The solutions of Hamiltonian equations are known to describe the underlying phase space of the mechanical system. Hamiltonian Monte Carlo is the sole use of the properties of solutions to the Hamiltonian equations in Bayesian statistics. 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 sptaio-temporal process, continuously varying with time, turns out to be nonparametric, nonstationary, nonseparable and non-Gaussian. Additionally, as the spatio-temporal lag goes to infinity, the lagged correlations converge to zero. We investigate the theoretical properties of the new spatio-temporal process, including its continuity and smoothness properties. In the Bayesian paradigm, we derive methods for complete Bayesian inference using MCMC techniques. The performance of our method has been compared with that of non-stationary Gaussian process (GP) using two simulation studies, where our method shows a significant improvement over the non-stationary GP. Further, application of our new model to two real data sets revealed encouraging performance.

翻译：哈密顿方程的解已知描述了机械系统的基础相空间。哈密顿蒙特卡洛在贝叶斯统计中仅利用了哈密顿方程解的性质。本文通过策略性地修改哈密顿方程，并利用高斯过程引入适当随机性，提出了一种新颖的时空模型。由此得到的随时间连续变化的时空过程表现为非参数、非平稳、不可分离且非高斯。此外，当时空滞后趋于无穷时，滞后相关性收敛于零。我们研究了这一新时空过程的理论性质，包括其连续性和光滑性。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，将我们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比非平稳高斯过程有显著改进。此外，将新模型应用于两个真实数据集，显示出令人鼓舞的性能。