Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators

Chapter 6Smoothing and regularization

As stated in Chapter 1, our view of functional data analysis involves the statistical analysis of sample paths that arise from one or more stochastic processes. The processes themselves are presumed to be random elements of some Hilbert space such as in a sense that will be precisely defined in Chapter 7. For now, it suffices to merely think of the collected data as being discretized readings from a sample of curves.

Discretization entails some loss of information. This may be sufficiently problematic in some instances to require remedial measures to recover some of what was lost. In addition, there may be contamination or distortions of the actual sample path values by noise or other sources of error. In such cases, it may be worthwhile to perform some preprocessing to filter out artifacts in the data that have arisen from extraneous sources. Such problems are not unique to fda and arise in a variety of statistical contexts. The methods that have evolved for their solution are generally referred to as smoothing or nonparametric smoothing techniques that will be the focus of this chapter.

6.1 Functional linear model

A conceptually simple smoothing problem arises from nonparametric regression analysis. In that setting, we have a real valued mean or regression function on that is discretely observed with additive random noise; i.e., the ...