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A New Spatio-Temporal Model Exploiting Hamiltonian Equations

Satyaki Mazumder, Sayantan Banerjee, Sourabh Bhattacharya

TL;DR

A novel spatio-temporal model is proposed using a strategic modification of the Hamiltonian equations, incorporating appropriate stochasticity via Gaussian processes, which turns out to be nonparametric, nonstationary, nonseparable and non-Gaussian.

Abstract

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.

A New Spatio-Temporal Model Exploiting Hamiltonian Equations

TL;DR

A novel spatio-temporal model is proposed using a strategic modification of the Hamiltonian equations, incorporating appropriate stochasticity via Gaussian processes, which turns out to be nonparametric, nonstationary, nonseparable and non-Gaussian.

Abstract

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.
Paper Structure (61 sections, 105 equations, 20 figures, 4 tables)

This paper contains 61 sections, 105 equations, 20 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Modified Hamiltonian equations and the proposed model
  3. Modification, the new leap-frog algorithm and the proposed model
  4. Remaining specification and analysis of the proposed model
  5. Connection to traditional Hamiltonian systems with Gaussian Process potentials
  6. Connection to Modified Parameterized Leapfrog HMC
  7. Calculation of likelihood functions
  8. Joint conditional density of the observed data
  9. Joint conditional density of latent data
  10. Complete likelihood combining observed and latent data
  11. Prior distributions
  12. Full conditional distributions
  13. Simulation studies
  14. Mixture of three Gaussian Processes
  15. Data generation
  16. ...and 46 more sections

Figures (20)

  • Figure 1: Higher the intensity of the color higher is the density. Black stars denote the true values. Most of the true values lie in the high density regions.
  • Figure 2: Higher the intensity of the color higher is the density. Black stars denote the true values. Most of the true values lie in the high density regions.
  • Figure 3: The map of Alaska and its surroundings. The red dots indicate the locations at which the data for the year 1950-2015 are considered. The positions indicated by the blue dots are those at which spatial predictions are made for the complete time series.
  • Figure 4: The highest probability density regions of the predictive densities and the true values of detrended temperatures (Alaska temperature data) for the year 2015 at 26 locations are depicted in this figure. The x-axis of of the figure indicates the locations, and at each location, color-coded densities are plotted in the vertical axis. The intensity of the color is proportional to the density. The true values are indicated by the black stars. Majority of the true values fall in the high density regions.
  • Figure 5: The highest probability density regions of the predictive densities and the true values of detrended temperatures (Alaska temperature data) for the years 2016 to 2021 at 16 locations are depicted in this figure. The x-axis of the figure indicates the locations, and at each location, color-coded densities are plotted in the vertical axis. The intensity of the color is proportional to the density. The true values are indicated by the black stars. Majority of the true values for all the 6 years fall in the high density regions.
  • ...and 15 more figures

Theorems & Definitions (8)

  • Remark 1
  • Remark 2
  • Definition 1: Definition of $M_s$
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7