1st Edition

Analysis of Variance for Functional Data

By Jin-Ting Zhang Copyright 2014
    410 Pages 80 B/W Illustrations
    by Chapman & Hall

    Despite research interest in functional data analysis in the last three decades, few books are available on the subject. Filling this gap, Analysis of Variance for Functional Data presents up-to-date hypothesis testing methods for functional data analysis. The book covers the reconstruction of functional observations, functional ANOVA, functional linear models with functional responses, ill-conditioned functional linear models, diagnostics of functional observations, heteroscedastic ANOVA for functional data, and testing equality of covariance functions. Although the methodologies presented are designed for curve data, they can be extended to surface data.

    Useful for statistical researchers and practitioners analyzing functional data, this self-contained book gives both a theoretical and applied treatment of functional data analysis supported by easy-to-use MATLAB® code. The author provides a number of simple methods for functional hypothesis testing. He discusses pointwise, L2-norm-based, F-type, and bootstrap tests.

    Assuming only basic knowledge of statistics, calculus, and matrix algebra, the book explains the key ideas at a relatively low technical level using real data examples. Each chapter also includes bibliographical notes and exercises. Real functional data sets from the text and MATLAB codes for analyzing the data examples are available for download from the author’s website.

    Introduction
    Functional Data
    Motivating Functional Data
    Why Is Functional Data Analysis Needed?
    Overview of the Book
    Implementation of Methodologies
    Options for Reading This Book

    Nonparametric Smoothers for a Single Curve
    Introduction
    Local Polynomial Kernel Smoothing
    Regression Splines
    Smoothing Splines
    P-Splines

    Reconstruction of Functional Data
    Introduction
    Reconstruction Methods
    Accuracy of LPK Reconstructions
    Accuracy of LPK Reconstruction in FLMs

    Stochastic Processes
    Introduction
    Stochastic Processes
    x2-Type Mixtures
    F-Type Mixtures
    One-Sample Problem for Functional Data

    ANOVA for Functional Data
    Introduction
    Two-Sample Problem
    One-Way ANOVA
    Two-Way ANOVA

    Linear Models with Functional Responses
    Introduction
    Linear Models with Time-Independent Covariates
    Linear Models with Time-Dependent Covariates

    Ill-Conditioned Functional Linear Models
    Introduction
    Generalized Inverse Method
    Reparameterization Method
    Side-Condition Method

    Diagnostics of Functional Observations
    Introduction
    Residual Functions
    Functional Outlier Detection
    Influential Case Detection
    Robust Estimation of Coefficient Functions
    Outlier Detection for a Sample of Functions

    Heteroscedastic ANOVA for Functional Data
    Introduction
    Two-Sample Behrens-Fisher Problems
    Heteroscedastic One-Way ANOVA
    Heteroscedastic Two-Way ANOVA

    Test of Equality of Covariance Functions
    Introduction
    Two-Sample Case
    Multi-Sample Case

    Bibliography

    Index

    Technical Proofs, Concluding Remarks, Bibliographical Notes, and Exercises appear at the end of most chapters.

    Biography

    Jin-Ting Zhang is an associate professor in the Department of Statistics and Applied Probability at the National University of Singapore. He has published extensively and has served on the editorial boards of several international statistical journals. He is the coauthor of Nonparametric Regression Methods for Longitudinal Data Analysis: Mixed-Effect Modelling Approaches and the coeditor of Advances in Statistics: Proceedings of the Conference in Honor of Professor Zhidong Bai on His 65th Birthday.

    "… a focused presentation of functional ANOVA and linear function-on-scalar regression problems using the ‘smooth first’ approach to estimation and inference. I would recommend this book to anyone interested in theoretical developments and hypothesis testing in this commonly encountered class of problems."
    —Jeff Goldsmith, Journal of the American Statistical Association, March 2014