Librería Portfolio

Features

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

Summary

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