Optimal filtering: 1 Filtering of stochastic processes
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Dordrecht [u.a.]
Kluwer
1999
|
| Schriftenreihe: | Mathematics and its applications
457 |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIII, 375 S. |
| ISBN: | 0792352866 |
Internformat
MARC
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Datensatz im Suchindex
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| adam_text | IMAGE 1
OPTIMAL FILTERING
VOLUME I: FILTERING OF STOCHASTIC PROCESSES
BY VLADIMIR FOMIN DEPARTMENT OF MATHEMATICS AND MECHANICS, ST PETERSBURG
STATE UNIVERSITY,
ST PETERSBURG, RUSSIA
KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON
IMAGE 2
CONTENTS
PREFACE XI
1 INTRODUCTION TO ESTIMATION AND FILTERING THEORY 1
1.1 BASIC NOTIONS OF PROBABILITY THEORY 1
1.1.1 RANDOM VARIABLES AND PROBABILITY SPACE 1
1.1.2 MEAN VALUES, COVARIATIONS AND DISTRIBUTION DENSITIES OF RANDOM
VARIABLES 3
1.1.3 CONDITIONAL MATHEMATICAL EXPECTATIONS AND CONDITIONAL DISTRIBUTION
DENSITIES 5
1.2 INTRODUCTION TO ESTIMATION THEORY 8
1.2.1 ADMISSIBLE ESTIMATION RULES, MEAN RISK, OPTIMAL AND BAYES
ESTIMATES 8
1.2.2 REMARKS 11
1.2.3 EMPIRICAL FUNCTIONAL METHOD 15
1.2.4 STOCHASTIC APPROXIMATION METHOD 17
1.2.5 RECURSIVE MODIFICATION OF THE LSM 20
1.2.6 ROBUST ESTIMATION 22
1.3 EXAMPLES OF ESTIMATION PROBLEMS 25
1.3.1 OPTIMAL ESTIMATION OF SIGNAL PARAMETERS 25
1.3.2 BAYES CLASSIFIER 26
1.3.3 PROBLEM OF SIGNAL DETECTION 30
1.3.4 APPROXIMATION OF A FUNCTION BY LINEAR COMBINATION OF KNOWN
FUNCTIONS 36
1.3.5 DETERMINISTIC PATTERN RECOGNITION 38
1.3.6 STOCHASTICAL PATTERN RECOGNITION 52
1.3.7 SELF-LEARNING PROBLEM 54
1.4 ESTIMATION AND FILTERING: SIMILARITY AND DISTINCTION 59
1.4.1 TIME SERIES FILTERING 60
1.4.2 FILTERING OF CONTINUOUS PROCESSES 62
V
IMAGE 3
VI
CONTENTS
1.4.3 MARKOV PROCESSES 66
1.5 BASIC NOTIONS OF FILTERING THEORY 68
1.5.1 CORRELATION OPERATORS OF STOCHASTIC PROCESSES 69
1.5.2 CORRELATION OPERATORS ADMITTING REGULARIZATION 71
1.5.3 GENERALIZED STOCHASTIC PROCESSES 73
1.5.4 LINEAR ALTERS 79
1.5.5 STATIONARY PROCESSES AND ALTERS 84
1.6 APPENDIX: PROOFS OF LEMMAS AND THEOREMS 94
1.6.1 PROOF OF THEOREM 1.1 94
1.6.2 PROOF OF THEOREM 1.2 95
1.6.3 PROOF OF LEMMA 1.6 97
1.6.4 PROOF OF LEMMA 1.1 99
1.6.5 PROOF OF LEMMA 1.2 100
1.6.6 PROOF OF LEMMA 1.3 100
1.6.7 PROOF OF LEMMA 1.4 102
1.6.8 PROOF OF LEMMA 1.5 104
1.6.9 COMMENTS 104
