Learning Nonlinear Dynamics: Improving the Estimation Efficiency and Reliability of Gaussian Process State-Space Models

2026-06-23

AI Summary

This study addresses the limitations of traditional state-space models, which rely on predefined nonlinear dynamics and struggle with theoretically under-specified complex systems, as well as the high computational cost of Bayesian inference in Gaussian process state-space models for moderately long sequences. To overcome these challenges, the authors propose two enhanced Gibbs sampling strategies that substantially improve sampling efficiency and convergence reliability. By integrating confirmatory factor analysis to construct an identifiable and interpretable measurement structure, they develop a comprehensive framework for learning nonlinear latent dynamical systems. Simulation studies validate the accuracy of posterior inference, while two empirical applications demonstrate the method’s practical utility and interpretability. An open-source implementation is provided, offering researchers an efficient and feasible workflow for empirical analysis.

Abstract

Understanding dynamic systems is a central goal in many scientific disciplines. State-space models provide a general framework for studying latent dynamic systems based on indirect observations. However, classical state-space methods require researchers to specify the parametric form of the system dynamics in advance, which can be challenging when the underlying processes are nonlinear and only partially explained by theory. Gaussian process state-space models address this by learning the system dynamics directly from data. However, estimating these models exactly can become computationally infeasible for moderately long time-series. In this paper, we therefore aim to improve the Bayesian estimation of approximate Gaussian process state-space models to make these models more accessible and facilitate the statistical learning of nonlinear dynamic systems in empirical research. To this end, we first propose two modifications to an existing Gibbs sampler for these models that considerably improve its sampling efficiency and convergence. Second, we use a confirmatory factor analysis measurement model, which reduces identifiability issues and allows researchers to impose a specific measurement structure on the model. Third, we provide a systematically validated software implementation of the model and sampler for applied use in empirical research. To validate the sampler, we conducted a simulation-based calibration which showed that the sampler converged reliably across many simulated data sets and produces well-calibrated posterior inferences. We further illustrate how the model can be applied and interpreted using two empirical examples. Together, these contributions provide a practical and validated workflow for learning nonlinear latent dynamics with Gaussian process state-space models.

Problem

• Gaussian process state-space models
• nonlinear dynamics
• Bayesian estimation
• computational efficiency
• latent dynamic systems

Innovation

• Gaussian process state-space models
• Bayesian inference
• Gibbs sampling
• confirmatory factor analysis
• nonlinear dynamics