Bayesian Signal Processing

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Bayesian Signal Processing
Classical, Modern, and
Particle Filtering Methods
James V. Candy
Lawrence Livermore National Laboratory
University of California Santa Barbara
2009 Copyright © 2009 by JohnWiley & Sons, Inc. All rights reserved

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关键词：Processing Bayesian processI Process Signal Bayesian Particle Filtering

Preface xiii
References to the Preface xix
Acknowledgments xxiii
1 Introduction 1
1.1 Introduction 1
1.2 Bayesian Signal Processing 1
1.3 Simulation-Based Approach to Bayesian Processing 4
1.4 Bayesian Model-Based Signal Processing 8
1.5 Notation and Terminology 12
References 14
Problems 15
2 Bayesian Estimation 19
2.1 Introduction 19
2.2 Batch Bayesian Estimation 19
2.3 Batch Maximum Likelihood Estimation 22
2.3.1 Expectation-Maximization Approach
to Maximum Likelihood 25
2.3.2 EM for Exponential Family of Distributions 30
2.4 Batch Minimum Variance Estimation 33
2.5 Sequential Bayesian Estimation 36
2.5.1 Joint Posterior Estimation 39
2.5.2 Filtering Posterior Estimation 41
2.6 Summary 43
References 44
Problems 45
vii
viii CONTENTS
3 Simulation-Based Bayesian Methods 51
3.1 Introduction 51
3.2 Probability Density Function Estimation 53
3.3 Sampling Theory 56
3.3.1 Uniform Sampling Method 58
3.3.2 Rejection Sampling Method 62
3.4 Monte Carlo Approach 64
3.4.1 Markov Chains 70
3.4.2 Metropolis-Hastings Sampling 71
3.4.3 RandomWalk Metropolis-Hastings Sampling 73
3.4.4 Gibbs Sampling 75
3.4.5 Slice Sampling 78
3.5 Importance Sampling 81
3.6 Sequential Importance Sampling 84
3.7 Summary 87
References 87
Problems 90
4 State–Space Models for Bayesian Processing 95
4.1 Introduction 95
4.2 Continuous-Time State–Space Models 96
4.3 Sampled-Data State–Space Models 100
4.4 Discrete-Time State–Space Models 104
4.4.1 Discrete Systems Theory 107
4.5 Gauss-Markov State–Space Models 112
4.5.1 Continuous-Time/Sampled-Data Gauss-Markov Models 112
4.5.2 Discrete-Time Gauss-Markov Models 114
4.6 Innovations Model 120
4.7 State–Space Model Structures 121
4.7.1 Time Series Models 121
4.7.2 State–Space and Time Series Equivalence Models 129
4.8 Nonlinear (Approximate) Gauss-Markov State–Space Models 135
4.9 Summary 139
References 140
Problems 141
5 Classical Bayesian State–Space Processors 147
5.1 Introduction 147
5.2 Bayesian Approach to the State–Space 147
5.3 Linear Bayesian Processor (Linear Kalman Filter) 150
5.4 Linearized Bayesian Processor (Linearized Kalman Filter) 160
5.5 Extended Bayesian Processor (Extended Kalman Filter) 167
CONTENTS ix
5.6 Iterated-Extended Bayesian Processor (Iterated-Extended
Kalman Filter) 174
5.7 Practical Aspects of Classical Bayesian Processors 182
5.8 Case Study: RLC Circuit Problem 186
5.9 Summary 191
References 191
Problems 193
6 Modern Bayesian State–Space Processors 197
6.1 Introduction 197
6.2 Sigma-Point (Unscented) Transformations 198
6.2.1 Statistical Linearization 198
6.2.2 Sigma-Point Approach 200
6.2.3 SPT for Gaussian Prior Distributions 205
6.3 Sigma-Point Bayesian Processor (Unscented Kalman Filter) 209
6.3.1 Extensions of the Sigma-Point Processor 218
6.4 Quadrature Bayesian Processors 218
6.5 Gaussian Sum (Mixture) Bayesian Processors 220
6.6 Case Study: 2D-Tracking Problem 224
6.7 Summary 230
References 231
