Sparse Simulation of Autoregressive Gaussian Processes

This study proposes a novel and improved numerical approximation of the simulation of Gaussian process autoregressive models. As a Bayesian nonparametric regression method, Gaussian process models offer the unique advantage of providing closed-form uncertainty quantification. When Gaussian process models are used for autoregressive models, the validation procedure requires the model’s simulation or multi-step-ahead prediction. However, simulating dynamical Gaussian process models is complex due to the intractable propagation of uncertain inputs through the nonlinear model. Numerical approximation, namely Monte Carlo simulation, is one of the most frequent options for simulating dynamical models based on Gaussian processes. The computational burden of Monte Carlo simulation algorithms increases cubically with data size, representing a challenge. This paper introduces a unified simulation framework invariant to sparse and variational approximations to obtain a static sample from the pseudo-point posterior. Furthermore, we propose an innovative method for simulating Gaussian process dynamical models. A single parameter is proposed to regulate the trade-off between computational complexity and algorithmic accuracy. This innovation demonstrates the potential to replace the conditionally independent Monte Carlo method with no additional computational burden, thereby enhancing estimates of latent responses. The proposed simulation method is demonstrated using two synthetic examples and a realistic case study.