Statistical Models for Robust Spline Smoothing

Specialeforsvar ved Helene Rytgaard

Titel: Statistical Models for Robust Spline Smoothing

Abstract: Statistical Models for Robust Spline Smoothing
(function space restrictions, L1 splines and generalizations)

The thesis considers one-dimensional curve data, where the objective is to estimate a functional relation describing the observations. In the first part, we go through the main parts of the classic approach to the problem, which leads to the theory of splines. We make a brief introduction to the dominating estimation methods within this area, but then moves on to a somewhat differing approach, aiming at providing a full statistical framework for spline smoothing. We investigate the duality between, on the one hand, the theory of inversion of differential operators and their corresponding Green's functions and, on the other hand, Gaussian processes and their covariance functions, and the two approaches are shown to give the same estimation results. Ultimately, this clears the way for us to formulate a full statistical model for spline smoothing taking a Bayesian viewpoint. Having that established, we can make various modifications of the model assumptions. We consider two main extensions. Firstly, we impose different shape restrictions on the functional form of the spline and show how these constrained problems can be solved. Secondly, we consider the so-called L1splines and aim to understand and describe these using the Laplace distribution. For this, we present some theory on the univariate, and also the multivariate, Laplace distribution.
Finally, we propose an iterative estimation method for solving the L1 spline problem, and we present some new extensions. The thesis comes together with an R-package, L1splines, that contains implementations of all the splines developed. 

Vejledere:  Bo Markussen / Lars Lau Rakét
Censor:      Asger Hobolth, Aarhus Universitet