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

Chapter 11Regression

In this final chapter, we focus on the fda-specific problem of functional linear regression. There are various ways to formulate this type of regression idea and we have chosen to study one of the more common situations that has appeared in the fda literature: namely, the case of a scalar-dependent variable and functional independent variable. This leads to a special case of the functional linear model introduced in Section 6.1 which makes it natural to investigate the performance of method of regularization estimation techniques in this setting. In Section 11.1, we describe the basic regression model that we pose for study and derive a penalized least-squares estimator for the corresponding coefficient function that was originally proposed in Crambes, Kneip, and Sarda (2009). Subsequent sections examine the large sample and optimality properties of this estimator.

11.1 A functional regression model

The basic premise is that we have a probability space and an associated second-order stochastic process that is jointly measurable in and with a square-integrable ...