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

Preface

This book aims to provide a compendium of the key mathematical concepts and results that are relevant for the theoretical development of functional data analysis (fda). As such, it is not intended to provide a general introduction to fda per se and, accordingly, we have not attempted to catalog the volumes of fda research work that have flowed at a brisk pace into the statistics literature over the past 15 years or so. Readers might therefore find it helpful to read the present text alongside other books on fda, such as Ramsay and Silverman (2005), which provide more thorough and practical developments of the topic.

This project grew out of our own struggle in acquiring the theoretical foundations for fda research in diverse fields of mathematics and statistics. With that in mind, the book strives to be self-contained. Rigorous proofs are provided for most of the results that we present. Nonetheless, a solid mathematics background at a graduate level is needed to be able to appreciate the content of the text. In particular, the reader is assumed to be familiar with linear algebra and real analysis and to have taken a course in measure theoretic probability. With this proviso, the material in the book would be suitable for a one-semester, special-topics class for advanced graduate students.

Functional data analysis is, from our perspective, the statistical analysis of sample path data observed from continuous time stochastic processes. Thus, we are dealing with random ...