Random fields estimation:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Hackensack, NJ [u.a.]
World Scientific
2005
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| Subjects: | |
| Online Access: | Table of contents only Inhaltsverzeichnis |
| Item Description: | Based partly on the author's earlier book: Random fields estimation theory. Harlow, Essex, England : Longman Scientific & Technical ; New York : Wiley, 1990. Includes bibliographical references (p. 363-369) and index |
| Physical Description: | XIII, 373 S. 24 cm |
| ISBN: | 9789812565365 9812565361 |
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| adam_text | Titel: Random fields estimation
Autor: Ramm, Alexander G.
Jahr: 2005
Contents
Preface vii
1. Introduction 1
2. Formulation of Basic Results 9
2.1 Statement of the problem................... 9
2.2 Formulation of the results (multidimensional case)..... 14
2.2.1 Basic results....................... 14
2.2.2 Generalizations..................... 17
2.3 Formulation of the results (one-dimensional case)...... 18
2.3.1 Basic results for the scalar equation.......... 19
2.3.2 Vector equations.................... 22
2.4 Examples of kernels of class R and solutions to the basic
equation............................. 25
2.5 Formula for the error of the optimal estimate........ 29
3. Numerical Solution of the Basic Integral
Equation in Distributions 33
3.1 Basic ideas ........................... 33
3.2 Theoretical approaches..................... 37
3.3 Multidimensional equation................... 43
3.4 Numerical solution based on the approximation of the kernel 46
3.5 Asymptotic behavior of the optimal filter as the white noise
component goes to zero.................... 54
3.6 A general approach....................... 57
÷ Random Fields Estimation Theory
4. Proofs 65
4.1 Proof of Theorem 2.1...................... 65
4.2 Proof of Theorem 2.2...................... 73
4.3 Proof of Theorems 2.4 and 2.5 ................ 79
4.4 Another approach ....................... 84
5. Singular Perturbation Theory for a Class of
Fredholm Integral Equations Arising in Random
Fields Estimation Theory 87
5.1 Introduction........................... 87
5.2 Auxiliary results........................ 90
5.3 Asymptotics in the case n = 1................. 93
5.4 Examples of asymptotical solutions: case n = 1....... 98
5.5 Asymptotics in the case n l................. 103
5.6 Examples of asymptotical solutions: case n 1....... 105
6. Estimation and Scattering Theory 111
6.1 The direct scattering problem ................ Ill
6.1.1 The direct scattering problem............. Ill
6.1.2 Properties of the scattering solution ......... 114
6.1.3 Properties of the scattering amplitude........ 120
6.1.4 Analyticity in k of the scattering solution...... 121
6.1.5 High-frequency behavior of the scattering solutions . 123
6.1.6 Fundamental relation between u+ and u-...... 127
6.1.7 Formula for det S(k) and state the Levinson Theorem 128
6.1.8 Completeness properties of the scattering solutions . 131
6.2 Inverse scattering problems.................. 134
6.2.1 Inverse scattering problems.............. 134
6.2.2 Uniqueness theorem for the inverse scattering problem 134
6.2.3 Necessary conditions for a function to be a scatterng
amplitude........................ 135
6.2.4 A Marchenko equation (M equation)......... 136
6.2.5 Characterization of the scattering data in the 3D in-
verse scattering problem................ 138
6.2.6 The Born inversion................... 141
6.3 Estimation theory and inverse scattering in R3....... 150
7. Applications 159
Contents xi
7.1 What is the optimal size of the domain on which the data
are to be collected?....................... 159
7.2 Discrimination of random fields against noisy background . 161
7.3 Quasioptimal estimates of derivatives of random functions . 169
7.3.1 Introduction....................... 169
