Flexible spatiotemporal models for environmental processes

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Environmental statistics methodology is concerned with diverse global measurements of natural phenomena. Some examples of problems which can be of interest in this field are: the modeling of mean concentrations of pollutants, the study of relationships between airborne pollution and lung diseases, and spatiotemporal maximum temperature trends over a region. Environmental processes commonly involve observations made at fixed locations across different instants in time. My research programme focuses on geostatistical processes, i.e. uni- and multivariate processes observed at locations that vary continuously across a region of interest. The main interest in geostatistics lies in predicting the process of interest at future times (temporal prediction) and at unobserved locations in space (spatial interpolation), while accounting for complex correlation structures. In the analysis of most spatiotemporal processes in environmental studies, observations present skewed distributions. Typically a single transformation of the data is used to approximate normality, and a stationary and isotropic (invariant under translation and rotation about the origin) Gaussian process (GP) is fitted to the transformed data. However, it can be shown that commonly-used transformations (e.g. log and square-root) induce non-stationarity in the data when considered on the original scale. Therefore, the use of transformations should be avoided as some sort of non-stationarity is induced to the original process. A key focus of my research programme over the next five years will be to develop models for uni- and multivariate spatiotemporal processes that do not require transformation of the data, but rather can be applied to the data on their original scale. I will investigate mixtures of distributions to model processes that show skewness and kurtosis greater than those of the skew normal distribution. Theoretical properties of the proposed models, such as the resultant covariance structure, kurtosis, and skewness, will be derived. These models will involve high-dimensional processes; I will therefore incorporate dimension-reduction techniques to the proposed models for large datasets. Inference will be performed under the Bayesian paradigm, and Markov chain Monte Carlo (MCMC) methods will be used to obtain samples from the posterior distribution. Packages in R will be made available to assist in the dissemination and implementation of the proposed models to a wide audience.