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E-raamat: Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators

(Professor, Department of Statistics, University of Michigan, USA), (Professor Emeritus, School of Mathematical and Statistical Sciences, Arizona State University, USA)
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Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear OperatorsSuurem pilt
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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 selfcontained 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 selfadjoint and non selfadjoint 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.
Preface xi
1 Introduction
1(14)
1.1 Multivariate analysis in a nutshell
2(11)
1.2 The path that lies ahead
13(2)
2 Vector and function spaces
15(46)
2.1 Metric spaces
16(4)
2.2 Vector and normed spaces
20(6)
2.3 Banach and Lp spaces
26(5)
2.4 Inner Product and Hilbert spaces
31(7)
2.5 The projection theorem and orthogonal decomposition
38(2)
2.6 Vector integrals
40(6)
2.7 Reproducing kernel Hilbert spaces
46(9)
2.8 Sobolev spaces
55(6)
3 Linear operator and functionals
61(30)
3.1 Operators
62(4)
3.2 Linear functionals
66(5)
3.3 Adjoint operator
71(3)
3.4 Nonnegative, square-root, and projection operators
74(3)
3.5 Operator inverses
77(6)
3.6 Frechet and Gateaux derivatives
83(4)
3.7 Generalized Gram-Schmidt decompositions
87(4)
4 Compact operators and singular value decomposition
91(38)
4.1 Compact operators
92(4)
4.2 Eigenvalues of compact operators
96(7)
4.3 The singular value decomposition
103(4)
4.4 Hilbert-Schmidt operators
107(6)
4.5 Trace class operators
113(3)
4.6 Integral operators and Mercer's Theorem
116(7)
4.7 Operators on an RKHS
123(3)
4.8 Simultaneous diagonalization of two nonnegative definite operators
126(3)
5 Perturbation theory
129(18)
5.1 Perturbation of self-adjoint compact operators
129(11)
5.2 Perturbation of general compact operators
140(7)
6 Smoothing and regularization
147(28)
6.1 Functional linear model
147(3)
6.2 Penalized least squares estimators
150(7)
6.3 Bias and variance
157(1)
6.4 A computational formula
158(3)
6.5 Regularization parameter selection
161(4)
6.6 Splines
165(10)
7 Random elements in a Hilbert space
175(36)
7.1 Probability measures on a Hilbert space
176(2)
7.2 Mean and covariance of a random element of a Hilbert space
178(6)
7.3 Mean-square continuous processes and the Karhunen--Loeve Theorem
184(6)
7.4 Mean-square continuous processes in L2 (E, B(E), μ)
190(5)
7.5 RKHS valued processes
195(3)
7.6 The closed span of a process
198(5)
7.7 Large sample theory
203(8)
8 Mean and covariance estimation
211(40)
8.1 Sample mean and covariance operator
212(2)
8.2 Local linear estimation
214(17)
8.3 Penalized least-squares estimation
231(20)
9 Principal components analysis
251(14)
9.1 Estimation via the sample covariance operator
253(2)
9.2 Estimation via local linear smoothing
255(6)
9.3 Estimation via penalized least squares
261(4)
10 Canonical correlation analysis
265(40)
10.1 CCA for random elements of a Hilbert space
267(7)
10.2 Estimation
274(7)
10.3 Prediction and regression
281(3)
10.4 Factor analysis
284(4)
10.5 MANOVA and discriminant analysis
288(6)
10.6 Orthogonal subspaces and partial cca
294(11)
11 Regression
305(22)
11.1 A functional regression model
305(3)
11.2 Asymptotic theory
308(10)
11.3 Minimax optimality
318(3)
11.4 Discretely sampled data
321(6)
References 327(4)
Index 331(3)
Notation Index 334
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.
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