---
title: Gaussian Process State-Space Models
url: https://www.emergentmind.com/topics/gaussian-process-state-space-models-gpssms
type: topic
---

# Gaussian Process State-Space Models

Gaussian Process State-Space Models (GPSSMs) are a flexible, nonparametric framework for learning, inference, and prediction in nonlinear dynamical systems. They combine the expressive power of Gaussian processes (GPs) for capturing unknown system dynamics with the probabilistic state-space modeling paradigm, enabling both uncertainty quantification and principled Bayesian reasoning across a range of real-world and synthetic application domains.

## 1. Model Structure and Foundational Concepts

A GPSSM models the evolution of a latent state $x_t$ via an unknown, possibly nonlinear, Markov transition:
\[
x_{t+1} = f(x_t) + v_t, \qquad y_t = g(x_t) + e_t
\]
where $f$ is modeled as a draw from a GP prior, $v_t$ and $e_t$ denote process and measurement noise, and $g$ is a measurement function, which may also be modeled as a GP. The GP prior,
\[
f \sim \mathcal{GP}(m(x), k(x, x')),
\]
imposes smoothness (or other structural) assumptions via the choice of kernel $k(\cdot, \cdot)$. This nonparametric prior enables GPSSMs to represent highly flexible, data-adaptive dynamics, as opposed to fixed parametric models.

Most classical and modern GPSSM formulations assume conditional independence between transition outputs. More recent work extends this to multi-output, output-dependent GPs using constructions such as the Linear Model of Coregionalization (LMC), enabling the model to capture dependencies between the channels of the transition vector [2212.07608].

## 2. Inference Methods: Marginalization, Variational, and Sampling Approaches

Inference in GPSSMs is challenging due to the strong temporal dependencies between the latent states and the high-dimensional, infinite-dimensional GP function space. The following taxonomy captures the main approaches:

- **Marginalization over the GP function**: For some formulations, $f$ can be marginalized analytically, yielding effective non-Markovian priors over state trajectories and reducing function inference to state trajectory sampling [1312.4852, 1907.06066].
- **Stochastic Approximation EM**: The particle stochastic approximation EM (PSAEM) framework combines a stochastic (noisy) EM auxiliary function with particle MCMC, specifically the particle Gibbs with ancestor sampling (PGAS), for efficient smoothing and M-step parameter updates [1312.4852].
- **Variational Inference**: The dominant approach in recent work, using either mean-field or more complex structured posteriors over the latent states, GP function values, and (optionally) variational distributions over inducing points to manage computational cost [1406.4905, 1705.10888, 1812.03580, 2302.09921].
- **Inducing Points and Sparse GP Approximations**: Inducing point variables $\mathbf{u}$ are introduced so that $p(f)$ can be factored through finite-dimensional Gaussian distributions over $u$; this makes inference and optimization scalable to long time series [1406.4905, 1812.03580, 2312.05910].
- **Particle MCMC and Sequential Monte Carlo**: When variational inference is infeasible or inaccurate (due to multimodal or highly non-Gaussian posteriors), PMCMC approaches such as particle Gibbs are used to target the joint smoothing distribution [1312.4852].

Tabular summary of prominent approaches:
| Inference Approach         | Key Elements                               | References               |
|---------------------------|--------------------------------------------|--------------------------|
| Particle EM (PSAEM + PGAS)| stochastic EM, PMCMC smoothing             | [1312.4852]              |
| Mean-field VI             | factorized q(x), stochastic gradient ascent| [1406.4905, 1812.03580]  |
| Free-form VI (SGHMC)      | non-factorized q(x, u), SGHMC sampling     | [2302.09921]             |
| EnKF-aided VI             | non-mean-field, recursive, EnKF            | [2312.05910, 2411.14679] |
| Recursive/Moment-matching | EKF/UKF/ADF, inducing management           | [2411.14679, 2510.15390] |

## 3. Model Extensions: Expressivity, Multi-output, and Nonstationarity

Standard GPSSMs, while highly flexible in the 1D case, suffer from limited expressivity in high-dimensional or nonstationary settings. Recent research has introduced several enhancements:

- **Transformed GPSSMs (TGPSSM)** use parametric, invertible normalizing flows $\mathcal{G}_F$ to “push forward” the GP prior, creating a more expressive stochastic process:
  \[
  \tilde{f}(x) = \mathcal{G}_F(f(x))
  \]
  which can represent non-Gaussian marginals and handle sharp transitions or heavy-tailed dynamics [2301.08843, 2309.01074, 2503.18309].

