Introduction to Functional Data Analysis - Kokoszka, Piotr

Résumé :

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.

Sommaire:

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