Adaptive algorithms and stochastic approximations:
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
1990
|
| Series: | Applications of mathematics
22 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 365 S. Ill. |
| ISBN: | 3540528946 0387528946 |
Staff View
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| 100 | 1 | |a Benveniste, Albert |d 1949- |e Verfasser |0 (DE-588)1308367600 |4 aut | |
| 240 | 1 | 0 | |a Algorithmes adaptatifs et approximations stochastiques |
| 245 | 1 | 0 | |a Adaptive algorithms and stochastic approximations |c Albert Benveniste ; Michel Métivier ; Pierre Priouret |
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Record in the Search Index
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| adam_text |
Contents
Introduction 1
Part I. Adaptive Algorithms: Applications 7
1. General Adaptive Algorithm Form 9
1.1 Introduction 9
1.2 Two Basic Examples and Their Variants 10
1.3 General Adaptive Algorithm Form and Main Assumptions 23
1.4 Problems Arising 29
1.5 Summary of the Adaptive Algorithm Form: Assumptions (A) 31
1.6 Conclusion 33
1.7 Exercises 34
1.8 Comments on the Literature 38
2. Convergence: the ODE Method 40
2.1 Introduction 40
2.2 Mathematical Tools: Informal Introduction 41
2.3 Guide to the Analysis of Adaptive Algorithms 48
2.4 Guide to Adaptive Algorithm Design 55
2.5 The Transient Regime 75
2.6 Conclusion 76
2.7 Exercises 76
2.8 Comments on the Literature 100
3. Rate of Convergence 103
3.1 Mathematical Tools: Informal Description 103
3.2 Applications to the Design of Adaptive Algorithms with
Decreasing Gain 110
3.3 Conclusions from Section 3.2 116
3.4 Exercises 116
3.5 Comments on the Literature 118
X Contents
4. Tracking Non-Stationary Parameters 120
4.1 Tracking Ability of Algorithms with Constant Gain 120
4.2 Multistep Algorithms 142
4.3 Conclusions 158
4.4 Exercises 158
4.5 Comments on the Literature 163
5. Sequential Detection; Model Validation 165
5.1 Introduction and Description of the Problem 166
5.2 Two Elementary Problems and their Solution 171
5.3 Central Limit Theorem and the Asymptotic Local Viewpoint 176
5.4 Local Methods of Change Detection 180
5.5 Model Validation by Local Methods 185
5.6 Conclusion 188
5.7 Annex: Proofs of Theorems 1 and 2 188
5.8 Exercises 191
5.9 Comments on the Literature 197
6. Appendices to Part I 199
6.1 Rudiments of Systems Theory 199
6.2 Second Order Stationary Processes 205
6.3 Kalman Filters 208
Part II. Stochastic Approximations: Theory 211
1. O.D.E. and Convergence A.S. for an Algorithm with
Locally Bounded Moments 213
1.1 Introduction of the General Algorithm 213
1.2 Assumptions Peculiar to Chapter 1 219
1.3 Decomposition of the General Algorithm 220
1.4 L2 Estimates 223
1.5 Approximation of the Algorithm by the Solution of the O.D.E. . 230
1.6 Asymptotic Analysis of the Algorithm 233
1.7 An Extension of the Previous Results 236
1.8 Alternative Formulation of the Convergence Theorem 238
1.9 A Global Convergence Theorem 239
1.10 Rate of I? Convergence of Some Algorithms 243
1.11 Comments on the Literature 249
Contents xi
2. Application to the Examples of Part I 251
2.1 Geometric Ergodicity of Certain Markov Chains 251
2.2 Markov Chains Dependent on a Parameter 0 259
2.3 Linear Dynamical Processes 265
2.4 Examples 270
2.5 Decision-Feedback Algorithms with Quantisation 276
2.6 Comments on the Literature 288
3. Analysis of the Algorithm in the General Case 289
3.1 New Assumptions and Control of the Moments 289
3.2 L" Estimates 293
3.3 Convergence towards the Mean Trajectory 298
3.4 Asymptotic Analysis of the Algorithm 301