- **Efficient Transformed GPSSMs (ETGPSSM)** further reduce complexity by coupling a single GP with dimension-specific or input-dependent flows, rather than independent GPs for each output. The resulting architecture both captures output correlation and significantly reduces parameter count and computational cost [2309.01074, 2503.18309].

- **Heterogeneous Multi-output Kernels**: To model multi-channel dynamics where each channel may exhibit different lengthscales, structure, or inputs, a block-diagonal kernel matrix is constructed so that each output uses its own kernel, input mapping, and hyperparameters:
  \[
  \mathbf{K}_{f_1 f_2} = \text{diag}_{d=1}^D K_0^{(d)}(\mathcal{Z}_1^{(d)}, \mathcal{Z}_2^{(d)}; \theta^{(d)})
  \]
  and the inducing point management is carried out independently per output [2510.15390].

- **Multi-resolution and Multi-timescale GPSSMs**: By decomposing the latent state into components modeled at different time resolutions/scales (e.g., fast vs. slow dynamics), more realistic long-sequence inference becomes feasible [2112.03230].

## 4. Online and Recursive Learning, Inducing Point Management

GPSSMs have been extended to handle streaming, online data via recursive and online inference:

- **Recursive Updates with Inducing Point Selection**: Recursive GPSSMs maintain a joint distribution over the state and a dynamically managed inducing set $u$, performing a prediction-correction cycle via first-order linearization or higher-order moment matching (EKF, UKF, ADF) [2411.14679, 2510.15390]. Inducing points are added based on novelty (conditional variance) and pruned by minimizing KL divergence, executed independently per output dimension in heterogeneous kernel settings.
  
- **Online Hyperparameter Adaptation**: Hyperparameters are updated online by recovering historical measurement information from the current filtering distribution, ensuring fast adaptation to nonstationarity without storing the full data history [2411.14679].

- **Ensemble Kalman Filtering (EnKF) Integration**: Non-mean-field variational inference can be realized by using the EnKF to propagate state uncertainty: particles are propagated and updated using GP predictions and Kalman gain from the emission model, resulting in efficient, closed-form approximations to the variational posterior and ELBO. This is well-suited for online scenarios [2312.05910, 2503.18309].

## 5. Expressivity, Limitations, and Stability

While GPSSMs provide uncertainty quantification and flexible function learning, their representational capacity and stability depend on choices in kernel functions and modeling:

- **Expressivity Limitations**: Standard GPSSMs with stationary kernels (SE, Matérn) cannot capture nonstationary or non-Gaussian state transitions; transformed GPSSMs with normalizing flows or LMC-based output dependency overcome this constraint [2301.08843, 2212.07608].
- **Stability and Boundedness**: For SE kernels, deterministic GPSSMs are globally uniformly ultimately bounded and do not admit unbounded trajectories; they always possess at least one equilibrium and have stability properties that can be analyzed analytically [1811.06646, 1811.06653]. For stochastic GPSSMs, mean square boundedness and positive recurrence hold for SE kernels, but the expressivity is limited to bounded dynamics, which is a fundamental modeling assumption.
- **Active Learning and Out-of-Distribution Detection**: Mutual information–based active learning (AL) strategies for input selection significantly enhance data efficiency in dynamical system learning [2108.00819]. Embedding domain knowledge via informed kernels improves prediction under scarce data and supports online out-of-distribution detection, critical for safety in robotics [2309.06655].

## 6. Application Domains and Empirical Performance

GPSSMs are widely applied for:

- System identification in robotics, control, and aerospace (e.g., quadrotors, UAVs, hypersonic vehicles)
- Forecasting, time-series modeling, and simulation in finance, neuroscience, and engineering
- Adaptive and robust model predictive control with uncertainty quantification
- Online and real-time learning for changing environments

Empirical evaluations across synthetic kink-functions, chaotic Lorenz systems, real-world actuator dynamics, and atmospheric testbeds consistently show that recent recursive, output-dependent, and normalizing flow–based GPSSMs achieve superior learning accuracy and runtime efficiency, often matching state-of-the-art offline methods with orders of magnitude less runtime and scaling robustly to high-dimensional, noisy, and partially observed systems [2510.15390, 2503.18309, 2411.14679].

