The solutions of Hamiltonian equations are known to describe the underlying phase space of a mechanical system. In this article, we propose a novel spatio-temporal model using a strategic modification of the Hamiltonian equations, incorporating appropriate stochasticity via Gaussian processes. The resultant spatio-temporal process, continuously varying with time, turns out to be nonparametric, non-stationary, non-separable, and non-Gaussian. Additionally, the lagged correlations converge to zero as the spatio-temporal lag goes to infinity. We investigate the theoretical properties of the new spatio-temporal process, including its continuity and smoothness properties. We derive methods for complete Bayesian inference using MCMC techniques in the Bayesian paradigm. The performance of our method has been compared with that of a non-stationary Gaussian process (GP) using two simulation studies, where our method shows a significant improvement over the non-stationary GP. Further, applying our new model to two real data sets revealed encouraging performance.

翻译：众所周知，哈密顿方程的解描述了力学系统的基本相空间。本文提出了一种新颖的时空模型，通过对哈密顿方程进行策略性修改，并借助高斯过程引入适当的随机性。由此产生的时空过程随时间连续变化，具有非参数、非平稳、非可分离及非高斯的特性。此外，其时滞相关性随着时空滞后趋于无穷大而收敛于零。我们研究了这一新型时空过程的理论性质，包括其连续性与光滑性。在贝叶斯范式下，我们推导了利用MCMC技术进行完全贝叶斯推断的方法。通过两项模拟研究，我们将所提方法与一种非平稳高斯过程（GP）进行了性能比较，结果显示我们的方法较非平稳GP有显著提升。进一步地，将新模型应用于两个真实数据集亦展现出令人鼓舞的性能。