Resumen de Spatio-temporal modeling of environmental processes derived from the economic activity

Gema Fernández-Avilés Calderón

• Spatial and spatio-temporal statistics have rapidly developed during the last twenty five years and nowadays can be considered a hot topic in the field of Statistics. In particular, the two last decades have witnessed an interesting progress both in theoretical developments and in practical studies.

  From the theoretical point of view, a hard work has been done to develop new non separable spatio-temporal covariance functions. Results from De Iaco, Cressie and Huang, Gneiting, Ma, Stein, and Porcu and Mateu, among others are a good example of the effort made in this area. But despite of the development of the discipline in the two last decades, there is still a long way to go. As an example, the usual approach in spatio-temporal Geostatistics continues to be based on the assumption that the spatial random field is stationary and isotropic; the justification being that with sparse data there is no reasonable alternative.

  This is the reason why in this doctoral thesis it has been compiled and extended a good collection on spatio-temporal and covariance and variogram models that can be used in a large range of real situations. R-codes have been developed for that models, so that there is no excuse for not using non-stationary or anisotropic covariance models in future research.

  Additionally, it is proposed a new compactly supported covariance model for the non stationary case.

  Regarding estimation matters, it is proposed Bevilacqua weighted composite likelihood method to estimate the parameters involved in covariance functions, due that it overcome the computational problems of the maximum likelihood and restricted likelihood procedures, and provides satisfactory results.

  As an alternative to the spatio-temporal modelling, we propose several functional strategies for spatio-temporal data. What is more, we have gone beyond the restrictive Goulard and Voltz (1993) approach (the functions are only known by a finite set of their points, and a parametric model was fitted to them for reconstructing the whole curve) and have assumed Giraldo (2009) perspective of the problem by carrying out a non-parametric fitting pre-process to the observed functions (in particular B-spline smoothing), where the smoothing parameter is chosen by Giraldo functional cross-validation.

  This new approach is in complete agreement with present trends in Functional Data Analysis, and in particular with non-parametric functional estimation methodology, and our proposal for doing kriging-based spatial prediction of random curves formally coincides with the functional kriging introduced in Goulard and Voltz. But the main novelty is that the non-parametric approach involves noticeable differences (for instance, data representation) and the additional problem of choosing the smoothing parameters (the keystone of non-parametric methods). The functional kriging predictor is based on the basic philosophy of functional data, that is, curves are single entities, rather than a sequence of individual observations.

  As for the practical part of the doctoral dissertation, it has been predicted by using spatial, spatio-temporal and functional strategies the level of carbon monoxide in Madrid City, the third most populous European city. As far as we know, it is the first time that these strategies have been used for predicting such a dangerous pollutant in Madrid. It must be said that levels of carbon monoxide is one of the problems that most worry Madrid citizens and authorities and that it is closely related to the traffic and economic activity.

  Surprisingly, functional strategies provide best predictions than the spatio-temporal ones, what encourages us to develop in depth this research line.