Introduction to Functional Data Analysis: 1st Edition (Hardback) book cover

Introduction to Functional Data Analysis

1st Edition

By Piotr Kokoszka, Matthew Reimherr

Chapman and Hall/CRC

290 pages

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Hardback: 9781498746342
pub: 2017-08-09
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pub: 2017-09-27
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Description

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.

Reviews

"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

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

About the Authors

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.

About the Series

Chapman & Hall/CRC Texts in Statistical Science

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Subject Categories

BISAC Subject Codes/Headings:
MAT029000
MATHEMATICS / Probability & Statistics / General
MAT029010
MATHEMATICS / Probability & Statistics / Bayesian Analysis