Reproducing kernel Hilbert spaces in probability and statistics:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston [u.a.]
Kluwer
2004
|
| Subjects: | |
| Online Access: | Publisher description Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references (p. 327-343) and index |
| Physical Description: | XXII, 355 S. graph. Darst. 25 cm |
| ISBN: | 9781402076794 |
Staff View
MARC
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| 100 | 1 | |a Berlinet, Alain |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Reproducing kernel Hilbert spaces in probability and statistics |c Alain Berlinet ; Christine Thomas-Agnan |
| 264 | 1 | |a Boston [u.a.] |b Kluwer |c 2004 | |
| 300 | |a XXII, 355 S. |b graph. Darst. |c 25 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references (p. 327-343) and index | ||
| 650 | 7 | |a Espaços de hilbert |2 larpcal | |
| 650 | 7 | |a Estatística |2 larpcal | |
| 650 | 4 | |a Hilbert, Espace de | |
| 650 | 4 | |a Noyaux (Mathématiques) | |
| 650 | 7 | |a Probabilidade |2 larpcal | |
| 650 | 4 | |a Probabilités | |
| 650 | 4 | |a Statistique mathématique | |
| 650 | 4 | |a Hilbert space | |
| 650 | 4 | |a Kernel functions | |
| 650 | 4 | |a Probabilities | |
| 650 | 4 | |a Mathematical statistics | |
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| 650 | 0 | 7 | |a Statistik |0 (DE-588)4056995-0 |2 gnd |9 rswk-swf |
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| 700 | 1 | |a Thomas-Agnan, Christine |e Verfasser |4 aut | |
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Record in the Search Index
| _version_ | 1804138477574422528 |
|---|---|
| adam_text | Contents
Preface
xiii
Acknowledgments
xvii
Introduction
xix
1.
THEORY
1
1.1
Introduction
1
1.2
Notation and basic definitions
3
1.3
Reproducing kernels and positive type functions
13
1.4
Basic properties of reproducing kernels
24
1.4.1
Sum of reproducing kernels
24
1.4.2
Restriction of the index set
25
1.4.3
Support of a reproducing kernel
26
1.4.4
Kernel of an operator
27
1.4.5
Condition for
Чк С
Kr.
30
1.4.6
Tensor products of RKHS
30
1.5
Separability. Continuity
31
1.6
Extensions
37
1.6.1
Schwartz kernels
37
1.6.2
Semi-kernels
40
1.7
Positive type operators
42
1.7.1
Continuous functions of positive type
42
1.7.2
Schwartz distributions of positive type or conditionally
of positive type
44
1.8
Exercises
48
viii RKHS
IN PROBABILITY AND STATISTICS
2.
RKHS AND STOCHASTIC PROCESSES
55
2.1
Introduction
55
2.2
Covariance function of a second order stochastic process
55
2.2.1
Case of ordinary stochastic processes
55
2.2.1.1
Case of generalized stochastic processes
56
2.2.2
Positivity
and covariance
57
2.2.2.1
Positive type functions and covariance functions
57
2.2.2.2
Generalized covariances and conditionally of positive
type functions
59
2.2.3
Hubert space generated by a process
62
2.3
Representation theorems
64
2.3.1
The
Loève
representation theorem
65
2.3.2
The Mercer representation theorem
68
2.3.3
The Karhunen representation theorem
70
2.3.4
Applications
72
2.4
Applications to stochastic filtering
75
2.4.1
Best Prediction
76
2.4.1.1
Best prediction and best linear prediction
76
2.4.1.2
Best linear unbiased prediction
79
2.4.2
Filtering and spline functions
80
2.4.2.1
No drift-no noise model and interpolating splines
82
2.4.2.2
Noise without drift model and smoothing splines
83
2.4.2.3
Complete model and partial smoothing splines
84
2.4.2.4
Case of
gaussian
processes
86
2.4.2.5
The Kriging models
88
2.4.2.6
Directions of generalization
94
2.5
Uniform Minimum Variance Unbiased Estimation
95
2.6
Density functional of
a gaussian
process and applications
to extraction and detection problems
97
2.6.1
Density functional of
a gaussian
process
97
2.6.2
Minimum variance unbiased estimation of the mean
value of
a gaussian
process with known covariance
100
2.6.3
Applications to extraction problems
102
2.6.4
Applications to detection problems
104
2.7
Exercises
105
3.