3.5 "Tube of Confidence" for an Infinite Horizon 305
3.6 Final Remark. Connections with the Results of Chapter 1 306
3.7 Comments on the Literature 306
4. Gaussian Approximations to the Algorithms 307
4.1 Process Distributions and their Weak Convergence 308
4.2 Diffusions. Gaussian Diffusions 312
4.3 The Process tf7(i) for an Algorithm with Constant Step Size . 314
4.4 Gaussian Approximation of the Processes f/'y(f) 321
4.5 Gaussian Approximation for Algorithms with Decreasing
Step Size 327
4.6 Gaussian Approximation and Asymptotic Behaviour
of Algorithms with Constant Steps 334
4.7 Remark on Weak Convergence Techniques 341
4.8 Comments on the Literature 341
5. Appendix to Part II: A Simple Theorem in the
"Robbins-Monro" Case 343
5.1 The Algorithm, the Assumptions and the Theorem 343
5.2 Proof of the Theorem 344
5.3 Variants 345
Bibliography 349
Subject Index to Part I 361
Subject Index to Part II 364 |
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| illustrated | Illustrated |
| indexdate | 2025-01-21T17:00:45Z |
| institution | BVB |
| isbn | 3540528946 0387528946 |
| language | English |
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| series | Applications of mathematics |
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| spelling | Benveniste, Albert 1949- Verfasser (DE-588)1308367600 aut Algorithmes adaptatifs et approximations stochastiques Adaptive algorithms and stochastic approximations Albert Benveniste ; Michel Métivier ; Pierre Priouret Berlin [u.a.] Springer 1990 XI, 365 S. Ill. txt rdacontent n rdamedia nc rdacarrier Applications of mathematics 22 Stochastische Approximation (DE-588)4183371-5 gnd rswk-swf Adaptives System (DE-588)4247928-9 gnd rswk-swf Algorithmus (DE-588)4001183-5 gnd rswk-swf Adaptives System (DE-588)4247928-9 s Stochastische Approximation (DE-588)4183371-5 s DE-604 Algorithmus (DE-588)4001183-5 s Métivier, Michel 1931-1988 Verfasser (DE-588)172256585 aut Priouret, Pierre 1939- Verfasser (DE-588)173702538 aut Applications of mathematics 22 (DE-604)BV000895226 22 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002624846&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Benveniste, Albert 1949- Métivier, Michel 1931-1988 Priouret, Pierre 1939- Adaptive algorithms and stochastic approximations Applications of mathematics Stochastische Approximation (DE-588)4183371-5 gnd Adaptives System (DE-588)4247928-9 gnd Algorithmus (DE-588)4001183-5 gnd |
| subject_GND | (DE-588)4183371-5 (DE-588)4247928-9 (DE-588)4001183-5 |
| title | Adaptive algorithms and stochastic approximations |
| title_alt | Algorithmes adaptatifs et approximations stochastiques |
| title_auth | Adaptive algorithms and stochastic approximations |
| title_exact_search | Adaptive algorithms and stochastic approximations |
| title_full | Adaptive algorithms and stochastic approximations Albert Benveniste ; Michel Métivier ; Pierre Priouret |
| title_fullStr | Adaptive algorithms and stochastic approximations Albert Benveniste ; Michel Métivier ; Pierre Priouret |
| title_full_unstemmed | Adaptive algorithms and stochastic approximations Albert Benveniste ; Michel Métivier ; Pierre Priouret |
| title_short | Adaptive algorithms and stochastic approximations |
| title_sort | adaptive algorithms and stochastic approximations |
| topic | Stochastische Approximation (DE-588)4183371-5 gnd Adaptives System (DE-588)4247928-9 gnd Algorithmus (DE-588)4001183-5 gnd |
| topic_facet | Stochastische Approximation Adaptives System Algorithmus |
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