Derivative Estimation in Noisy Data; an Additive Penalty P-Spline Approach

Abstract

In many situations it is of primary interest to estimate the rate of change of the relationship between response and explanatory variables. In this thesis derivative estimation using spline smoothing is explored. A review of derivative estimates found as a by product of several popular spline smoothing techniques is provided. Concerns with these estimates are raised and an additive penalty method utilising the attractive properties of P-Splines is introduced. This approach is shown to improve on semiparametric and P-Spline derivative estimates in simulated smoothing scenarios. Variability bands for derivative estimates are developed for the additive penalty and P-Spline methods with these tested for coverage and precision in further simulations. Motivating examples in environmental, biomedical and astronomical applications are revisited throughout the thesis.