Statistical modeling and inference for spatial and spatio-temporal data - Liu, Jialuo, author; Wang, Haonan, advisor; Breidt, F. Jay, committee member; Kokoszka, Piotr S., committee member; Luo, Rockey J., committee member (Colorado State University. Libraries, 2019)
Spatio-temporal processes with a continuous index in space and time are encountered in many scientific disciplines such as climatology, environmental sciences, and public health. A fundamental component for modeling such spatio-temporal processes is the covariance function, which is traditionally assumed to be stationary. While convenient, this stationarity assumption can be unrealistic in many situations. In the first part of this dissertation, we develop a new class of locally stationary spatio-temporal covariance functions. A novel spatio-temporal expanding distance (STED) asymptotic framework is proposed to study the properties of statistical inference. The STED asymptotic framework is established on a fixed spatio-temporal domain, aiming to characterize spatio-temporal processes that are globally nonstationary in a rescaled fixed domain and locally stationary in a distance expanding domain. The utility of STED is illustrated by establishing the asymptotic properties of the maximum likelihood estimation for a general class of spatio-temporal covariance functions, as well as a simulation study which suggests sound finite-sample properties. Then, we address the problem of simultaneous estimation of the mean and covariance functions for continuously indexed spatio-temporal processes. A flexible spatio-temporal model with partially linear regression in the mean function and local stationarity in the covariance function is proposed. We study a profile likelihood method for estimation in the presence of spatio-temporally correlated errors. Specifically, for the nonparametric component, we employ a family of bimodal kernels to alleviate bias, which may be of independent interest for semiparametric spatial statistics. The theoretical properties of our profile likelihood estimation, including consistency and asymptotic normality, are established. A simulation study is conducted and corroborates our theoretical findings, while a health hazard data example further illustrates the methodology. Maximum likelihood method for irregularly spaced spatial datasets is computationally intensive, as it involves the manipulation of sizable dense covariance matrices. Finding the exact likelihood is generally impractical, especially for large datasets. In the third part, we present an approximation to the Gaussian log-likelihood function using Krylov subspace methods. This method reduces the computational complexity from O(N³) operations to O(N²) for dense matrices and further to quasi-linear if matrices are sparse. Specifically, we implement the conjugate gradient method to solve linear systems iteratively and use Monte Carlo method and Gauss quadrature rule to obtain a stochastic estimator of the log-determinant. We give conditions to ensure consistency of the estimators. Simulation studies have been conducted to explore various important computational aspects including complexity, accuracy and efficiency. We also apply our proposed method to estimate the spatial structure of a big LiDAR dataset.
Keywords: random fields; spatial statistics; Gaussian process; spatio-temporal statistics; semiparametric modeling