Topics In Bivariate Spline Smoothing

Abstract

Penalized spline methods have been popular since the work of Eilers and Marx (1996). Recent years saw extensive theoretical studies and a wide range of applications of penalized splines. In this dissertation, we consider penalized splines for smoothing two-dimensional data. In Chapter 2, we propose a new spline smoother, the sandwich smoother, for smoothing data on a rectangular grid. Univariate P-spline smoothers are applied simultaneously along both coordinates. The sandwich smoother has a tensor product structure that simplifies an asymptotic analysis and it can be fast computed. We derive a local central limit theorem for the sandwich smoother, with simple expressions for the asymptotic bias and variance, by showing that the sandwich smoother is asymptotically equivalent to a bivariate kernel regression estimator with a product kernel. As far as we are aware, this is the first central limit theorem for a bivariate spline estimator of any type. Our simulation study shows that the sandwich smoother is orders of magnitude faster to compute than other bivariate spline smoothers, even when the latter are computed using a fast GLAM (Generalized Linear Array Model) algorithm, and comparable to them in terms of mean squared integrated errors. One important application of the sandwich smoother is to estimate covariance functions in functional data analysis. In this application, our numerical results show that the sandwich smoother is orders of magnitude faster than local linear regression. In Chapter 3, based on the sandwich smoother, we propose a fast covariance function estimation method (FACE) for smoothing high-dimensional functional data. We show that our method overcomes the computational difficulty of common bivariate smoothers for smoothing high-dimensional covariance operators, and in particular we derive a fast algorithm for selecting the smoothing parameter. We also show that through FACE we can simultaneously obtain the smoothed covariance operator and its associated eigenfunctions. For functional principal component analysis, we derive a fast method for calculating the principal scores. A simulation study is done to illustrate the computational speed of FACE. Although not a focus of this dissertation, we present in Appendix A a theoretical study of the local asymptotics of P-splines for the univariate case. In this work we derived the local asymptotic distribution of P-splines at both an interior point and near the boundary. Some of the results in the work are used in studying the sandwich smoother.