Function Estimation and Functional Data Analysis

The roughness penalty method is widely used in function estimation, and is closely related to methods of regularization well known in numerical analysis. Some background to the development of this method is discussed. The versatility of the method is illustrated by its application to an unusual smoothing problem, involving the estimation of a branching system of curves. The second main focus of the paper is on problems where the data are themselves functions, an area known as functional data analysis. The roughness penalty method is a key component of extensions to the functional context of principal components analysis and canonical correlation analysis. These techniques are described and contrasted, and are illustrated by reference to a set of data on the development of human gait.