Title:
Reproducing kernel Hilbert spaces in probability and statistics
Personal Author:
Publication Information:
Boston, MA : Kluwer Academic, 2004
ISBN:
9781402076794
Added Author:
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Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
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Searching... | 30000010119388 | QA322.4 B474 2004 | Open Access Book | Book | Searching... |
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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
Preface | p. xiii |
Acknowledgments | p. xvii |
Introduction | p. xix |
1. Theory | p. 1 |
1.1 Introduction | p. 1 |
1.2 Notation and basic definitions | p. 3 |
1.3 Reproducing kernels and positive type functions | p. 13 |
1.4 Basic properties of reproducing kernels | p. 24 |
1.4.1 Sum of reproducing kernels | p. 24 |
1.4.2 Restriction of the index set | p. 25 |
1.4.3 Support of a reproducing kernel | p. 26 |
1.4.4 Kernel of an operator | p. 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 RKHS | p. 30 |
1.5 Separability. Continuity | p. 31 |
1.6 Extensions | p. 37 |
1.6.1 Schwartz kernels | p. 37 |
1.6.2 Semi-kernels | p. 40 |
1.7 Positive type operators | p. 42 |
1.7.1 Continuous functions of positive type | p. 42 |
1.7.2 Schwartz distributions of positive type or conditionally of positive type | p. 44 |
1.8 Exercises | p. 48 |
2. Rkhs and Stochastic Processes | p. 55 |
2.1 Introduction | p. 55 |
2.2 Covariance function of a second order stochastic process | p. 55 |
2.2.1 Case of ordinary stochastic processes | p. 55 |
2.2.1.1 Case of generalized stochastic processes | p. 56 |
2.2.2 Positivity and covariance | p. 57 |
2.2.2.1 Positive type functions and covariance functions | p. 57 |
2.2.2.2 Generalized covariances and conditionally of positive type functions | p. 59 |
2.2.3 Hilbert space generated by a process | p. 62 |
2.3 Representation theorems | p. 64 |
2.3.1 The Loeve representation theorem | p. 65 |
2.3.2 The Mercer representation theorem | p. 68 |
2.3.3 The Karhunen representation theorem | p. 70 |
2.3.4 Applications | p. 72 |
2.4 Applications to stochastic filtering | p. 75 |
2.4.1 Best Prediction | p. 76 |
2.4.1.1 Best prediction and best linear prediction | p. 76 |
2.4.1.2 Best linear unbiased prediction | p. 79 |
2.4.2 Filtering and spline functions | p. 80 |
2.4.2.1 No drift-no noise model and interpolating splines | p. 82 |
2.4.2.2 Noise without drift model and smoothing splines | p. 83 |
2.4.2.3 Complete model and partial smoothing splines | p. 84 |
2.4.2.4 Case of gaussian processes | p. 86 |
2.4.2.5 The Kriging models | p. 88 |
2.4.2.6 Directions of generalization | p. 94 |
2.5 Uniform Minimum Variance Unbiased Estimation | p. 95 |
2.6 Density functional of a gaussian process and applications to extraction and detection problems | p. 97 |
2.6.1 Density functional of a gaussian process | p. 97 |
2.6.2 Minimum variance unbiased estimation of the mean value of a gaussian process with known covariance | p. 100 |
2.6.3 Applications to extraction problems | p. 102 |
2.6.4 Applications to detection problems | p. 104 |
2.7 Exercises | p. 105 |
3. Nonparametric Curve Estimation | p. 109 |
3.1 Introduction | p. 109 |
3.2 A brief introduction to splines | p. 110 |
3.2.1 Abstract Interpolating splines | p. 111 |
3.2.2 Abstract smoothing splines | p. 116 |
3.2.3 Partial and mixed splines | p. 118 |
3.2.4 Some concrete splines | p. 121 |
3.2.4.1 D[superscript m] splines | p. 121 |
3.2.4.2 Periodic D[superscript m] splines | p. 122 |
3.2.4.3 L splines | p. 123 |
3.2.4.4 [alpha]-splines, thin plate splines and Duchon's rotation invariant splines | p. 123 |
3.2.4.5 Other splines | p. 124 |
3.3 Random interpolating splines | p. 125 |
3.4 Spline regression estimation | p. 125 |
3.4.1 Least squares spline estimators | p. 126 |
3.4.2 Smoothing spline estimators | p. 127 |
3.4.3 Hybrid splines | p. 128 |
3.4.4 Bayesian models | p. 129 |