NONPARAMETRIC CURVE ESTIMATION
109
3.1
Introduction
109
3.2
A brief introduction to splines
110
3.2.1
Abstract Interpolating splines 111
3.2.2
Abstract smoothing splines
116
3.2.3
Partial and mixed splines
118
3.2.4
Some concrete splines
121
3.2.4.1
Dm splines
121
Contents jx
3.2.4.2
Periodic Dm
splines
122
3.2.4.3
L
splines
123
3.2.4.4
a-splines, thin plate splines and Duchon s rotation
invariant splines
123
3.2.4.5
Other splines
124
3.3
Random interpolating splines
125
3.4
Spline regression estimation
125
3.4.1
Least squares spline estimators
126
3.4.2
Smoothing spline estimators
127
3.4.3
Hybrid splines
128
3.4.4
Bayesian models
129
3.5
Spline density estimation
132
3.6
Shape restrictions in curve estimation
134
3.7
Unbiased density estimation
135
3.8
Kernels and higher order kernels
136
3.9
Local approximation of functions
143
3.10
Local polynomial smoothing of statistical
funcţionale
148
3.10.1
Density estimation in selection bias models.
150
3.10.2
Hazard functions
152
3.10.3
Reliability and econometric functions.
154
3.11
Kernels of order (m,
р)
155
3.11.1
Definition of Kq-based hierarchies
158
3.11.2
Computational aspects
160
3.11.3
Sequences of hierarchies
165
3.11.4
Optimality properties of higher order kernels
167
3.11.5
The multiple kernel method
171
3.11.6
The estimation procedure for the density and its
derivatives
172
3.12
Exercises
175
4.
MEASURES AND RANDOM MEASURES
185
4.1
Introduction
185
4.1.1
Dirac measures
186
4.1.2
General approach
190
4.1.3
The example of moments
192
4.2
Measurability of RKHS-valued variables
194
4.3
Gaussian measure on RKHS
196
4.3.1
Gaussian measure and
gaussian
process
196
4.3.2
Construction of
gaussian
measures
198
4.4
Weak convergence in
Рг(Ћ)
199
4.4.1
Weak convergence criterion
202
4.5
Integration of %-valued random variables
202
4.5.1
Notation. Definitions
203
x
RKHS IN PROBABILITY AND STATISTICS
4.5.2
Integrability of X and of {Xt
:
t
Є
E}.
205
4.6
Inner products on sets of measures
210
4.7
Inner product and weak topology
214
4.8
Application to normal approximation
218
4.9
Random measures
220
4.9.1
The empirical measure as ft-valued variable
223
4.9.1.1
Integrable
kernels
224
4.9.1.2
Estimation of
1μ
228
4.9.2
Convergence of random measures
232
4.10
Exercises
234
5.
MISCELLANEOUS APPLICATIONS
241
5.1
Introduction
241
5.2
Law of Iterated Logarithm
241
5.3
Learning and decision theory
245
5.3.1
Binary classification with RKHS
245
5.3.2
Support Vector Machine
248
5.4
ANOVA in function spaces
249
5.4.1
ANOVA decomposition of a function on a product
domain
249
5.4.2
Tensor product smoothing splines
252
5.4.3
Regression with tensor product splines
254
5.5
Strong approximation in RKHS
255
5.6
Generalized method of moments
259
5.7
Exercises
262
6.
COMPUTATIONAL ASPECTS
265
6.1
Kernel of a given normed space
266
6.1.1
Kernel of a finite dimensional space
266
6.1.2
Kernel of some subspaces
266
6.1.3
Decomposition principle
267
6.1.4
Kernel of a class of periodic functions
268
6.1.5
A family of Beppo-Levi spaces
270
6.1.6
Sobolev spaces endowed with a variety of norms
276
6.1.6.1
First family of norms
277
6.1.6.2
Second family of norms
285
6.2
Norm and space corresponding to a given reproducing
kernel
288
6.3
Exercises
289
Contents xi
7.
A COLLECTION OF EXAMPLES
293
7.1
Introduction
293
7.2
Using the characterization theorem
293
7.2.1
Case of finite X
294
7.2.2
Case of countably infinite X
294
7.2.3
Using any mapping from
E
into some pre-Hilbert
space
295
7.3
Factorizable kernels
295
7.4
Examples of spaces, norms and kernels
299
Appendix
344
Introduction to Sobolev spaces
345
A.I Schwartz-distributions or generalized functions
345
A.
1.1
Spaces and their topology
345
A.
1.2
Weak-derivative or derivative in the sense of distributions
346
A.
