Jacob Shields is the first Ph.D. student to earn his doctorate degree in the Department of Statistics at Indiana University. The degree was introduced in 2010. Jacob joined the program in Fall 2012. He presented his dissertation defense (title and abstract below) on November 15, 2017. He was hooded by his advisor, Professor Chunfeng Huang, during the commencement ceremony on December 16, 2017.
Intrinsic Random Functions on Spheres: Theory, Methods, and Application
Global-scale phenomena, of which there are a multitude of important applications ranging from climate science to epidemiology, can be viewed as random processes on the sphere. Many popular methods are based on the assumption that the underlying process is stationary, requiring rotation invariance. This assumption is often deemed unrealistic in practice. In our research, we introduce a class of non-stationary processes, the intrinsic random functions (IRFs), on spheres. The idea behind IRFs is to find a class of non-stationary processes that is closely related to stationarity. We show that low-frequency truncation plays an essential role. Based on this, we develop methods to estimate the degree of non-stationarity and associated covariance parameters for processes on the sphere, which are demonstrated through simulation studies. In addition, IRF based Kriging is constructed and shown to be equivalent to the spline smoothing formula in the corresponding reproducing kernel Hilbert space. Finally, we apply these methods through the analysis of temperature anomaly data.