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.

Multivariate data is prevalent in modern applications, yet it often presents significant analytical challenges. Factor models can offer an effective tool to address issues associated with large-scale datasets. In this dissertation, we propose two novel Bayesian factors models. These models are designed to effectively reduce the dimensionality of the data, as the number of latent factors is typically much smaller than that of the observation vectors. Therefore, our proposed models can achieve substantial dimension reduction. Our first model is for spatiotemporal areal data. In this case, the region of interest is divided into subregions, and at each time point, there is one univariate observation per subregion. Our model writes the vector of observations at each time point in a factor model form as the product of a vector of factor loadings and a vector of common factors plus a vector of error. Our model assumes that the common factor evolves through time according to a dynamic linear model. To represent the spatial relationships among subregions, each column of the factor loadings matrix is assigned intrinsic conditional autoregressive (ICAR) priors. Therefore, we call our approach the Dynamic ICAR Spatiotemporal Factor Models (DIFM). Our second model, Bayesian Clustering Factor Model (BCFM) assumes latent factors and clusters are present in the data. We apply Gaussian mixture models on common factors to discover clusters. For both models, we develop MCMC to explore the posterior distribution of the parameters. To select the number of factors and, in the case of clustering methods, the number of clusters, we develop model selection criteria that utilize the Laplace-Metropolis estimator of the predictive density and BIC with integrated likelihood. / Doctor of Philosophy / Understanding large-scale datasets has emerged as one of the most significant challenges for researchers recently. This is particularly true for datasets that are inherently complex and nontrivial to analyze. In this dissertation, we present two novel classes of Bayesian factor models for two classes of complex datasets. Frequently, the number of factors is much smaller than the number of variables, and therefore factor models can be an effective approach to handle multivariate datasets. First, we develop Dynamic ICAR Spatiotemporal Factor Model (DIFM) for datasets collected on a partition of a spatial domain of interest over time. The DIFM accounts for the spatiotemporal correlation and provides predictions of future trends. Second, we develop Bayesian Clustering Factor Model (BCFM) for multivariate data that cluster in a space of dimension lower than the dimension of the vector of observations. BCFM enables researchers to identify different characteristics of the subgroups, offering valuable insights into their underlying structure.

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.