1.3
Facts about Fourier transforms
346
A.2 Sobolev spaces
346
A.2.1 Absolute continuity of functions of one variable
346
A.2.2 Sobolev space with
non
negative integer exponent
347
A.2.3 Sobolev space with real exponent
348
A.2.4 Periodic Sobolev space
349
A.3 Beppo-Levi spaces
349
Index
353
|
| any_adam_object | 1 |
| author | Berlinet, Alain Thomas-Agnan, Christine |
| author_facet | Berlinet, Alain Thomas-Agnan, Christine |
| author_role | aut aut |
| author_sort | Berlinet, Alain |
| author_variant | a b ab c t a cta |
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| ctrlnum | (OCoLC)53285165 (DE-599)BVBBV035216465 |
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| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.733 |
| dewey-search | 515/.733 |
| dewey-sort | 3515 3733 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV035216465 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T21:28:49Z |
| institution | BVB |
| isbn | 9781402076794 |
| language | English |
| lccn | 2003064182 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017022616 |
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| owner_facet | DE-739 DE-11 DE-91G DE-BY-TUM |
| physical | XXII, 355 S. graph. Darst. 25 cm |
| publishDate | 2004 |
| publishDateSearch | 2004 |
| publishDateSort | 2004 |
| publisher | Kluwer |
| record_format | marc |
| spelling | Berlinet, Alain Verfasser aut Reproducing kernel Hilbert spaces in probability and statistics Alain Berlinet ; Christine Thomas-Agnan Boston [u.a.] Kluwer 2004 XXII, 355 S. graph. Darst. 25 cm txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references (p. 327-343) and index Espaços de hilbert larpcal Estatística larpcal Hilbert, Espace de Noyaux (Mathématiques) Probabilidade larpcal Probabilités Statistique mathématique Hilbert space Kernel functions Probabilities Mathematical statistics Kernfunktion (DE-588)4163607-7 gnd rswk-swf Statistik (DE-588)4056995-0 gnd rswk-swf Hilbert-Raum (DE-588)4159850-7 gnd rswk-swf Hilbert-Raum (DE-588)4159850-7 s Kernfunktion (DE-588)4163607-7 s Statistik (DE-588)4056995-0 s 1\p DE-604 b DE-604 Thomas-Agnan, Christine Verfasser aut Erscheint auch als Online-Ausgabe 978-1-4419-9096-9 http://www.loc.gov/catdir/enhancements/fy0819/2003064182-d.html Publisher description Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017022616&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Berlinet, Alain Thomas-Agnan, Christine Reproducing kernel Hilbert spaces in probability and statistics Espaços de hilbert larpcal Estatística larpcal Hilbert, Espace de Noyaux (Mathématiques) Probabilidade larpcal Probabilités Statistique mathématique Hilbert space Kernel functions Probabilities Mathematical statistics Kernfunktion (DE-588)4163607-7 gnd Statistik (DE-588)4056995-0 gnd Hilbert-Raum (DE-588)4159850-7 gnd |
| subject_GND | (DE-588)4163607-7 (DE-588)4056995-0 (DE-588)4159850-7 |
| title | Reproducing kernel Hilbert spaces in probability and statistics |
| title_auth | Reproducing kernel Hilbert spaces in probability and statistics |
| title_exact_search | Reproducing kernel Hilbert spaces in probability and statistics |
| title_full | Reproducing kernel Hilbert spaces in probability and statistics Alain Berlinet ; Christine Thomas-Agnan |
| title_fullStr | Reproducing kernel Hilbert spaces in probability and statistics Alain Berlinet ; Christine Thomas-Agnan |
| title_full_unstemmed | Reproducing kernel Hilbert spaces in probability and statistics Alain Berlinet ; Christine Thomas-Agnan |
| title_short | Reproducing kernel Hilbert spaces in probability and statistics |
| title_sort | reproducing kernel hilbert spaces in probability and statistics |
| topic | Espaços de hilbert larpcal Estatística larpcal Hilbert, Espace de Noyaux (Mathématiques) Probabilidade larpcal Probabilités Statistique mathématique Hilbert space Kernel functions Probabilities Mathematical statistics Kernfunktion (DE-588)4163607-7 gnd Statistik (DE-588)4056995-0 gnd Hilbert-Raum (DE-588)4159850-7 gnd |
| topic_facet | Espaços de hilbert Estatística Hilbert, Espace de Noyaux (Mathématiques) Probabilidade Probabilités Statistique mathématique Hilbert space Kernel functions Probabilities Mathematical statistics Kernfunktion Statistik Hilbert-Raum |
| url | http://www.loc.gov/catdir/enhancements/fy0819/2003064182-d.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017022616&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT berlinetalain reproducingkernelhilbertspacesinprobabilityandstatistics AT thomasagnanchristine reproducingkernelhilbertspacesinprobabilityandstatistics |