Lagrangian Spatio-Temporal Covariance Functions for Multivariate Nonstationary Random Fields

Abstract

The modeling of spatio-temporal and multivariate spatial random fields has been an important and growing area of research due to the increasing availability of spacetime-referenced data in a large number of scientific applications. In geostatistics, the covariance function plays a crucial role in describing the spatio-temporal dependence in the data and is key to statistical modeling, inference, stochastic simulation and prediction. Therefore, the development of flexible covariance models, which can accomodate the inherent variability of the real data, is necessary for an advantageous modeling of random fields. This thesis is composed of four significant contributions in the development and applications of new covariance models for stationary multivariate spatial processes, and nonstationary spatial and spatio-temporal processes. The first focus of the thesis is on modeling of stationary multivariate spatial random fields through flexible multivariate covariance functions. Chapter 2 proposes a semiparametric approach for multivariate covariance function estimation with flexible specification of the cross-covariance functions via their spectral representations. The proposed method is applied to model and predict the bivariate data of particulate matter concentration (PM2.5) and wind speed (WS) in the United States. Chapter 3 introduces a parametric class of multivariate covariance functions with asymmetric cross-covariance functions. The proposed covariance model is applied to analyze the asymmetry and perform prediction in a trivariate data of PM2.5, WS and relative humidity (RH) in the United States. The second focus of the thesis is on nonstationary spatial and spatio-temporal random fields. Chapter 4 presents a space deformation method which imparts nonstationarity to any stationary covariance function. The proposed method utilizes the functional data registration algorithm and classical multidimensional scaling to estimate the spatial deformation. The application of the proposed method is demonstrated on a precipitation data. Finally, chapter 5 proposes a parametric class of time-varying spatio-temporal covariance functions, which are nonstationary in time. The proposed class is a time-varying generalization of an existing nonseparable stationary class of spatio-temporal covariance functions. The proposed time-varying model is then used to study the seasonality effect and perform space-time predictions in the daily PM2.5 data from Oregon, United States.