In the analysis of spatio-temporal data, statistical inference based on the assumption that the data follow a Gaussian distribution is ubiquitous due to its many attractive properties. However, data collected from different fields of science, such as meteorology, astronomy, biology and medicine, rarely meet the assumption of Gaussianity. One option is to apply a monotonic transformation to the data such that the transformed data have a distribution that is close to Gaussian. In this thesis, we focus on a flexible two-parameter family of transformations, the Tukey g-and-h (TGH) transformation. This family of transformations has the desirable properties that the two parameters g∈T and h≥0 involved control skewness and tail-heaviness of the distribution, respectively. Applying the TGH transformation to a standard univariate normal distribution results in the univariate TGH distribution. Extensions to the multivariate case and to a spatial process were developed recently. In this thesis, motivated by the need to exploit wind as renewable energy, we tackle the challenges of modeling big spatio-temporal data that are non-Gaussian by applying the TGH transformation to different types of Gaussian processes: spatial (random field), temporal (time series), spatio-temporal, and their multivariate extensions. We explore various aspects of spatio-temporal data modeling techniques using transformed Gaussian processes with the TGH transformation. First, we use the TGH transformation to generate non-Gaussian spatial data with the Matérn covariance function, and study the effect of non-Gaussianity on Gaussian likelihood inference for the parameters in the Matérn covariance function via a sophisticatedly designed simulation study. Second, we build two autoregressive time series models using the TGH transformation. One of the models is applied to a data set of observational wind speeds and shows advantaged in accurate forecasting; the other model is used to fit wind speed data from a climate model on a grid covering Saudi Arabia and to Gaussianize the data for each location separately. Third, we develop a parsimonious spatio-temporal model for time series data on a spatial grid and utilize the aforementioned Gaussianized climate model wind speed data to fit the latent Gaussian spatio-temporal process. Finally, we discuss issues under a unified framework of modeling multivariate trans-Gaussian processes and adopt one of the TGH autoregressive models to build a stochastic generator for global wind speed.
Biography: Yuan Yan is PhD student of Prof. Marc Genton in the Spatio-Temporal Statistics and Data Science group at KAUST. She received her M.Sc. in Statistics from KAUST in 2014 and Bachelor's degree in Computing Mathematics from the City University of Hong Kong in 2012. Her research interests include spatio-temporal statistics and functional data analysis with current research focusing on transformed Gaussian processes.