## 7. Mathematical and Algorithmic Summary

Central equations and concepts:
- **Marginal likelihood**:
  \[
  p(y_{0:T} | \theta) = \int p(y_{0:T} | x_{0:T}, \theta) p(x_{0:T} | \theta) dx_{0:T}
  \]
  where $x_{0:T}$ are latent states, $f$ is marginalized.
- **SAEM surrogate update** [1312.4852]:
  \[
  \hat{Q}_k(\theta) = (1-\gamma_k)\hat{Q}_{k-1}(\theta) + \gamma_k \log p(y_{0:T}, x_{0:T}^{[k]} | \theta)
  \]
- **Variational lower bound (ELBO)** [1406.4905, 2302.09921]:
  \[
  \mathcal{L} = \mathbb{E}_{q(\cdot)} \left[ \log\frac{p(y_{1:T}, x_{0:T}, f_{1:T}, u)}{q(x_{0:T}, f_{1:T}, u)} \right]
  \]
  with structured mean-field or free-form $q$.
- **Normalizing Flow–transformed GP prior** [2301.08843]:
  \[
  \tilde{f}(x) = \mathcal{G}_F(f(x)), \qquad \pi(\tilde{f}) = p(f) \prod_{j=1}^{J-1} |\det(\frac{\partial \mathcal{G}_{\theta_j}}{\partial \cdot})|^{-1}
  \]
- **EnKF state update** [2312.05910]:
  \[
  \begin{aligned}
  \bar{x}_t^{(n)} &=  f(x_{t-1}^{(n)}) + v_t^{(n)} \\
  G_t &= \bar{P}_t C^{\top}(C\bar{P}_tC^{\top} + R)^{-1} \\
  x_t^{(n)} &= \bar{x}_t^{(n)} + G_t(y_t + e_t^{(n)} - C\bar{x}_t^{(n)})
  \end{aligned}
  \]
- **Moment Matching for Heterogeneous Outputs** [2510.15390]:
  \[
  K_{f_1f_2} = \text{diag}_{d} K_0^{(d)}(\mathcal{Z}_1^{(d)}, \mathcal{Z}_2^{(d)};\theta^{(d)})
  \]

## References

- [1312.4852] Särkkä, Solin, & Hartikainen, "Identification of Gaussian Process State-Space Models with Particle Stochastic Approximation EM"
- [1406.4905] Frigola, Chen, & Rasmussen, "Variational Gaussian Process State-Space Models"
- [1705.10888] Doerr, Daniel, & Schölkopf, "Identification of Gaussian Process State Space Models"
- [1811.06646], [1811.06653] Beckers & Hirche, "Stability of Gaussian Process State Space Models"; "Equilibrium Distributions and Stability Analysis of GPSSMs"
- [1812.03580] Eleftheriadis et al., "Closed-form Inference and Prediction in GPSSMs"
- [2108.00819] Zhang et al., "Active Learning in Gaussian Process State Space Model"
- [2112.03230] Dohmatob et al., "Traversing Time with Multi-Resolution Gaussian Process State-Space Models"
- [2212.07608] Chen et al., "Output-Dependent Gaussian Process State-Space Model"
- [2301.08843] Lin et al., "Towards Flexibility and Interpretability of Gaussian Process State-Space Model"
- [2302.09921] Chang et al., "Free-Form Variational Inference for GPSSMs"
- [2309.01074], [2503.18309] Lin et al., "Towards Efficient Modeling and Inference in Multi-Dimensional GPSSMs"; "Efficient Transformed GPSSMs for Non-Stationary High-Dimensional Systems"
- [2312.05910] Chen et al., "Ensemble Kalman Filtering Meets GPSSM for Non-Mean-Field and Online Inference"
- [2411.14679] Lin et al., "Recursive Gaussian Process State Space Model"
- [2510.15390] Lin et al., "Recursive Inference for Heterogeneous Multi-Output GPSSMs with Arbitrary Moment Matching"
- [2309.06655] Piga et al., "Out of Distribution Detection via Domain-Informed GPSSMs"
- [1907.06066] Gustafsson, "The Use of Gaussian Processes in System Identification"

These models form the methodological backbone for modern nonlinear system identification where robustness, flexibility, and uncertainty estimates are essential.

Source: https://www.emergentmind.com/topics/gaussian-process-state-space-models-gpssms