Reproducing kernel Hilbert spaces in probability and statistics /

Reproducing kernel Hilbert spaces in probability and statistics /

Saved in:
Bibliographic Details
Author / Creator:Berlinet, A.
Imprint:Boston : Kluwer Academic, c2004.
Description:xxii, 355 p. : ill. ; 25 cm.
Language:English
Subject:
Hilbert space.
Kernel functions.
Probabilities.
Mathematical statistics.
Hilbert space.
Kernel functions.
Mathematical statistics.
Probabilities.
Format: Print Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/5058289
Hidden Bibliographic Details
Other authors / contributors:Thomas-Agnan, Christine.
ISBN:1402076797
Notes:Includes bibliographical references (p. 327-343) and index.
Table of Contents:
  • Preface
  • Acknowledgments
  • Introduction
  • 1.. Theory
  • 1.1. Introduction
  • 1.2. Notation and basic definitions
  • 1.3. Reproducing kernels and positive type functions
  • 1.4. Basic properties of reproducing kernels
  • 1.4.1. Sum of reproducing kernels
  • 1.4.2. Restriction of the index set
  • 1.4.3. Support of a reproducing kernel
  • 1.4.4. Kernel of an operator
  • 1.4.5. Condition for H[subscript K] [subset or is implied by] H[subscript R]
  • 1.4.6. Tensor products of RKHS
  • 1.5. Separability. Continuity
  • 1.6. Extensions
  • 1.6.1. Schwartz kernels
  • 1.6.2. Semi-kernels
  • 1.7. Positive type operators
  • 1.7.1. Continuous functions of positive type
  • 1.7.2. Schwartz distributions of positive type or conditionally of positive type
  • 1.8. Exercises
  • 2.. Rkhs and Stochastic Processes
  • 2.1. Introduction
  • 2.2. Covariance function of a second order stochastic process
  • 2.2.1. Case of ordinary stochastic processes
  • 2.2.1.1. Case of generalized stochastic processes
  • 2.2.2. Positivity and covariance
  • 2.2.2.1. Positive type functions and covariance functions
  • 2.2.2.2. Generalized covariances and conditionally of positive type functions
  • 2.2.3. Hilbert space generated by a process
  • 2.3. Representation theorems
  • 2.3.1. The Loeve representation theorem
  • 2.3.2. The Mercer representation theorem
  • 2.3.3. The Karhunen representation theorem
  • 2.3.4. Applications
  • 2.4. Applications to stochastic filtering
  • 2.4.1. Best Prediction
  • 2.4.1.1. Best prediction and best linear prediction
  • 2.4.1.2. Best linear unbiased prediction
  • 2.4.2. Filtering and spline functions
  • 2.4.2.1. No drift-no noise model and interpolating splines
  • 2.4.2.2. Noise without drift model and smoothing splines
  • 2.4.2.3. Complete model and partial smoothing splines
  • 2.4.2.4. Case of gaussian processes
  • 2.4.2.5. The Kriging models
  • 2.4.2.6. Directions of generalization
  • 2.5. Uniform Minimum Variance Unbiased Estimation
  • 2.6. Density functional of a gaussian process and applications to extraction and detection problems
  • 2.6.1. Density functional of a gaussian process
  • 2.6.2. Minimum variance unbiased estimation of the mean value of a gaussian process with known covariance
  • 2.6.3. Applications to extraction problems
  • 2.6.4. Applications to detection problems
  • 2.7. Exercises
  • 3.. Nonparametric Curve Estimation
  • 3.1. Introduction
  • 3.2. A brief introduction to splines
  • 3.2.1. Abstract Interpolating splines
  • 3.2.2. Abstract smoothing splines
  • 3.2.3. Partial and mixed splines
  • 3.2.4. Some concrete splines
  • 3.2.4.1. D[superscript m] splines
  • 3.2.4.2. Periodic D[superscript m] splines
  • 3.2.4.3. L splines
  • 3.2.4.4. [alpha]-splines, thin plate splines and Duchon's rotation invariant splines
  • 3.2.4.5. Other splines
  • 3.3. Random interpolating splines
  • 3.4. Spline regression estimation
