Functional principal components

Abstract

This chapter introduces one of the most fundamental concepts of FDA, that of the functional principal components (FPC’s). FPC’s allow us to reduce the dimension of infinitely dimensional functional data to a small finite dimension in an optimal way. In Sections 3.1 and 3.2, we introduce the FPC’s from two angles, as coordinates maximizing variability, and as an optimal orthonormal basis. In Section 3.3, we identify the FPC’s with the eigenfunctions of the covariance operator, and show how its eigenvalues decompose the variance of the functional data. We conclude with Section 3.4 which explains how to compute the FPC’s in the R package fda.