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.

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