2 OPTIMAL FILTERING OF STOCHASTIC PROCESSES IN THE CONTEXT OF THE
WIENER-KOLMOGOROV THEORY 111
2.1 LINEAR FILTERING OF STOCHASTIC PROCESSES 113
2.1.1 STATEMENT OF THE PROBLEM 113
2.1.2 STRUCTURE OF OPTIMAL FILTER 115
2.1.3 METHODS OF FACTORIZATION 119
2.1.4 RECURRENCE FORM OF OPTIMAL FILTER 121
2.1.5 PHYSICALLY REALIZABLE AND UNREALIZABLE ALTERS 122
2.2 FILTERING OF STATIONARY PROCESSES 124
2.2.1 WIENER-KOLMOGOROV PROBLEM OF OPTIMAL FILTERING . . .. 124 2.2.2
FILTERING SPECTRAL METHOD FOR STATIONARY TIME SERIES . . . 130 2.2.3
FACTORIZATION OF SPECTRAL DENSITY OF TIME SERIES 136
2.2.4 STRUCTURE OF OPTIMAL STATIONARY FILTER 140
2.3 RECURSIVE FILTERING 144
2.3.1 KALMAN-BUCY FILTER 144
2.3.2 RECURRENCE FORMS OF OPTIMAL FILTER 149
2.3.3 RECURRENCE FORM OF WIENER-KOLMOGOROV FILTER FOR TIME SERIES 158
2.4 LINEAR ALTERS MAXIMIZING A SIGNAL TO NOISE RATIO 165
2.4.1 SETTING OF A PROBLEM 166
2.4.2 MAXIMIZATION OF SIGNAL TO NOISE RATIO 173
2.5 APPENDIX: PROOFS OF LEMMAS AND THEOREMS 180
2.5.1 PROOF OF THEOREM 2.1 180
2.5.2 PROOF OF THEOREM 2.2 182
2.5.3 PROOF OF THEOREM 2.3 182
2.5.4 PROOF OF LEMMA 2.1 183
IMAGE 4
CONTENTS VII
2.5.5 PROOF OF THEOREM 2.4 184
2.5.6 PROOF OF LEMMA 2.2 185
2.5.7 PROOF OF LEMMA 2.3 . . . 187
2.5.8 PROOF OF THEOREM 2.5 188
2.5.9 PROOF OF THEOREM 2.6 194
2.5.10 PROOF OF THEOREM 2.7 194
2.5.11 PROOF OF THEOREM 2.8 198
2.5.12 PROOF OF THEOREM 2.9 201
2.5.13 PROOF OF LEMMA 2.4 201
2.5.14 PROOF OF LEMMA 2.5 202
2.5.15 PROOF OF LEMMA 2.6 203
2.5.16 PROOF OF LEMMA 2.7 203
2.5.17 PROOF OF THEOREM 2.10 204
2.5.18 PROOF OF THEOREM 2.11 205
2.5.19 PROOF OF THEOREM 2.12 206
2.6 BIBLIOGRAPHICAL COMMENTS 207
3 ABSTRACT OPTIMAL FILTERING THEORY 213
3.1 RANDOM ELEMENTS 213
3.1.1 BASIC AND GENERALIZED ELEMENTS 214
3.1.2 RANDOM ELEMENTS WITH VALUES IN EXTENDED SPACE . . .. 217 3.1.3
STOCHASTIC PROCESSES AS GENERALIZED ELEMENTS 219
3.1.4 POSSIBLE GENERALIZATION OF THE CONCEPT OF RANDOM ELEMENT 221 3.2
LINEAR STABLE ESTIMATION 221
3.2.1 STATEMENT OF LINEAR ESTIMATION PROBLEM 221
3.2.2 SOLUTION OF STABLE OPTIMAL FILTERING PROBLEM 223
3.2.3 EXAMPLE OF ESTIMATION PROBLEM 225
3.3 RESOLUTION SPACE AND RELATIVE FINITARY TRANSFORMATIONS 230
3.3.1 HUBERT RESOLUTION SPACE 231
3.3.2 FINITARY OPERATORS IN HUBERT RESOLUTION SPACE 232
3.3.3 EXAMPLE: INTEGRAL OPERATORS ON L2 (R) 234
3.4 EXTENDED RESOLUTION SPACE AND LINEAR TRANSFORMATIONS IN IT . . . 236
3.4.1 SPACE EXTENSION EQUIPPED WITH TIME STRUCTURE 236
