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.
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
