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底层相空间。本文提出了一种新颖的时空模型，该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，被证明是非参数、非平稳、非可分离且非高斯的。此外，当时空滞后趋于无穷大时，其滞后相关性收敛于零。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。我们在贝叶斯范式下，推导了使用MCMC技术进行完整贝叶斯推断的方法。通过两项模拟研究，我们将所提方法的性能与非平稳高斯过程进行了比较，结果表明我们的方法相较于非平稳高斯过程有显著提升。此外，将我们的新模型应用于两个真实数据集也展现了令人鼓舞的性能。