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

Tailen Hsing Professor, Department of Statistics, University of Michigan, USA. Professor Hsing is a fellow of International Statistical Institute and of the Institute of Mathematical Statistics. He has published numerous papers on subjects ranging from bioinformatics to extreme value theory, functional data analysis, large sample theory and processes with long memory. Randall Eubank Professor Emeritus, School of Mathematical and Statistical Sciences, Arizona State University, USA. Professor Eubank is well know and respected in the functional data analysis (FDA) field. He has published numerous papers on the subject and is a regular invited speaker at key meetings.

• Preface xi 1 Introduction 11.1 Multivariate analysis in a nutshell 21.2 The path that lies ahead 132 Vector and function spaces 152.1 Metric spaces 162.2 Vector and normed spaces 202.3 Banach and Lp spaces 262.4 Inner Product and Hilbert spaces 312.5 The projection theorem and orthogonal decomposition 382.6 Vector integrals 402.7 Reproducing kernel Hilbert spaces 462.8 Sobolev spaces 553 Linear operator and functionals 613.1 Operators 623.2 Linear functionals 663.3 Adjoint operator 713.4 Nonnegative, square-root, and projection operators 743.5 Operator inverses 773.6 Fréchet and Gâteaux derivatives 833.7 Generalized Gram–Schmidt decompositions 874 Compact operators and singular value decomposition 914.1 Compact operators 924.2 Eigenvalues of compact operators 964.3 The singular value decomposition 1034.4 Hilbert–Schmidt operators 1074.5 Trace class operators 1134.6 Integral operators and Mercer’s Theorem 1164.7 Operators on an RKHS 1234.8 Simultaneous diagonalization of two nonnegative definite operators 1265 Perturbation theory 1295.1 Perturbation of self-adjoint compact operators 1295.2 Perturbation of general compact operators 1406 Smoothing and regularization 1476.1 Functional linear model 1476.2 Penalized least squares estimators 1506.3 Bias and variance 1576.4 A computational formula 1586.5 Regularization parameter selection 1616.6 Splines 1657 Random elements in a Hilbert space 1757.1 Probability measures on a Hilbert space 1767.2 Mean and covariance of a random element of a Hilbert space 1787.3 Mean-square continuous processes and the Karhunen–Lòeve Theorem 1847.4 Mean-square continuous processes in L2 (E,B(E), mu) 1907.5 RKHS valued processes 1957.6 The closed span of a process 1987.7 Large sample theory 2038 Mean and covariance estimation 2118.1 Sample mean and covariance operator 2128.2 Local linear estimation 2148.3 Penalized least-squares estimation 2319 Principal components analysis 2519.1 Estimation via the sample covariance operator 2539.2 Estimation via local linear smoothing 2559.3 Estimation via penalized least squares 26110 Canonical correlation analysis 26510.1 CCA for random elements of a Hilbert space 26710.2 Estimation 27410.3 Prediction and regression 28110.4 Factor analysis 28410.5 MANOVA and discriminant analysis 28810.6 Orthogonal subspaces and partial cca 29411 Regression 30511.1 A functional regression model 30511.2 Asymptotic theory 30811.3 Minimax optimality 31811.4 Discretely sampled data 321References 327Index 331Notation Index 334

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