Novel methodologies in multivariate spatial statistics

Multivariate data indexed by spatial coordinates has become ubiquitous in climate, environmental and geo-sciences. This has promoted sustained interest in modeling, estimation and simulation of spatial stochastic processes through multivariate Gaussian random fields. The goal of this dissertation is to advance the theory and methods used in multivariate spatial and spatio-temporal statistics. In particular through the novel constructions of p-variate covariance functions, spatio-temporal frameworks for modeling natural phenomena in the cryospheric sciences and fast simulation of large vector-valued spatial Gaussian random fields. Chapter 1 reviews current methods and important concepts in spatial statistics. Relevant terminology and theorems are introduced in the context of univariate, multivariate and spatio-temporal random fields that are used in subsequent chapters. Two univariate covariance functions that are given particular attention are the generalized Cauchy and Matern as their multivariate forms arise in Chapters 2 and 3. Chapter 1 concludes with a discussion of open problems in the field. Chapter 2 focuses on modeling, estimation and prediction of vector-valued Gaussian random fields which exhibit long range dependence in space. In particular, a novel multivariate construction of the generalized Cauchy covariance is presented. Each process component belongs to the generalized Cauchy class of covariance functions which may accommodate both short and long range dependent processes. The quality of the proposed model is compared through predictive error metrics to the celebrated multivariate Matern in several simulation studies and a real-world trivariate climatological data set. Our proposed model is shown to have superior performance. The concluding section includes a discussion of results and future research directions. In Chapter 3 a geostatistical framework for modeling complex spatio-temporal processes is introduced. The modeling framework accounts for both macro-trends and micro-ergodic behavior in space and time. This framework finds particular resonance in cryospheric sciences where access to monitoring sites severely limits data collection in the Arctic regions; it is applied to the real-world problem of accurately modeling and forecasting Active Layer Thickness in sub-arctic regions. Several models of increasing complexity are proposed and compared based on predictive error metrics. Significant improvements in time-forward prediction for the various error metrics are achieved through the utilization of nonseparable spatio-temporal models over naive methods. Chapter 4 focuses on fast and exact simulation of isotropic multivariate Gaussian random fields through multivariate circulant embedding. Compared to standard simulation methods the proposed approach significantly reduces computational complexity. This allows one to simulate large multivariate Gaussian random fields in linearithmic rather than cubic time. A two-part synthesis process is proposed where embedding and simulation phases are decoupled. Examples of simulated bivariate Gaussian random fields are included for common covariance structures. Computational complexity and future directions are further discussed in the concluding sections of the chapter.