Introduction to Functional Data Analysis

Piotr Kokoszka is a professor of statistics at Colorado State University. His research interests include functional data analysis, with emphasis on dependent data structures, and applications to geosciences and finance. He is a coauthor of the monograph Inference for Functional Data with Applications (with L. Horváth). He is an associate editor of several journals, including Computational Statistics and Data Analysis, Journal of Multivariate Analysis, Journal of Time Series Analysis, and Scandinavian Journal of Statistics.Matthew Reimherr is an assistant professor of statistics at Pennsylvania State University. His research interests include functional data analysis, with emphasis on longitudinal studies and applications to genetics and public health. He is an associate editor of Statistical Modeling.

• First steps in the analysis of functional data Basis expansions Sample mean and covariance Principal component functions Analysis of BOA stock returns Diffusion tensor imaging Problems Further topics in exploratory FDA Derivatives Penalized smoothing Curve alignment Further reading Problems Mathematical framework for functional data Square integrable functions Random functions Linear transformations Scalar- on - function regressionExamples Review of standard regression theory Difficulties specific to functional regression Estimation through a basis expansion Estimation with a roughness penalty Regression on functional principal components Implementation in the refund package Nonlinear scalar-on-function regression Problems Functional response models Least squares estimation and application to angular motionPenalized least squares estimation Functional regressors Penalized estimation in the refund package Estimation based on functional principal components Test of no effectVerification of the validity of a functional linear model Extensions and further reading Problems Functional generalized linear models Background Scalar-on-function GLM's Functional response GLM Implementation in the refund package Application to DTI Further reading Problems Sparse FDA Introduction Mean function estimationCovariance function estimation Sparse functional PCA Sparse functional regression Problems Functional time seriesFundamental concepts of time series analysis Functional autoregressive process Forecasting with the Hyndman-Ullah methodForecasting with multivariate predictors Long-run covariance function Testing stationarity of functional time series Generation and estimation of the FAR(1) model using package fda Conditions for the existence of the FAR(1) process Further reading and other topics Problems Spatial functional data and modelsFundamental concepts of spatial statistics Functional spatial fields Functional kriging Mean function estimation Implementation in the R package geofd Other topics and further reading Problems Elements of Hilbert space theoryHilbert space Projections and orthonormal sets Linear operators Basics of spectral theory Tensors ProblemsRandom functions Random elements in metric spaces Expectation and covariance in a Hilbert space Gaussian functions and limit theorems Functional principal components ProblemsInference from a random sampleConsistency of sample mean and covariance functions Estimated functional principal components Asymptotic normality Hypothesis testing about the mean Confidence bands for the mean Application to BOA cumulative returns Proof of Theorem Problems

"This well-written book provides a great and intuitive introduction to functional data analysis (FDA) which has emerged as an important area in statistics and found tons of scientific applications...This book succeeds at introducing this novel statistical concept and methodology while keeps the level of mathematical and statistical sophistication required to understand at the level of an introductory graduate-level course, which makes for pleasant reading. A nice feature of the book is its strong focus on implementation using R, which makes it a great candidate of textbooks or reference books for (master-level) graduate students and applied researchers...Some unique features of this book as compared to existing ones include (1) its strong focus on implementation using R; (2) chapters on Sparse FDA, generalized functional linear models, functional time series, and spatial functional data; (3) well-designed exercises that can be used as homework problems."~Xianyang Zhang, Texas A&M University"The main advantage of the book is its emphasis introducing the material through realistic examples and computational tools, while also providing mathematical guidance for the methodologies. Also, important topics like functional time series and spatial functional data are not adequately covered in comparable texts like Ramsay and Silverman, Ramsay and Hooker, Ferraty and Vieu, and Hsing and Eubank. In that respect, the book offers additional and practically relevant material and perspective."~Debashis Paul, University of California, Davis"The classic tools from the field of functional data analysis are introduced comprehensively and immediately put into a framework of potential application. I would probably advise any reader that is new to functional data analysis to start by reading this book."~Claudia Klüppelberg, Technische Universität München"Being more advanced and up to date than the Ramsay and Silverman, it complements various topics that are just briefly mentioned or not covered at all by Ramsay and Silverman."~Laura Sangali, Politecnico di Milano"This book by Kokoszka and Reimherr provides a nice mix of foundational material, accessible theory, and practical examples (including much R code). It is a valuable addition to the FDA literature, and is perhaps an ideal choice of a course textbook for either an undergraduate or graduate course in FDA, whereas several of the other textbooks are more valuable as references for researchers and practitioners than as tutorials for learners. At the end of each chapter is a nice variety of problems that instructors could use for homework assignments." ~David Benner Hitchcock