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Reproducing Kernel Hilbert Spaces in Probability and Statistics - Alain Berlinet

Reproducing Kernel Hilbert Spaces in Probability and Statistics

Hardcover Published: 31st December 2003
ISBN: 9781402076794
Number Of Pages: 355

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The reproducing kernel Hilbert space construction is a bijection or transform theory which associates a positive definite kernel (gaussian processes) with a Hilbert space offunctions. Like all transform theories (think Fourier), problems in one space may become transparent in the other, and optimal solutions in one space are often usefully optimal in the other. The theory was born in complex function theory, abstracted and then accidently injected into Statistics; Manny Parzen as a graduate student at Berkeley was given a strip of paper containing his qualifying exam problem- It read "reproducing kernel Hilbert space"- In the 1950's this was a truly obscure topic. Parzen tracked it down and internalized the subject. Soon after, he applied it to problems with the following fla- vor: consider estimating the mean functions of a gaussian process. The mean functions which cannot be distinguished with probability one are precisely the functions in the Hilbert space associated to the covariance kernel of the processes. Parzen's own lively account of his work on re- producing kernels is charmingly told in his interview with H. Joseph Newton in Statistical Science, 17, 2002, p. 364-366. Parzen moved to Stanford and his infectious enthusiasm caught Jerry Sacks, Don Ylvisaker and Grace Wahba among others. Sacks and Ylvis- aker applied the ideas to design problems such as the following. Sup- pose (XdO

