Nonstationary, Nonparametric, Nonseparable Bayesian Spatio-Temporal Modeling Using Kernel Convolution of Order Based Dependent Dirichlet Process

TL;DR

This paper introduces a flexible Bayesian spatio-temporal model using kernel convolution of dependent Dirichlet processes, capable of capturing complex nonstationary and nonseparable dependencies, with efficient inference via transdimensional MCMC.

Contribution

It develops a novel nonstationary, nonparametric Bayesian model for space-time data using kernel convolution of dependent Dirichlet processes, addressing infinite series truncation and variable dimension inference.

Findings

Model outperforms previous methods on simulated data.

Effective in fitting real spatial and spatio-temporal datasets.

Provides theoretical bounds on series truncation error.

Abstract

In this article, using kernel convolution of order based dependent Dirichlet process (Griffin and Steel (2006)) we construct a nonstationary, nonseparable, nonparametric space-time process, which, as we show, satisfies desirable properties, and includes the stationary, separable, parametric processes as special cases. We also investigate the smoothness properties of our proposed model. Since our model entails an infinite random series, for Bayesian model fitting purpose we must either truncate the series or more appropriately consider a random number of summands, which renders the model dimension a random variable. We attack the variable dimensionality problem using Transdimensional Transformation based Markov Chain Monte Carlo introduced by Das and Bhattacharya (2019b), which can update all the variables and also change dimensions in a single block using essentially a single random variable drawn from some arbitrary density defined on a relevant support. For the sake of completeness we also address the problem of truncating the infinite series by providing a uniform bound on the error incurred by truncating the infinite series. We illustrate the effectiveness of our model and methodologies on a simulated data set and demonstrate that our approach significantly outperforms that of Fuentes and Reich (2013) which is based on principles somewhat similar to ours. We also fit two real, spatial and spatio-temporal datasets with our approach and obtain quite encouraging results in both the cases.