7.3.2 Estimates of the derivatives.............. 170
7.3.3 Derivatives of random functions............ 172
7.3.4 Finding critical points................. 180
7.3.5 Derivatives of random fields.............. 181
7.4 Stable summation of orthogonal series and integrals with
randomly perturbed coefficients................ 182
7.4.1 Introduction....................... 182
7.4.2 Stable summation of series............... 184
7.4.3 Method of multipliers.................. 185
7.5 Resolution ability of linear systems.............. 185
7.5.1 Introduction....................... 185
7.5.2 Resolution ability of linear systems.......... 187
7.5.3 Optimization of resolution ability........... 191
7.5.4 A general definition of resolution ability....... 196
7.6 Ill-posed problems and estimation theory .......... 198
7.6.1 Introduction....................... 198
7.6.2 Stable solution of ill-posed problems......... 205
7.6.3 Equations with random noise............. 216
7.7 A remark on nonlinear (polynomial) estimates........ 230
8. Auxiliary Results 233
8.1 Sobolev spaces and distributions............... 233
8.1.1 A general imbedding theorem............. 233
8.1.2 Sobolev spaces with negative indices......... 236
8.2 Eigenfunction expansions for elliptic selfadjoint operators . 241
8.2.1 Resoluion of the identity and integral representation
of selfadjoint operators................. 241
8.2.2 Differentiation of operator measures ......... 242
8.2.3 Carleman operators................... 246
8.2.4 Elements of the spectral theory of elliptic operators
in L2(Rr)........................ 249
8.3 Asymptotics of the spectrum of linear operators....... 260
8.3.1 Compact operators................... 260
8.3.1.1 Basic definitions................ 260
xii Random Fields Estimation Theory
8.3.1.2 Minimax principles and estimates of eigen-
values and singular values.......... 262
8.3.2 Perturbations preserving asymptotics of the spectrum
of compact operators.................. 265
8.3.2.1 Statement of the problem .......... 265
8.3.2.2 A characterization of the class of linear com-
pact operators................. 266
8.3.2.3 Asymptotic equivalence of s-values of two op-
erators ..................... 268
8.3.2.4 Estimate of the remainder.......... 270
8.3.2.5 Unbounded operators............. 274
8.3.2.6 Asymptotics of eigenvalues.......... 275
8.3.2.7 Asymptotics of eigenvalues (continuation) . 283
8.3.2.8 Asymptotics of s-values............ 284
8.3.2.9 Asymptotics of the spectrum for quadratic
forms...................... 287
8.3.2.10 Proof of Theorem 2.3............. 293
8.3.3 Trace class and Hilbert-Schmidt operators...... 297
8.3.3.1 Trace class operators............. 297
8.3.3.2 Hilbert-Schmidt operators.......... 298
8.3.3.3 Determinants of operators.......... 299
8.4 Elements of probability theory ................ 300
8.4.1 The probability space and basic definitions...... 300
8.4.2 Hubert space theory.................. 306
8.4.3 Estimation in Hubert space L2(O ,U, P) ....... 310
8.4.4 Homogeneous and isotropic random fields...... 312
8.4.5 Estimation of parameters ............... 315
8.4.6 Discrimination between hypotheses.......... 317
8.4.7 Generalized random fields............... 319
8.4.8 Kaiman filters...................... 320
Appendix A Analytical Solution of the Basic
Integral Equation for a Class of One-Dimensional
Problems 325
A.l Introduction........................... 326
A.2 Proofs.............................. 329
Appendix B Integral Operators Basic in Random
Fields Estimation Theory 337
Contents xiii
B.l Introduction.......................... 337
B.2 Reduction of the basic integral equation to a boundary-value
problem............................. 341
B.3 Isomorphism property..................... 349
B.4 Auxiliary material....................... 354
Bibliographical Notes 359
Bibliography 363
Symbols 371
Index 373
|
| adam_txt |
Titel: Random fields estimation
Autor: Ramm, Alexander G.
