Recursive Gaussian Process State Space Model

TL;DR

This work addresses the challenge of online learning for Gaussian Process State-Space Models (GPSSMs) under limited memory and nonstationary data. It introduces a Recursive GPSSM (RGPSSM) that uses a two-step EKF-like Bayesian update with first-order linearization, augmented inducing points, and online hyperparameter optimization to enable domain-adaptive, recursive inference without predefining the operating domain. A dynamic inducing-point management strategy (addition/removal) maintains computational tractability, while an information-source-based online hyperparameter update aligns the GP prior with evolving dynamics; a Cholesky-stable variant further enhances numerical reliability. The framework generalizes several existing online GP methods, supports multi-output extensions, and demonstrates superior accuracy and adaptability across synthetic and real-world datasets. These contributions offer a practical, scalable approach for online GP-based system identification and control in nonstationary, data-limited environments.

Abstract

Learning dynamical models from data is not only fundamental but also holds great promise for advancing principle discovery, time-series prediction, and controller design. Among various approaches, Gaussian Process State-Space Models (GPSSMs) have recently gained significant attention due to their combination of flexibility and interpretability. However, for online learning, the field lacks an efficient method suitable for scenarios where prior information regarding data distribution and model function is limited. To address this issue, this paper proposes a recursive GPSSM method with adaptive capabilities for both operating domains and Gaussian process (GP) hyperparameters. Specifically, we first utilize first-order linearization to derive a Bayesian update equation for the joint distribution between the system state and the GP model, enabling closed-form and domain-independent learning. Second, an online selection algorithm for inducing points is developed based on informative criteria to achieve lightweight learning. Third, to support online hyperparameter optimization, we recover historical measurement information from the current filtering distribution. Comprehensive evaluations on both synthetic and real-world datasets demonstrate the superior accuracy, computational efficiency, and adaptability of our method compared to state-of-the-art online GPSSM techniques.

Figures (5)

• Figure 1: Schematic diagram of the RGPSSM algorithmic workflow.
• Figure 2: Online learning results for NASCAR$^\circledR$ dynamics. Top: filtered state trajectory; Bottom: filtering and prediction.
• Figure 3: Evolution of the GP hyperparameters during RGPSSM learning
• Figure 4: Evolution of the system states and their estimation, where the initial values of the GP hyperparameters are $l_1 = l_2 = 5$ and $\sigma_1^2 = 10$. The shadows indicate 95% confidence intervals.
• Figure 5: Evolution of inducing-point set and prediction error over time.

Theorems & Definitions (2)

• Definition 1: First-order linearization
• Remark 4.1