Problems 233
7 Particle-Based Bayesian State–Space Processors 237
7.1 Introduction 237
7.2 Bayesian State–Space Particle Filters 237
7.3 Importance Proposal Distributions 242
7.3.1 Minimum Variance Importance Distribution 242
7.3.2 Transition Prior Importance Distribution 245
7.4 Resampling 246
7.4.1 Multinomial Resampling 249
7.4.2 Systematic Resampling 251
7.4.3 Residual Resampling 251
7.5 State–Space Particle Filtering Techniques 252
7.5.1 Bootstrap Particle Filter 253
7.5.2 Auxiliary Particle Filter 261
7.5.3 Regularized Particle Filter 264
7.5.4 MCMC Particle Filter 266
7.5.5 Linearized Particle Filter 270
7.6 Practical Aspects of Particle Filter Design 272
7.6.1 Posterior Probability Validation 273
7.6.2 Model Validation Testing 277
7.7 Case Study: Population Growth Problem 285
7.8 Summary 289
x CONTENTS
References 290
Problems 293
8 Joint Bayesian State/Parametric Processors 299
8.1 Introduction 299
8.2 Bayesian Approach to Joint State/Parameter Estimation 300
8.3 Classical/Modern Joint Bayesian State/Parametric Processors 302
8.3.1 Classical Joint Bayesian Processor 303
8.3.2 Modern Joint Bayesian Processor 311
8.4 Particle-Based Joint Bayesian State/Parametric Processors 313
8.5 Case Study: Random Target Tracking Using a Synthetic
Aperture Towed Array 318
8.6 Summary 327
References 328
Problems 330
9 Discrete Hidden Markov Model Bayesian Processors 335
9.1 Introduction 335
9.2 Hidden Markov Models 335
9.2.1 Discrete-Time Markov Chains 336
9.2.2 Hidden Markov Chains 337
9.3 Properties of the Hidden Markov Model 339
9.4 HMM Observation Probability: Evaluation Problem 341
9.5 State Estimation in HMM: The Viterbi Technique 345
9.5.1 Individual Hidden State Estimation 345
9.5.2 Entire Hidden State Sequence Estimation 347
9.6 Parameter Estimation in HMM:
The EM/Baum-Welch Technique 350
9.6.1 Parameter Estimation with State Sequence Known 352
9.6.2 Parameter Estimation with State Sequence Unknown 354
9.7 Case Study: Time-Reversal Decoding 357
9.8 Summary 362
References 363
Problems 365
10 Bayesian Processors for Physics-Based Applications 369
10.1 Optimal Position Estimation for the Automatic Alignment 369
10.1.1 Background 369
10.1.2 Stochastic Modeling of Position Measurements 372
10.1.3 Bayesian Position Estimation and Detection 374
10.1.4 Application: Beam Line Data 375
10.1.5 Results: Beam Line (KDP Deviation) Data 377
10.1.6 Results: Anomaly Detection 379
CONTENTS xi
10.2 Broadband Ocean Acoustic Processing 382
10.2.1 Background 382
10.2.2 Broadband State–Space Ocean Acoustic
Propagators 384
10.2.3 Broadband Bayesian Processing 389
10.2.4 Broadband BSP Design 393
10.2.5 Results 395
10.3 Bayesian Processing for Biothreats 397
10.3.1 Background 397
10.3.2 Parameter Estimation 400
10.3.3 Bayesian Processor Design 401
10.3.4 Results 403
10.4 Bayesian Processing for the Detection of Radioactive Sources 404
10.4.1 Background 404
10.4.2 Physics-Based Models 404
10.4.3 Gamma-Ray Detector Measurements 407
10.4.4 Bayesian Physics-Based Processor 410
10.4.5 Physics-Based Bayesian Deconvolution Processor 412
10.4.6 Results 415
References 417
Appendix A Probability & Statistics Overview 423
A.1 Probability Theory 423
A.2 Gaussian Random Vectors 429
A.3 Uncorrelated Transformation:
Gaussian Random Vectors 430
References 430
Index 431

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