Title: Topics in Spatial Dependence Functions: nonparametric variogram estimation, circular Matérn covariance function and Gaussian Markov random field approximation.
Spatial statistics plays an important role in various fields, where data are observed with location information, including environmental science, geography and epidemiology. Such data are usually modeled through spatial random process with dependence functions as the focus of the study. In this research, we propose new tools for exploring spatial dependence in the following directions. (1) A nonparametric valid variogram estimation is developed through an inversion of its spectral function. The proposed variogram estimator is guaranteed to be a conditionally negative definite function. (2) A circular Matérn covariance function on the circle is re-solved through a stochastic partial differential equation approach. A proper probability space is investigated, which is required for such construction of random processes on the circle. (3) The connection between Gaussian random field (GF) and Gaussian Markov random field (GMRF) on the circle is studied and we propose the approximation of GF by the GMRF. This will reduce the computational complexity.