3.4.2 LINEAR TRANSFORMATIONS IN T-EXTENSION OF HUBERT SPACE . 237 3.4.3
EXAMPLE: LINEAR DIFFERENTIAL OPERATORS 240
3.5 ABSTRACT VERSION OF THE WIENER-KOLMOGOROV FILTERING THEORY . . 245
3.5.1 ELEMENTARY ( FINITE-DIMENSIONAL ) FILTERING PROBLEM . . . 246
3.5.2 STATEMENT OF PROBLEM OF OPTIMAL ESTIMATION 249
3.5.3 GENERALIZATION OF ESTIMATION PROBLEM 250
3.5.4 SOLUBILITY OF ESTIMATION PROBLEM 251
3.5.5 LOCAL ESTIMATION 252
3.6 OPTIMAL ESTIMATION IN DISCRETE RESOLUTION SPACE 253
3.6.1 SAMPLING TIME STRUCTURE OF RESOLUTION SPACE 254
3.6.2 FINITARY OPERATORS ON DISCRETE RESOLUTION SPACE 255
IMAGE 5
VIII CONTENTS
3.6.3 UNPREDICTED FILTERING PROBLEM 259
3.6.4 GENERALIZED OPTIMAL FILTERING PROBLEM 260
3.6.5 EXISTENCE OF OPTIMAL FILTER AND ITS STRUCTURE 261
3.6.6 APPROXIMATION OF OPTIMAL FILTER 262
3.7 SPECTRAL FACTORIZATION 264
3.7.1 FACTORIZATION OF POSITIVE DEFINITE OPERATORS 264
3.7.2 STANDARD SPECTRAL FACTORIZATION 265
3.8 OPTIMAL FILTER STRUCTURE FOR DISCRETE TIME CASE 267
3.8.1 BODE-SHANNON REPRESENTATION OF WEIGHT OPERATOR . . .. 267 3.8.2
BODE-SHANNON INTERPRETATION OF OPTIMAL FILTER ACTION . . 269 3.8.3
CONSTRUCTION OF OPTIMIZING SEQUENCE 269
3.8.4 RECURSIVE REPRESENTATION OF OPTIMAL FILTER 271
3.9 ABSTRACT WIENER PROBLEM 272
3.9.1 OPTIMAL FILTERING AND GENERAL MULTI-CRITERIA PROBLEM . . 273 3.9.2
SINGLE CRITERION ABSTRACT WIENER PROBLEM 274
3.9.3 ROBUSTNESS IN MINIMIZATION OF QUADRATIC FUNCTIONAL . . . 278 3.9.4
ABSTRACT LINEAR-QUADRATIC PROBLEM OF OPTIMAL CONTROL 284 3.9.5
LINEAR-QUADRATIC CONTROL AND SPECTRAL FACTORIZATION . . . 285 3.10
APPENDIX: PROOFS OF LEMMAS AND THEOREMS 287
3.10.1 PROOF OF THEOREM 3.1 287
3.10.2 PROOF OF THEOREM 3.2 287
3.10.3 PROOF OF LEMMA 3.1 288
3.10.4 PROOF OF LEMMA 3.2 288
3.10.5 PROOF OF LEMMA 3.3 288
3.10.6 PROOF OF LEMMA 3.4 289
3.10.7 PROOF OF LEMMA 3.5 289
3.10.8 PROOF OF THEOREM 3.3 289
3.10.9 PROOF OF THEOREM 3.4 290
3.10.10PROOF OF THEOREM 3.5 292
3.10.11 PROOF OF THEOREM 3.6 292
3.10.12 PROOF OF THEOREM 3.7 293
3.11 BIBLIOGRAPHICAL COMMENTS 294
4 NONLINEAR FILTERING OF TIME SERIES 297
4.1 STATEMENT OF NONLINEAR OPTIMAL FILTERING PROBLEM 298
4.1.1 FILTERING OF STOCHASTIC TIME SERIES 299
4.1.2 GEOMETRIE INTERPRETATION OF OPTIMAL FILTERING PROBLEM . 302 4.1.3
FILTERING OF NONSTOCHASTIC TIME SERIES 303
4.2 OPTIMAL FILTERING OF CONDITIONALLY GAUSSIAN TIME SERIES 304
4.2.1 CONDITIONALLY MARKOV TIME SERIES 304
4.2.2 CONDITIONALLY GAUSSIAN TIME SERIES 305
