Stochastic Models for Smoothing Splines: A Bayesian Approach

Abstract

Flexible data regression is an important tool for capturing complicated trends in data. One approach is penalized smoothing splines, where there are several mainstream methods. A weakness is, however, the quantification of uncertainty. We will in thesis present two mainstream smoothing spline methods, P-splines and O'Sullivan splines, and the RW2 model; a Bayesian hierarchical model based on a latent field. The Bayesian priors are specified by a stochastic Poisson equation, and spline estimates are approximated along a finite element Galerkin approach. We evaluate the three methods using integrated nested Laplace approximations (INLA) for a full Bayesian analysis supplying credible bands. The methods give fairly similar results and we investigate the theoretical motivates behind the methods. As an extension of the Bayesian models, the smoothing parameter is incorporated in latent field. This gives an adaptive smoothing method, which better estimates jumps and quick curvature changes. Further, the close relationship between O'Sullivan splines smoothing splines is discussed, revealing O'Sullivan splines to be a finite element Petrov-Galerkin approximation of smoothing splines. The main results are the possibility of credible bands, the extension to adaptive smoothing and the finite element understanding of O'Sullivan splines.