Jahr: 2005
Contents
Preface vii
1. Introduction 1
2. Formulation of Basic Results 9
2.1 Statement of the problem. 9
2.2 Formulation of the results (multidimensional case). 14
2.2.1 Basic results. 14
2.2.2 Generalizations. 17
2.3 Formulation of the results (one-dimensional case). 18
2.3.1 Basic results for the scalar equation. 19
2.3.2 Vector equations. 22
2.4 Examples of kernels of class R and solutions to the basic
equation. 25
2.5 Formula for the error of the optimal estimate. 29
3. Numerical Solution of the Basic Integral
Equation in Distributions 33
3.1 Basic ideas . 33
3.2 Theoretical approaches. 37
3.3 Multidimensional equation. 43
3.4 Numerical solution based on the approximation of the kernel 46
3.5 Asymptotic behavior of the optimal filter as the white noise
component goes to zero. 54
3.6 A general approach. 57
÷ Random Fields Estimation Theory
4. Proofs 65
4.1 Proof of Theorem 2.1. 65
4.2 Proof of Theorem 2.2. 73
4.3 Proof of Theorems 2.4 and 2.5 . 79
4.4 Another approach . 84
5. Singular Perturbation Theory for a Class of
Fredholm Integral Equations Arising in Random
Fields Estimation Theory 87
5.1 Introduction. 87
5.2 Auxiliary results. 90
5.3 Asymptotics in the case n = 1. 93
5.4 Examples of asymptotical solutions: case n = 1. 98
5.5 Asymptotics in the case n l. 103
5.6 Examples of asymptotical solutions: case n 1. 105
6. Estimation and Scattering Theory 111
6.1 The direct scattering problem . Ill
6.1.1 The direct scattering problem. Ill
6.1.2 Properties of the scattering solution . 114
6.1.3 Properties of the scattering amplitude. 120
6.1.4 Analyticity in k of the scattering solution. 121
6.1.5 High-frequency behavior of the scattering solutions . 123
6.1.6 Fundamental relation between u+ and u-. 127
6.1.7 Formula for det S(k) and state the Levinson Theorem 128
6.1.8 Completeness properties of the scattering solutions . 131
6.2 Inverse scattering problems. 134
6.2.1 Inverse scattering problems. 134
6.2.2 Uniqueness theorem for the inverse scattering problem 134
6.2.3 Necessary conditions for a function to be a scatterng
amplitude. 135
6.2.4 A Marchenko equation (M equation). 136
6.2.5 Characterization of the scattering data in the 3D in-
verse scattering problem. 138
6.2.6 The Born inversion. 141
6.3 Estimation theory and inverse scattering in R3. 150
7. Applications 159
Contents xi
7.1 What is the optimal size of the domain on which the data
are to be collected?. 159
7.2 Discrimination of random fields against noisy background . 161
7.3 Quasioptimal estimates of derivatives of random functions . 169
7.3.1 Introduction. 169
7.3.2 Estimates of the derivatives. 170
7.3.3 Derivatives of random functions. 172
7.3.4 Finding critical points. 180
7.3.5 Derivatives of random fields. 181
7.4 Stable summation of orthogonal series and integrals with
randomly perturbed coefficients. 182
7.4.1 Introduction. 182
7.4.2 Stable summation of series. 184
7.4.3 Method of multipliers. 185
7.5 Resolution ability of linear systems. 185
7.5.1 Introduction. 185
7.5.2 Resolution ability of linear systems. 187
7.5.3 Optimization of resolution ability. 191
7.5.4 A general definition of resolution ability. 196
7.6 Ill-posed problems and estimation theory . 198
7.6.1 Introduction. 198
7.6.2 Stable solution of ill-posed problems. 205
7.6.3 Equations with random noise. 216
7.7 A remark on nonlinear (polynomial) estimates. 230
8. Auxiliary Results 233
8.1 Sobolev spaces and distributions. 233
8.1.1 A general imbedding theorem. 233
8.1.2 Sobolev spaces with negative indices. 236
8.2 Eigenfunction expansions for elliptic selfadjoint operators . 241
8.2.1 Resoluion of the identity and integral representation
of selfadjoint operators. 241
8.2.2 Differentiation of operator measures . 242
8.2.3 Carleman operators. 246
8.2.4 Elements of the spectral theory of elliptic operators
in L2(Rr). 249
8.3 Asymptotics of the spectrum of linear operators. 260
8.3.1 Compact operators. 260
8.3.1.1 Basic definitions. 260
xii Random Fields Estimation Theory
8.3.1.2 Minimax principles and estimates of eigen-
