 | Candy J.V. Bayesian signal processing: classical, modern, and particle filtering methods. - Hoboken: Wiley: IEEE, 2009. - xxiii, 445 p.: ill., map. - (Adaptive and learning systems for signal processing, communications, and control). - Incl. bibl. ref. - Ind.: p.431-445. - ISBN 978-0-470-18094-5
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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
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 Random Walk 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 Kaiman
Filter) ............................................... 160
5.5 Extended Bayesian Processor (Extended Kaiman
Filter) ............................................... 167
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 Kaiman
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
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
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.l 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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