# Equilibrium Distributions and Stability Analysis of Gaussian Process State Space Models

arXiv: https://arxiv.org/abs/1811.06653 · HTML: https://mlsanity.com/paper/1811.06653/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Nov 2018.

## Built on by (most cited first, 4 of 4)

- [Almost Surely Stable Deep Dynamics](https://mlsanity.com/paper/2103.14722/) (2021)
- [Gaussian Process Port-Hamiltonian Systems: Bayesian Learning with Physics Prior](https://mlsanity.com/paper/2305.09017/) (2023)
- [Towards safe Bayesian optimization with Wiener kernel regression](https://mlsanity.com/paper/2411.02253/) (2024)
- [Wiener Chaos in Kernel Regression: Towards Untangling Aleatoric and Epistemic Uncertainty](https://mlsanity.com/paper/2312.07387/) (2023)

## Abstract (arXiv)

Gaussian Process State Space Models (GP-SSM) are a data-driven stochastic model class suitable to represent nonlinear dynamics. They have become increasingly popular in non-parametric modeling approaches since they provide not only a prediction of the system behavior but also an accuracy of the prediction. For the application of these models, the analysis of fundamental system properties is required. In this paper, we analyze equilibrium distributions and stability properties of the GP-SSM. The computation of equilibrium distributions is based on the numerical solution of a Fredholm integral equation of the second kind and is suitable for any covariance function. Besides, we show that the GP-SSM with squared exponential covariance function is always mean square bounded and there exists a set which is positive recurrent.