In the analysis of spatio-temporal data, statistical inference based on the Gaussian assumption is ubiquitous due to its many attractive properties. However, data collected from different fields of science 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 has the desirable properties that the two parameters g ∈ R and h ≥ 0 involved control skewness and tail-heaviness of the distribution, respectively. Applying the TGH transformation to a standard 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 via a sophisticatedly designed simulation study. Second, we build two autoregressive time series models using the TGH transformation. One model is applied to a dataset 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 gridded locations covering Saudi Arabia and to Gaussianize the data for each location. 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.

Real-time sensing brings the proliferation of big data that contains rich information of complex systems. It is well known that real-world systems show high levels of nonlinear and nonstationary behaviors in the presence of extraneous noise. This brings significant challenges for human experts to visually inspect the integrity and performance of complex systems from the collected data. My research goal is to develop innovative methodologies for modeling and optimizing complex systems, and create enabling technologies for real-world applications. Specifically, my research focuses on Mining Dynamic Recurrences in Nonlinear and Nonstationary Systems for Feature Extraction, Process Monitoring and Fault Diagnosis. This research will enable and assist in (i) sensor-driven modeling, monitoring and optimization of complex systems; (ii) integrating product design with system design of nonlinear dynamic processes; and (iii) creating better prediction/diagnostic tools for real-world complex processes. My research accomplishments include the following. (1) Feature Extraction and Analysis: I proposed a novel multiscale recurrence analysis to not only delineate recurrence dynamics in complex systems, but also resolve the computational issues for the large-scale datasets. It was utilized to identify heart failure subjects from the 24-hour heart rate variability (HRV) time series and control the quality of mobile-phone-based electrocardiogram (ECG) signals. (2) Modeling and Prediction: I proposed the design of stochastic sensor network to allow a subset of sensors at varying locations within the network to transmit dynamic information intermittently, and a new approach of sparse particle filtering to model spatiotemporal dynamics of big data in the stochastic sensor network. It may be noted that the proposed algorithm is very general and can be potentially applicable for stochastic sensor networks in a variety of disciplines, e.g., environmental sensor network and battlefield surveillance network. (3) Monitoring and Control: Process monitoring of dynamic transitions in complex systems is more concerned with aperiodic recurrences and heterogeneous types of recurrence variations. However, traditional recurrence analysis treats all recurrence states homogeneously, thereby failing to delineate heterogeneous recurrence patterns. I developed a new approach of heterogeneous recurrence analysis for complex systems informatics, process monitoring and anomaly detection. (4) Simulation and Optimization: Another research focuses on fractal-based simulation to study spatiotemporal dynamics on fractal surfaces of high-dimensional complex systems, and further optimize spatiotemporal patterns. This proposed algorithm is applied to study the reaction-diffusion modeling on fractal surfaces and real-world 3D heart surfaces.