| Abstract: | In the field of spatial statistics, it is often desirable to generate a smooth surface for a region over which only noisy observations of the surface are available at some locations, or even across time. Kriging and kernel estimations are two of the most popular methods. However, these two methods become problematic when the domain is not regular, such as when it is rectangular or convex. Bivariate B-splines developed by mathematicians provide a useful nonparametric tool in bivariate surface modeling. They inherit several appealing properties of univariate B-splines and are applicable in various modeling problems. More importantly, bivariate B-splines have advantages over kriging and kernel estimation when dealing with complicated domains. The purpose of this dissertation is to develop a nonparametric surface fitting method by using bivariate B-splines that can handle complex spatial domains. The dissertation consists of four parts. The first part of this dissertation explains the challenges of smoothing over complicated domains and reviews existing methods. The second part introduces bivariate B-splines and explains its properties and implementation techniques. The third and fourth parts discuss application of the bivariate B-splines in two nonparametric spatial surface fitting problems. In particular, the third part develops a penalized B-splines method to reconstruct a smooth surface from noisy observations. A numerical algorithm is derived, implemented, and applied to simulated and real data. The fourth part develops a reduced rank mixed-effects model for functional principal components analysis of sparsely observed spatial data. A numerical algorithm is used to implement the method and tested on simulated and real data. The electronic version of this dissertation is accessible from http://hdl.handle.net/1969.1/150949 |
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