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Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators

Book Description

Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators provides a uniquely broad compendium of the key mathematical concepts and results that are relevant for the theoretical development of functional data analysis (FDA).

The self-contained treatment of selected topics of functional analysis and operator theory includes reproducing kernel Hilbert spaces, singular value decomposition of compact operators on Hilbert spaces and perturbation theory for both self-adjoint and non self-adjoint operators. The probabilistic foundation for FDA is described from the perspective of random elements in Hilbert spaces as well as from the viewpoint of continuous time stochastic processes. Nonparametric estimation approaches including kernel and regularized smoothing are also introduced. These tools are then used to investigate the properties of estimators for the mean element, covariance operators, principal components, regression function and canonical correlations. A general treatment of canonical correlations in Hilbert spaces naturally leads to FDA formulations of factor analysis, regression, MANOVA and discriminant analysis.

This book will provide a valuable reference for statisticians and other researchers interested in developing or understanding the mathematical aspects of FDA. It is also suitable for a graduate level special topics course.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright
  4. Dedication
  5. Preface
  6. Chapter 1: Introduction
    1. 1.1 Multivariate analysis in a nutshell
    2. 1.2 The path that lies ahead
  7. Chapter 2: Vector and function spaces
    1. 2.1 Metric spaces
    2. 2.2 Vector and normed spaces
    3. 2.3 Banach and <img xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:svg="http://www.w3.org/2000/svg" xmlns:ibooks="http://vocabulary.itunes.apple.com/rdf/ibooks/vocabulary-extensions-1.0" src="images/c02-math-0311.png" alt="c02-math-285" style="vertical-align:middle;"></img> spaces spaces
    4. 2.4 Inner Product and Hilbert spaces
    5. 2.5 The projection theorem and orthogonal decomposition
    6. 2.6 Vector integrals
    7. 2.7 Reproducing kernel Hilbert spaces
    8. 2.8 Sobolev spaces
  8. Chapter 3: Linear operator and functionals
    1. 3.1 Operators
    2. 3.2 Linear functionals
    3. 3.3 Adjoint operator
    4. 3.4 Nonnegative, square-root, and projection operators
    5. 3.5 Operator inverses
    6. 3.6 Fréchet and Gâteaux derivatives
    7. 3.7 Generalized Gram–Schmidt decompositions
  9. Chapter 4: Compact operators and singular value decomposition
    1. 4.1 Compact operators
    2. 4.2 Eigenvalues of Compact Operators
    3. 4.3 The Singular Value Decomposition
    4. 4.4 Hilbert–Schmidt Operators
    5. 4.5 Trace class operators
    6. 4.6 Integral operators and Mercer's Theorem
    7. 4.7 Operators on an RKHS
    8. 4.8 Simultaneous diagonalization of two nonnegative definite operators
  10. Chapter 5: Perturbation theory
    1. 5.1 Perturbation of self-adjoint compact operators
    2. 5.2 Perturbation of general compact operators
  11. Chapter 6: Smoothing and regularization
    1. 6.1 Functional linear model
    2. 6.2 Penalized least squares estimators
    3. 6.3 Bias and variance
    4. 6.4 A computational formula
    5. 6.5 Regularization parameter selection
    6. 6.6 Splines
  12. Chapter 7: Random elements in a Hilbert space
    1. 7.1 Probability measures on a Hilbert space
    2. 7.2 Mean and covariance of a random element of a Hilbert space
    3. 7.3 Mean-square continuous processes and the Karhunen–Lòeve Theorem
    4. 7.4 Mean-square continuous processes in <img xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:svg="http://www.w3.org/2000/svg" xmlns:ibooks="http://vocabulary.itunes.apple.com/rdf/ibooks/vocabulary-extensions-1.0" src="images/c07-math-0407.png" alt="c07-math-368" style="vertical-align:middle;"></img>
    5. 7.5 RKHS valued processes
    6. 7.6 The closed span of a process
    7. 7.7 Large sample theory
  13. Chapter 8: Mean and covariance estimation
    1. 8.1 Sample mean and covariance operator
    2. 8.2 Local linear estimation
    3. 8.3 Penalized least-squares estimation
  14. Chapter 9: Principal components analysis
    1. 9.1 Estimation via the sample covariance operator
    2. 9.2 Estimation via local linear smoothing
    3. 9.3 Estimation via penalized least squares
  15. Chapter 10: Canonical correlation analysis
    1. 10.1 CCA for random elements of a Hilbert space
    2. 10.2 Estimation
    3. 10.3 Prediction and regression
    4. 10.4 Factor analysis
    5. 10.5 MANOVA and discriminant analysis
    6. 10.6 Orthogonal subspaces and partial cca
  16. Chapter 11: Regression
    1. 11.1 A functional regression model
    2. 11.2 Asymptotic theory
    3. 11.3 Minimax optimality
    4. 11.4 Discretely sampled data
  17. References
  18. Index
  19. Notation Index
  20. Wiley Series in Probability and Statistics
  21. End User License Agreement