posterior mean ${\mathbb{E}}[{f}({x}) | {x}, {{\mathcal{D}}_{}} ]$ and variance ${\mathbb{V}}[{f}({x}) | {x}, {{\mathcal{D}}_{}} ]$ of ${f}$ given $({x}, {{\mathcal{D}}_{}})$ are expressed as follows: where

Figures (1)

  • Figure 1: Trajectories of the state and input by the proposed method. The blue solid and dashed lines denote the reference signal ${ {[}{{r}({t})}{]}_{2} }$ and ${ {[}{{r}({t})}{]}_{2} } \pm {\overline{r}}$, respectively. The red lines in the upper and lower figures indicate the state ${ {[}{{\xi}({t})}{]}_{2} }$ and control input ${{u({t})}}$, respectively.

Theorems & Definitions (17)

  • Theorem 1: The posterior mean and variance of ${f}$
  • proof
  • Remark 1: Contributions of Theorem \ref{['thm:GP_fm']}
  • Theorem 2: The cumulative distribution function of ${f}$
  • proof
  • Remark 2: Contributions of Theorem \ref{['thm:GP_prob_fm']}
  • Corollary 1: The probability of ${f}$ being in an interval
  • proof
  • Remark 3: Autonormalized importance sampling
  • Proposition 1: First property of the proposed ${q}( {h}_{1:{D}} )$
  • ...and 7 more