Differential operators in functional data analysis

Abstract

The derivatives of functional observations have played a strong role from the beginning of this book. For example, for growth curves and handwriting coordinate functions we chose to work with acceleration directly, and for temperature profiles we considered functions (π/6)² D Temp+D ³Temp. We used D ² β to construct a measure of curvature in an estimated regression function β to regularize or smooth the estimate, and applied this same principle in functional principal components analysis, canonical correlation, and other types of linear models. Thus, derivatives can be used both as the object of inquiry and as tools for stabilizing solutions.