Prefacep. xiii
Acknowledgmentsp. xvii
Introductionp. xix
Theoryp. 1
Introductionp. 1
Notation and basic definitionsp. 3
Reproducing kernels and positive type functionsp. 13
Basic properties of reproducing kernelsp. 24
Sum of reproducing kernelsp. 24
Restriction of the index setp. 25
Support of a reproducing kernelp. 26
Kernel of an operatorp. 27
Condition for H[subscript K] [subset or is implied by] H[subscript R]p. 30
Tensor products of RKHSp. 30
Separability. Continuityp. 31
Extensionsp. 37
Schwartz kernelsp. 37
Semi-kernelsp. 40
Positive type operatorsp. 42
Continuous functions of positive typep. 42
Schwartz distributions of positive type or conditionally of positive typep. 44
Exercisesp. 48
Rkhs and Stochastic Processesp. 55
Introductionp. 55
Covariance function of a second order stochastic processp. 55
Case of ordinary stochastic processesp. 55
Case of generalized stochastic processesp. 56
Positivity and covariancep. 57
Positive type functions and covariance functionsp. 57
Generalized covariances and conditionally of positive type functionsp. 59
Hilbert space generated by a processp. 62
Representation theoremsp. 64
The Loeve representation theoremp. 65
The Mercer representation theoremp. 68
The Karhunen representation theoremp. 70
Applicationsp. 72
Applications to stochastic filteringp. 75
Best Predictionp. 76
Best prediction and best linear predictionp. 76
Best linear unbiased predictionp. 79
Filtering and spline functionsp. 80
No drift-no noise model and interpolating splinesp. 82
Noise without drift model and smoothing splinesp. 83
Complete model and partial smoothing splinesp. 84
Case of gaussian processesp. 86
The Kriging modelsp. 88
Directions of generalizationp. 94
Uniform Minimum Variance Unbiased Estimationp. 95
Density functional of a gaussian process and applications to extraction and detection problemsp. 97
Density functional of a gaussian processp. 97
Minimum variance unbiased estimation of the mean value of a gaussian process with known covariancep. 100
Applications to extraction problemsp. 102
Applications to detection problemsp. 104
Exercisesp. 105
Nonparametric Curve Estimationp. 109
Introductionp. 109
A brief introduction to splinesp. 110
Abstract Interpolating splinesp. 111
Abstract smoothing splinesp. 116
Partial and mixed splinesp. 118
Some concrete splinesp. 121
D[superscript m] splinesp. 121
Periodic D[superscript m] splinesp. 122
L splinesp. 123
[alpha]-splines, thin plate splines and Duchon's rotation invariant splinesp. 123
Other splinesp. 124
Random interpolating splinesp. 125
Spline regression estimationp. 125
Least squares spline estimatorsp. 126
Smoothing spline estimatorsp. 127
Hybrid splinesp. 128
Bayesian modelsp. 129
Spline density estimationp. 132
Shape restrictions in curve estimationp. 134
Unbiased density estimationp. 135
Kernels and higher order kernelsp. 136
Local approximation of functionsp. 143
Local polynomial smoothing of statistical functionalsp. 148
Density estimation in selection bias modelsp. 150
Hazard functionsp. 152
Reliability and econometric functionsp. 154
Kernels of order (m, p)p. 155
Definition of K[subscript o]-based hierarchiesp. 158
Computational aspectsp. 160
Sequences of hierarchiesp. 165
Optimality properties of higher order kernelsp. 167
The multiple kernel methodp. 171
The estimation procedure for the density and its derivativesp. 172
Exercisesp. 175
Measures and Random Measuresp. 185
Introductionp. 185
Dirac measuresp. 186
General approachp. 190
The example of momentsp. 192
Measurability of RKHS-valued variablesp. 194
Gaussian measure on RKHSp. 196
Gaussian measure and gaussian processp. 196
Construction of gaussian measuresp. 198
Weak convergence in Pr(H)p. 199
Weak convergence criterionp. 202
Integration of H-valued random variablesp. 202
Notation. Definitionsp. 203
Integrability of X and of {X[superscript t] : t [set mempership] E}p. 205
Inner products on sets of measuresp. 210
Inner product and weak topologyp. 214
Application to normal approximationp. 218
Random measuresp. 220
The empirical measure as H-valued variablep. 223
Integrable kernelsp. 224
Estimation of I[subscript mu]p. 228
Convergence of random measuresp. 232
Exercisesp. 234
Miscellaneous Applicationsp. 241
Introductionp. 241
Law of Iterated Logarithmp. 241
Learning and decision theoryp. 245
Binary classification with RKHSp. 245
Support Vector Machinep. 248
ANOVA in function spacesp. 249
ANOVA decomposition of a function on a product domainp. 249
Tensor product smoothing splinesp. 252
Regression with tensor product splinesp. 254
Strong approximation in RKHSp. 255
Generalized method of momentsp. 259
Exercisesp. 262
Computational Aspectsp. 265
Kernel of a given normed spacep. 266
Kernel of a finite dimensional spacep. 266
Kernel of some subspacesp. 266
Decomposition principlep. 267
Kernel of a class of periodic functionsp. 268
A family of Beppo-Levi spacesp. 270
Sobolev spaces endowed with a variety of normsp. 276
First family of normsp. 277
Second family of normsp. 285
Norm and space corresponding to a given reproducing kernelp. 288
Exercisesp. 289
A Collection of Examplesp. 293
Introductionp. 293
Using the characterization theoremp. 293
Case of finite Xp. 294
Case of countably infinite Xp. 294
Using any mapping from E into some pre-Hilbert spacep. 295
Factorizable kernelsp. 295
Examples of spaces, norms and kernelsp. 299
Appendixp. 344
Introduction to Sobolev spacesp. 345
Schwartz-distributions or generalized functionsp. 345
Spaces and their topologyp. 345
Weak-derivative or derivative in the sense of distributionsp. 346
Facts about Fourier transformsp. 346
Sobolev spacesp. 346
Absolute continuity of functions of one variablep. 346
Sobolev space with non negative integer exponentp. 347
Sobolev space with real exponentp. 348
Periodic Sobolev spacep. 349
Beppo-Levi spacesp. 349
Indexp. 353
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9781402076794
ISBN-10: 1402076797
Audience: General
Format: Hardcover
Language: English
Number Of Pages: 355
Published: 31st December 2003
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 24.18 x 16.31  x 2.77
Weight (kg): 0.78