4.2.3 RECURRENCE RELATIONS FOR OPTIMAL ESTIMATES 308
4.2.4 EXAMPLE: FILTERING OF CONDITIONAL GAUSSIAN TIME SERIES . 310 4.3
CONNECTION OF LINEAR AND NONLINEAR FILTERING PROBLEMS 313
IMAGE 6
CONTENTS IX
4.3.1 COMPLETE SEQUENCES OF OPTIMAL ESTIMATES 315
4.3.2 COMPLETE SEQUENCES OF OPTIMAL ESTIMATES 316
4.3.3 POLYNOMIAL ESTIMATES 317
4.4 MINIMAX FILTERING 319
4.4.1 STATEMENT OF OPTIMAL FILTERING PROBLEM 320
4.4.2 EXAMPLE 1: WHITE NOISE 322
4.4.3 EXAMPLE 2: BOUNDED DISTURBANCE 323
4.4.4 OPERATOR LINEAR-QUADRATIC PROBLEM 324
4.4.5 RECURSIVENESS IN LINEAR-QUADRATIC PROBLEM 325
4.4.6 KALMAN-BUCY FILTER OPTIMALITY 330
4.4.7 PROPERTIES OF KALMAN-BUCY FILTER 333
4.5 PROOFS OF LEMMAS AND THEOREMS 334
4.5.1 PROOF OF LEMMA 4.1 334
4.5.2 PROOF OF THEOREM 4.1 335
4.5.3 PROOF OF THEOREM 4.2 336
4.5.4 PROOF OF THEOREM 4.3 337
4.5.5 PROOF OF THEOREM 4.4 340
4.5.6 PROOF OF LEMMA 4.2 342
4.5.7 PROOF OF THEOREM 4.5 342
4.5.8 PROOF OF LEMMA 4.3 344
4.5.9 PROOF OF LEMMA 4.4 344
4.5.10 PROOF OF THEOREM 4.6 345
4.5.11 PROOF OF LEMMA 4.5 345
4.5.12 PROOF OF THEOREM 4.7 346
4.6 BIBLIOGRAPHICAL COMMENTS 347
REFERENCES 349
NOTATION 367
INDEX 372
|
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| spelling | Fomin, Vladimir N. 1937-2000 Verfasser (DE-588)131358073 aut Optimal filtering 1 Filtering of stochastic processes Vladimir Fomin Dordrecht [u.a.] Kluwer 1999 XIII, 375 S. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications 457 Mathematics and its applications ... (DE-604)BV012513823 1 Mathematics and its applications 457 (DE-604)BV008163334 457 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008494060&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Fomin, Vladimir N. 1937-2000 Optimal filtering Mathematics and its applications |
| title | Optimal filtering |
| title_auth | Optimal filtering |
| title_exact_search | Optimal filtering |
| title_full | Optimal filtering 1 Filtering of stochastic processes Vladimir Fomin |
| title_fullStr | Optimal filtering 1 Filtering of stochastic processes Vladimir Fomin |
| title_full_unstemmed | Optimal filtering 1 Filtering of stochastic processes Vladimir Fomin |
| title_short | Optimal filtering |
| title_sort | optimal filtering filtering of stochastic processes |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008494060&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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