3.5 Spline density estimation | p. 132 |
3.6 Shape restrictions in curve estimation | p. 134 |
3.7 Unbiased density estimation | p. 135 |
3.8 Kernels and higher order kernels | p. 136 |
3.9 Local approximation of functions | p. 143 |
3.10 Local polynomial smoothing of statistical functionals | p. 148 |
3.10.1 Density estimation in selection bias models | p. 150 |
3.10.2 Hazard functions | p. 152 |
3.10.3 Reliability and econometric functions | p. 154 |
3.11 Kernels of order (m, p) | p. 155 |
3.11.1 Definition of K[subscript o]-based hierarchies | p. 158 |
3.11.2 Computational aspects | p. 160 |
3.11.3 Sequences of hierarchies | p. 165 |
3.11.4 Optimality properties of higher order kernels | p. 167 |
3.11.5 The multiple kernel method | p. 171 |
3.11.6 The estimation procedure for the density and its derivatives | p. 172 |
3.12 Exercises | p. 175 |
4. Measures and Random Measures | p. 185 |
4.1 Introduction | p. 185 |
4.1.1 Dirac measures | p. 186 |
4.1.2 General approach | p. 190 |
4.1.3 The example of moments | p. 192 |
4.2 Measurability of RKHS-valued variables | p. 194 |
4.3 Gaussian measure on RKHS | p. 196 |
4.3.1 Gaussian measure and gaussian process | p. 196 |
4.3.2 Construction of gaussian measures | p. 198 |
4.4 Weak convergence in Pr(H) | p. 199 |
4.4.1 Weak convergence criterion | p. 202 |
4.5 Integration of H-valued random variables | p. 202 |
4.5.1 Notation. Definitions | p. 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 measures | p. 210 |
4.7 Inner product and weak topology | p. 214 |
4.8 Application to normal approximation | p. 218 |
4.9 Random measures | p. 220 |
4.9.1 The empirical measure as H-valued variable | p. 223 |
4.9.1.1 Integrable kernels | p. 224 |
4.9.1.2 Estimation of I[subscript mu] | p. 228 |
4.9.2 Convergence of random measures | p. 232 |
4.10 Exercises | p. 234 |
5. Miscellaneous Applications | p. 241 |
5.1 Introduction | p. 241 |
5.2 Law of Iterated Logarithm | p. 241 |
5.3 Learning and decision theory | p. 245 |
5.3.1 Binary classification with RKHS | p. 245 |
5.3.2 Support Vector Machine | p. 248 |
5.4 ANOVA in function spaces | p. 249 |
5.4.1 ANOVA decomposition of a function on a product domain | p. 249 |
5.4.2 Tensor product smoothing splines | p. 252 |
5.4.3 Regression with tensor product splines | p. 254 |
5.5 Strong approximation in RKHS | p. 255 |
5.6 Generalized method of moments | p. 259 |
5.7 Exercises | p. 262 |
6. Computational Aspects | p. 265 |
6.1 Kernel of a given normed space | p. 266 |
6.1.1 Kernel of a finite dimensional space | p. 266 |
6.1.2 Kernel of some subspaces | p. 266 |
6.1.3 Decomposition principle | p. 267 |
6.1.4 Kernel of a class of periodic functions | p. 268 |
6.1.5 A family of Beppo-Levi spaces | p. 270 |
6.1.6 Sobolev spaces endowed with a variety of norms | p. 276 |
6.1.6.1 First family of norms | p. 277 |
6.1.6.2 Second family of norms | p. 285 |
6.2 Norm and space corresponding to a given reproducing kernel | p. 288 |
6.3 Exercises | p. 289 |
7. A Collection of Examples | p. 293 |
7.1 Introduction | p. 293 |
7.2 Using the characterization theorem | p. 293 |
7.2.1 Case of finite X | p. 294 |
7.2.2 Case of countably infinite X | p. 294 |
7.2.3 Using any mapping from E into some pre-Hilbert space | p. 295 |
7.3 Factorizable kernels | p. 295 |
7.4 Examples of spaces, norms and kernels | p. 299 |
Appendix | p. 344 |
Introduction to Sobolev spaces | p. 345 |
A.1 Schwartz-distributions or generalized functions | p. 345 |
A.1.1 Spaces and their topology | p. 345 |
A.1.2 Weak-derivative or derivative in the sense of distributions | p. 346 |
A.1.3 Facts about Fourier transforms | p. 346 |
A.2 Sobolev spaces | p. 346 |
A.2.1 Absolute continuity of functions of one variable | p. 346 |
A.2.2 Sobolev space with non negative integer exponent | p. 347 |
A.2.3 Sobolev space with real exponent | p. 348 |
A.2.4 Periodic Sobolev space | p. 349 |
A.3 Beppo-Levi spaces | p. 349 |
Index | p. 353 |