Cover image for Reproducing kernel Hilbert spaces in probability and statistics
Title:
Reproducing kernel Hilbert spaces in probability and statistics
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Publication Information:
Boston, MA : Kluwer Academic, 2004
ISBN:
9781402076794

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30000010119388 QA322.4 B474 2004 Open Access Book Book
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Summary

Summary

The book covers theoretical questions including the latest extension of the formalism, and computational issues and focuses on some of the more fruitful and promising applications, including statistical signal processing, nonparametric curve estimation, random measures, limit theorems, learning theory and some applications at the fringe between Statistics and Approximation Theory. It is geared to graduate students in Statistics, Mathematics or Engineering, or to scientists with an equivalent level.


Table of Contents

Prefacep. xiii
Acknowledgmentsp. xvii
Introductionp. xix
1. Theoryp. 1
1.1 Introductionp. 1
1.2 Notation and basic definitionsp. 3
1.3 Reproducing kernels and positive type functionsp. 13
1.4 Basic properties of reproducing kernelsp. 24
1.4.1 Sum of reproducing kernelsp. 24
1.4.2 Restriction of the index setp. 25
1.4.3 Support of a reproducing kernelp. 26
1.4.4 Kernel of an operatorp. 27
1.4.5 Condition for H[subscript K] [subset or is implied by] H[subscript R]p. 30
1.4.6 Tensor products of RKHSp. 30
1.5 Separability. Continuityp. 31
1.6 Extensionsp. 37
1.6.1 Schwartz kernelsp. 37
1.6.2 Semi-kernelsp. 40
1.7 Positive type operatorsp. 42
1.7.1 Continuous functions of positive typep. 42
1.7.2 Schwartz distributions of positive type or conditionally of positive typep. 44
1.8 Exercisesp. 48
2. Rkhs and Stochastic Processesp. 55
2.1 Introductionp. 55
2.2 Covariance function of a second order stochastic processp. 55
2.2.1 Case of ordinary stochastic processesp. 55
2.2.1.1 Case of generalized stochastic processesp. 56
2.2.2 Positivity and covariancep. 57
2.2.2.1 Positive type functions and covariance functionsp. 57
2.2.2.2 Generalized covariances and conditionally of positive type functionsp. 59
2.2.3 Hilbert space generated by a processp. 62
2.3 Representation theoremsp. 64
2.3.1 The Loeve representation theoremp. 65
2.3.2 The Mercer representation theoremp. 68
2.3.3 The Karhunen representation theoremp. 70
2.3.4 Applicationsp. 72
2.4 Applications to stochastic filteringp. 75
2.4.1 Best Predictionp. 76
2.4.1.1 Best prediction and best linear predictionp. 76
2.4.1.2 Best linear unbiased predictionp. 79
2.4.2 Filtering and spline functionsp. 80
2.4.2.1 No drift-no noise model and interpolating splinesp. 82
2.4.2.2 Noise without drift model and smoothing splinesp. 83
2.4.2.3 Complete model and partial smoothing splinesp. 84
2.4.2.4 Case of gaussian processesp. 86
2.4.2.5 The Kriging modelsp. 88
2.4.2.6 Directions of generalizationp. 94
2.5 Uniform Minimum Variance Unbiased Estimationp. 95
2.6 Density functional of a gaussian process and applications to extraction and detection problemsp. 97
2.6.1 Density functional of a gaussian processp. 97
2.6.2 Minimum variance unbiased estimation of the mean value of a gaussian process with known covariancep. 100
2.6.3 Applications to extraction problemsp. 102
2.6.4 Applications to detection problemsp. 104
2.7 Exercisesp. 105
3. Nonparametric Curve Estimationp. 109
3.1 Introductionp. 109
3.2 A brief introduction to splinesp. 110
3.2.1 Abstract Interpolating splinesp. 111
3.2.2 Abstract smoothing splinesp. 116
3.2.3 Partial and mixed splinesp. 118
3.2.4 Some concrete splinesp. 121
3.2.4.1 D[superscript m] splinesp. 121
3.2.4.2 Periodic D[superscript m] splinesp. 122
3.2.4.3 L splinesp. 123
3.2.4.4 [alpha]-splines, thin plate splines and Duchon's rotation invariant splinesp. 123