  • 3.4.1. Least squares spline estimators
  • 3.4.2. Smoothing spline estimators
  • 3.4.3. Hybrid splines
  • 3.4.4. Bayesian models
  • 3.5. Spline density estimation
  • 3.6. Shape restrictions in curve estimation
  • 3.7. Unbiased density estimation
  • 3.8. Kernels and higher order kernels
  • 3.9. Local approximation of functions
  • 3.10. Local polynomial smoothing of statistical functionals
  • 3.10.1. Density estimation in selection bias models
  • 3.10.2. Hazard functions
  • 3.10.3. Reliability and econometric functions
  • 3.11. Kernels of order (m, p)
  • 3.11.1. Definition of K[subscript o]-based hierarchies
  • 3.11.2. Computational aspects
  • 3.11.3. Sequences of hierarchies
  • 3.11.4. Optimality properties of higher order kernels
  • 3.11.5. The multiple kernel method
  • 3.11.6. The estimation procedure for the density and its derivatives
  • 3.12. Exercises
  • 4.. Measures and Random Measures
  • 4.1. Introduction
  • 4.1.1. Dirac measures
  • 4.1.2. General approach
  • 4.1.3. The example of moments
  • 4.2. Measurability of RKHS-valued variables
  • 4.3. Gaussian measure on RKHS
  • 4.3.1. Gaussian measure and gaussian process
  • 4.3.2. Construction of gaussian measures
  • 4.4. Weak convergence in Pr(H)
  • 4.4.1. Weak convergence criterion
  • 4.5. Integration of H-valued random variables
  • 4.5.1. Notation. Definitions
  • 4.5.2. Integrability of X and of {{X[superscript t] : t [set mempership] E}}
  • 4.6. Inner products on sets of measures
  • 4.7. Inner product and weak topology
  • 4.8. Application to normal approximation
  • 4.9. Random measures
  • 4.9.1. The empirical measure as H-valued variable
  • 4.9.1.1. Integrable kernels
  • 4.9.1.2. Estimation of I[subscript mu]
  • 4.9.2. Convergence of random measures
  • 4.10. Exercises
  • 5.. Miscellaneous Applications
  • 5.1. Introduction
  • 5.2. Law of Iterated Logarithm
  • 5.3. Learning and decision theory
  • 5.3.1. Binary classification with RKHS
  • 5.3.2. Support Vector Machine
  • 5.4. ANOVA in function spaces
  • 5.4.1. ANOVA decomposition of a function on a product domain
  • 5.4.2. Tensor product smoothing splines
  • 5.4.3. Regression with tensor product splines
  • 5.5. Strong approximation in RKHS
  • 5.6. Generalized method of moments
  • 5.7. Exercises
  • 6.. Computational Aspects
  • 6.1. Kernel of a given normed space
  • 6.1.1. Kernel of a finite dimensional space
  • 6.1.2. Kernel of some subspaces
  • 6.1.3. Decomposition principle
  • 6.1.4. Kernel of a class of periodic functions
  • 6.1.5. A family of Beppo-Levi spaces
  • 6.1.6. Sobolev spaces endowed with a variety of norms
  • 6.1.6.1. First family of norms
  • 6.1.6.2. Second family of norms
  • 6.2. Norm and space corresponding to a given reproducing kernel
  • 6.3. Exercises
  • 7.. A Collection of Examples
  • 7.1. Introduction
  • 7.2. Using the characterization theorem
  • 7.2.1. Case of finite X
  • 7.2.2. Case of countably infinite X
  • 7.2.3. Using any mapping from E into some pre-Hilbert space
  • 7.3. Factorizable kernels
  • 7.4. Examples of spaces, norms and kernels
  • Appendix
  • Introduction to Sobolev spaces
  • A.1. Schwartz-distributions or generalized functions
  • A.1.1. Spaces and their topology
  • A.1.2. Weak-derivative or derivative in the sense of distributions
  • A.1.3. Facts about Fourier transforms
  • A.2. Sobolev spaces
  • A.2.1. Absolute continuity of functions of one variable
  • A.2.2. Sobolev space with non negative integer exponent
  • A.2.3. Sobolev space with real exponent
  • A.2.4. Periodic Sobolev space
  • A.3. Beppo-Levi spaces
  • Index
Purchased from The Ed & Martha Vanderwicken Book Fund bookplate

Similar Items