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

Cover -- Contents -- Preface -- Chapter 1 Introduction -- 1.1 Multivariate analysis in a nutshell -- 1.2 The path that lies ahead -- Chapter 2 Vector and function spaces -- 2.1 Metric spaces -- 2.2 Vector and normed spaces -- 2.3 Banach and Lp spaces -- 2.4 Inner Product and Hilbert spaces -- 2.5 The projection theorem and orthogonal decomposition -- 2.6 Vector integrals -- 2.7 Reproducing kernel Hilbert spaces -- 2.8 Sobolev spaces -- Chapter 3 Linear operator and functionals -- 3.1 Operators -- 3.2 Linear functionals -- 3.3 Adjoint operator -- 3.4 Nonnegative, square-root, and projection operators -- 3.5 Operator inverses -- 3.6 Fréchet and Gâteaux derivatives -- 3.7 Generalized Gram-Schmidt decompositions -- Chapter 4 Compact operators and singular value decomposition -- 4.1 Compact operators -- 4.2 Eigenvalues of compact operators -- 4.3 The singular value decomposition -- 4.4 Hilbert-Schmidt operators -- 4.5 Trace class operators -- 4.6 Integral operators and Mercer's Theorem -- 4.7 Operators on an RKHS -- 4.8 Simultaneous diagonalization of two nonnegative definite operators -- Chapter 5 Perturbation theory -- 5.1 Perturbation of self-adjoint compact operators -- 5.2 Perturbation of general compact operators -- Chapter 6 Smoothing and regularization -- 6.1 Functional linear model -- 6.2 Penalized least squares estimators -- 6.3 Bias and variance -- 6.4 A computational formula -- 6.5 Regularization parameter selection -- 6.6 Splines -- Chapter 7 Random elements in a Hilbert space -- 7.1 Probability measures on a Hilbert space -- 7.2 Mean and covariance of a random element of a Hilbert space -- 7.3 Mean-square continuous processes and the Karhunen-Lòeve Theorem -- 7.4 Mean-square continuous processes in L2(E, B(E),μ) -- 7.5 RKHS valued processes -- 7.6 The closed span of a process -- 7.7 Large sample theory.

Chapter 8 Mean and covariance estimation -- 8.1 Sample mean and covariance operator -- 8.2 Local linear estimation -- 8.3 Penalized least-squares estimation -- Chapter 9 Principal components analysis -- 9.1 Estimation via the sample covariance operator -- 9.2 Estimation via local linear smoothing -- 9.3 Estimation via penalized least squares -- Chapter 10 Canonical correlation analysis -- 10.1 CCA for random elements of a Hilbert space -- 10.2 Estimation -- 10.3 Prediction and regression -- 10.4 Factor analysis -- 10.5 MANOVA and discriminant analysis -- 10.6 Orthogonal subspaces and partial cca -- Chapter 11 Regression -- 11.1 A functional regression model -- 11.2 Asymptotic theory -- 11.3 Minimax optimality -- 11.4 Discretely sampled data -- References -- Index -- Notation Index -- Wiley Series in Probability and Statistics -- EULA.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.