Kokoszka P., Reimherr M. Introduction to functional data analysis

Boca Raton: CRC Press, 2017. — 290 p. — (Texts in Statistical Science Series). — ISBN: 978-1-498-74634-2.Systematically develops core methodology of functional data analysis
Covers recent developments, including sparsely observed and dependent functions
Rigorously develops requisite mathematical concepts
Uses R for numerical examples and provides a dedicated R package
Each chapter contains theoretical and data analytic problems
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 Ph.D. 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
Analysts of BOA stock returns
Diffusion tensor imaging
Chapter 1 problems
Further topics in exploratory FDA
Derivatives
Penalized smoothing
Curve alignment
Further reading
Chapter 2 problems
Mathematical framework for functional data
Square integrable functions
Random functions
Linear transformations
Scalar-on-function regression
Examples
Review of standard regression theory
Difficulties spceine to functional regression
Estimation through a basts expansion
Estimation with a roughness penalty
Regression on functional principal components
Implementation in the rofund package
Nonlinear scalar on function regression
Chapter 4 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
Chapter 5 Problems
Functional generalized linear modelsSealar-on-function GLM's
Functional response GLM
Implementation in the refund package
Application to DTI
Further reading
Chapter 6 problems
Sparse FDAMean function estimation
Covariance function estimaition
Sparse functional PCA
Sparse functional regression
Chapter 7 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(l) model using package fda
Conditions for the exitence of the FAR(l) proccess
Further reading and other topics
Chapter 8 problems
Spatial functional data and models
Fundamental concepts of spatial statistics
Functional spatial fields
Functional kriging
Mean function estimation
Implementation in the R package qeofd
Other topics and further reading
Chapter 9 problems
Elements of Hilbert space theory
Hilbert space
Projections and orihonornial sets
Linear operators
Basics of spectral theory
Tensors
Chapter 10 problems
Random functions
Random elements in metric space
Expectation and covariance in a Hilbert space
Gaussian functions and limit theorems
Functional principal components
Chapter 11 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 12.2.1
Chapter 12 problems