Introduction to Functional Data Analysis

Auteur

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.

Texte du rabat

The book provides an introduction to functional data analysis (FDA), useful to students and researchers. FDA is now generally viewed as a fundamental subfield of statistics. FDA methods have been applied to science, business and engineering.

Résumé

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.

Contenu

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