Introduction to Functional Data Analysis

Introduction to Functional Data Analysis provides a concise textbook introduction to the field. It explains how to analyze functional data, both at exploratory and inferential levels. It also provides a systematic and accessible exposition of the methodology and the required mathematical framework. The book can be used as textbook for a semester-long course on FDA for advanced undergraduate or MS statistics majors, as well as for MS and PhD students in other disciplines, including applied mathematics, environmental science, public health, medical research, geophysical sciences and economics. It can also be used for self-study and as a reference for researchers in those fields who wish to acquire solid understanding of FDA methodology and practical guidance for its implementation. Each chapter contains plentiful examples of relevant R code and theoretical and data analytic problems. The material of the book can be roughly divided into four parts of approximately equal length: 1) basic concepts and techniques of FDA, 2) functional regression models, 3) sparse and dependent functional data, and 4) introduction to the Hilbert space framework of FDA. The book assumes advanced undergraduate background in calculus, linear algebra, distributional probability theory, foundations of statistical inference, and some familiarity with R programming. Other required statistics background is provided in scalar settings before the related functional concepts are developed. Most chapters end with references to more advanced research for those who wish to gain a more in-depth understanding of a specific topic.

Table of Contents:
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 regression Examples 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 motion Penalized least squares estimation Functional regressors Penalized estimation in the refund package Estimation based on functional principal components Test of no effect Verification 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 estimation Covariance function estimation Sparse functional PCA Sparse functional regression Problems Functional time series Fundamental concepts of time series analysis Functional autoregressive process Forecasting with the Hyndman-Ullah method Forecasting 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 models Fundamental 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 theory Hilbert space Projections and orthonormal sets Linear operators Basics of spectral theory Tensors Problems Random functions Random elements in metric spaces Expectation and covariance in a Hilbert space Gaussian functions and limit theorems Functional principal components Problems Inference from a random sample Consistency 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