Fitting of spatio-temporal empirical covariograms: An application to air pollution data

Modelling spatio-temporal dependencies resulting from dynamic processes that evolve in both space and time is essential in many scientific fields. Spatio-temporal kriging is one of the space-time procedures that has progressed the most over the last few years. Kriging predictions strongly depend on the covariance function associated to the stochastic process under study. Therefore, the choice of such a covariance function, which is usually based on empirical covariance, is a core aspect in the prediction procedure. As empirical covariance is not necessarily a permissible covariance function, it is necessary to fit a valid covariance model. Due to the complexity of these valid models in the spatio-temporal case, visualising them is of great help, at least when selecting the set of candidate models to represent the spatio-temporal dependencies suggested by the empirical covariogram.

                In this article we focus on the visualisation of stationary non-separable covariance functions and how they change as their main parameters take different values. More specifically, we centre our attention on the pioneer sum-product model, the models based on mixed forms, a classical Cressie-Huang model, a Gneiting model based on completely monotone and Bernstein functions and an innovative proposal that allows for negative values. We use proprietary codes for visualisation purposes. In order to illustrate the usefulness of visualisation when choosing the appropriate non-separable spatio-temporal covariance model, we focus on an important pollution problem, namely the level of carbon monoxide, in one of the largest cities in Europe, Madrid, a city with one of the best monitoring nets in the world.