values and singular values. 262
8.3.2 Perturbations preserving asymptotics of the spectrum
of compact operators. 265
8.3.2.1 Statement of the problem . 265
8.3.2.2 A characterization of the class of linear com-
pact operators. 266
8.3.2.3 Asymptotic equivalence of s-values of two op-
erators . 268
8.3.2.4 Estimate of the remainder. 270
8.3.2.5 Unbounded operators. 274
8.3.2.6 Asymptotics of eigenvalues. 275
8.3.2.7 Asymptotics of eigenvalues (continuation) . 283
8.3.2.8 Asymptotics of s-values. 284
8.3.2.9 Asymptotics of the spectrum for quadratic
forms. 287
8.3.2.10 Proof of Theorem 2.3. 293
8.3.3 Trace class and Hilbert-Schmidt operators. 297
8.3.3.1 Trace class operators. 297
8.3.3.2 Hilbert-Schmidt operators. 298
8.3.3.3 Determinants of operators. 299
8.4 Elements of probability theory . 300
8.4.1 The probability space and basic definitions. 300
8.4.2 Hubert space theory. 306
8.4.3 Estimation in Hubert space L2(O ,U, P) . 310
8.4.4 Homogeneous and isotropic random fields. 312
8.4.5 Estimation of parameters . 315
8.4.6 Discrimination between hypotheses. 317
8.4.7 Generalized random fields. 319
8.4.8 Kaiman filters. 320
Appendix A Analytical Solution of the Basic
Integral Equation for a Class of One-Dimensional
Problems 325
A.l Introduction. 326
A.2 Proofs. 329
Appendix B Integral Operators Basic in Random
Fields Estimation Theory 337
Contents xiii
B.l Introduction. 337
B.2 Reduction of the basic integral equation to a boundary-value
problem. 341
B.3 Isomorphism property. 349
B.4 Auxiliary material. 354
Bibliographical Notes 359
Bibliography 363
Symbols 371
Index 373 |
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| id | DE-604.BV022295363 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:53:13Z |
| indexdate | 2024-07-09T20:54:22Z |
| institution | BVB |
| isbn | 9789812565365 9812565361 |
| language | English |
| lccn | 2006299148 |
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| spelling | Ramm, Alexander G. Verfasser aut Random fields estimation Alexander G. Ramm Hackensack, NJ [u.a.] World Scientific 2005 XIII, 373 S. 24 cm txt rdacontent n rdamedia nc rdacarrier Based partly on the author's earlier book: Random fields estimation theory. Harlow, Essex, England : Longman Scientific & Technical ; New York : Wiley, 1990. Includes bibliographical references (p. 363-369) and index Random fields Estimation theory Stochastische Analysis (DE-588)4132272-1 gnd rswk-swf Zufälliges Feld (DE-588)4191094-1 gnd rswk-swf Schätztheorie (DE-588)4121608-8 gnd rswk-swf Zufälliges Feld (DE-588)4191094-1 s Schätztheorie (DE-588)4121608-8 s Stochastische Analysis (DE-588)4132272-1 s DE-604 http://www.loc.gov/catdir/toc/fy0702/2006299148.html Table of contents only HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015505446&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ramm, Alexander G. Random fields estimation Random fields Estimation theory Stochastische Analysis (DE-588)4132272-1 gnd Zufälliges Feld (DE-588)4191094-1 gnd Schätztheorie (DE-588)4121608-8 gnd |
| subject_GND | (DE-588)4132272-1 (DE-588)4191094-1 (DE-588)4121608-8 |
| title | Random fields estimation |
| title_auth | Random fields estimation |
| title_exact_search | Random fields estimation |
| title_exact_search_txtP | Random fields estimation |
| title_full | Random fields estimation Alexander G. Ramm |
| title_fullStr | Random fields estimation Alexander G. Ramm |
| title_full_unstemmed | Random fields estimation Alexander G. Ramm |
| title_short | Random fields estimation |
| title_sort | random fields estimation |
| topic | Random fields Estimation theory Stochastische Analysis (DE-588)4132272-1 gnd Zufälliges Feld (DE-588)4191094-1 gnd Schätztheorie (DE-588)4121608-8 gnd |
| topic_facet | Random fields Estimation theory Stochastische Analysis Zufälliges Feld Schätztheorie |
| url | http://www.loc.gov/catdir/toc/fy0702/2006299148.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015505446&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT rammalexanderg randomfieldsestimation |