3.2.4.5 Other splinesp. 124
3.3 Random interpolating splinesp. 125
3.4 Spline regression estimationp. 125
3.4.1 Least squares spline estimatorsp. 126
3.4.2 Smoothing spline estimatorsp. 127
3.4.3 Hybrid splinesp. 128
3.4.4 Bayesian modelsp. 129
3.5 Spline density estimationp. 132
3.6 Shape restrictions in curve estimationp. 134
3.7 Unbiased density estimationp. 135
3.8 Kernels and higher order kernelsp. 136
3.9 Local approximation of functionsp. 143
3.10 Local polynomial smoothing of statistical functionalsp. 148
3.10.1 Density estimation in selection bias modelsp. 150
3.10.2 Hazard functionsp. 152
3.10.3 Reliability and econometric functionsp. 154
3.11 Kernels of order (m, p)p. 155
3.11.1 Definition of K[subscript o]-based hierarchiesp. 158
3.11.2 Computational aspectsp. 160
3.11.3 Sequences of hierarchiesp. 165
3.11.4 Optimality properties of higher order kernelsp. 167
3.11.5 The multiple kernel methodp. 171
3.11.6 The estimation procedure for the density and its derivativesp. 172
3.12 Exercisesp. 175
4. Measures and Random Measuresp. 185
4.1 Introductionp. 185
4.1.1 Dirac measuresp. 186
4.1.2 General approachp. 190
4.1.3 The example of momentsp. 192
4.2 Measurability of RKHS-valued variablesp. 194
4.3 Gaussian measure on RKHSp. 196
4.3.1 Gaussian measure and gaussian processp. 196
4.3.2 Construction of gaussian measuresp. 198
4.4 Weak convergence in Pr(H)p. 199
4.4.1 Weak convergence criterionp. 202
4.5 Integration of H-valued random variablesp. 202
4.5.1 Notation. Definitionsp. 203
4.5.2 Integrability of X and of {{X[superscript t] : t [set mempership] E}}p. 205
4.6 Inner products on sets of measuresp. 210
4.7 Inner product and weak topologyp. 214
4.8 Application to normal approximationp. 218
4.9 Random measuresp. 220
4.9.1 The empirical measure as H-valued variablep. 223
4.9.1.1 Integrable kernelsp. 224
4.9.1.2 Estimation of I[subscript mu]p. 228
4.9.2 Convergence of random measuresp. 232
4.10 Exercisesp. 234
5. Miscellaneous Applicationsp. 241
5.1 Introductionp. 241
5.2 Law of Iterated Logarithmp. 241
5.3 Learning and decision theoryp. 245
5.3.1 Binary classification with RKHSp. 245
5.3.2 Support Vector Machinep. 248
5.4 ANOVA in function spacesp. 249
5.4.1 ANOVA decomposition of a function on a product domainp. 249
5.4.2 Tensor product smoothing splinesp. 252
5.4.3 Regression with tensor product splinesp. 254
5.5 Strong approximation in RKHSp. 255
5.6 Generalized method of momentsp. 259
5.7 Exercisesp. 262
6. Computational Aspectsp. 265
6.1 Kernel of a given normed spacep. 266
6.1.1 Kernel of a finite dimensional spacep. 266
6.1.2 Kernel of some subspacesp. 266
6.1.3 Decomposition principlep. 267
6.1.4 Kernel of a class of periodic functionsp. 268
6.1.5 A family of Beppo-Levi spacesp. 270
6.1.6 Sobolev spaces endowed with a variety of normsp. 276
6.1.6.1 First family of normsp. 277
6.1.6.2 Second family of normsp. 285
6.2 Norm and space corresponding to a given reproducing kernelp. 288
6.3 Exercisesp. 289
7. A Collection of Examplesp. 293
7.1 Introductionp. 293
7.2 Using the characterization theoremp. 293
7.2.1 Case of finite Xp. 294
7.2.2 Case of countably infinite Xp. 294
7.2.3 Using any mapping from E into some pre-Hilbert spacep. 295
7.3 Factorizable kernelsp. 295
7.4 Examples of spaces, norms and kernelsp. 299
Appendixp. 344
Introduction to Sobolev spacesp. 345
A.1 Schwartz-distributions or generalized functionsp. 345
A.1.1 Spaces and their topologyp. 345
A.1.2 Weak-derivative or derivative in the sense of distributionsp. 346
A.1.3 Facts about Fourier transformsp. 346
A.2 Sobolev spacesp. 346
A.2.1 Absolute continuity of functions of one variablep. 346
A.2.2 Sobolev space with non negative integer exponentp. 347
A.2.3 Sobolev space with real exponentp. 348
A.2.4 Periodic Sobolev spacep. 349
A.3 Beppo-Levi spacesp. 349
Indexp. 353