Bayesian Estimation and Tracking
A Practical Guide
1. Edition June 2012
400 Pages, Hardcover
Professional Book
Short Description
This book presents a practical approach to estimation methods that are designed to provide a clear path to programming all algorithms. The coverage gives readers a firm understanding of Bayesian estimation methods and their interrelatedness. Starting with fundamental principles of Bayesian theory, the book presents a conceptually cohesive roadmap that begins with fundamental principles and leads directly to derivations of the Gaussian estimation methods currently in use within a Bayesian framework. Thoroughly classroom-tested for the past few years, this book serves as both a graduate-level text and as a professional reference.
A practical approach to estimating and tracking dynamic systems in real-worl applications
Much of the literature on performing estimation for non-Gaussian systems is short on practical methodology, while Gaussian methods often lack a cohesive derivation. Bayesian Estimation and Tracking addresses the gap in the field on both accounts, providing readers with a comprehensive overview of methods for estimating both linear and nonlinear dynamic systems driven by Gaussian and non-Gaussian noices.
Featuring a unified approach to Bayesian estimation and tracking, the book emphasizes the derivation of all tracking algorithms within a Bayesian framework and describes effective numerical methods for evaluating density-weighted integrals, including linear and nonlinear Kalman filters for Gaussian-weighted integrals and particle filters for non-Gaussian cases. The author first emphasizes detailed derivations from first principles of eeach estimation method and goes on to use illustrative and detailed step-by-step instructions for each method that makes coding of the tracking filter simple and easy to understand.
Case studies are employed to showcase applications of the discussed topics. In addition, the book supplies block diagrams for each algorithm, allowing readers to develop their own MATLAB(r) toolbox of estimation methods.
Bayesian Estimation and Tracking is an excellent book for courses on estimation and tracking methods at the graduate level. The book also serves as a valuable reference for research scientists, mathematicians, and engineers seeking a deeper understanding of the topics.
Acknowledgments
List of Figures xi
List of Tables xxi
Part I. Prelininaries
1. Introduction 3
1.1 Bayesian Inference 5
1.2 Bayesian Hierarchy of Estimation Methods 7
1.3 Scope of this Text 8
1.4 Modeling and Simulation with Matlab(r) 13
2. Preliminary Mathematical Concepts 19
2.1 A Very Brief Overview of Matrix Linear Algebra 20
2.2 Vector Point Generators 27
2.3 Approximating Nonlinear Multidimensional Functions with Multidimensional Arguments 32
2.4 Overview of Multivariate Statistics 47
3. General Concepts of Bayesian Estimation 69
3.1 Bayesian Estimation 70
3.2 Point Estimators 72
3.3 Introduction to Recursive Bayesian Filtering of Probability Density Functions 76
3.4 Introduction to Recursive Bayesian Estimation of the State Mean and Covariance 81
3.5 Discussion of General Estimation Methods 88
4. Case Studies: Preliminary Discussions 93
4.1 The Overall Simulation/Estimation/Evaluation Process 94
4.2 A Scenario Simulator for Tracking a Constant-Velocity Target Through a DIFAR Buoy Field 97
4.3 DIFAR Buoy Signal Processing 102
4.4 The DIFAR Likelihood Function 111
Part II. The Gaussian Assumption: A Family of Kalman Filter Estimators
5. The Gaussian Noise Case: Multidimensional Integration of Gaussian-Weighted Distributions 119
5.1 Summary of Important Results From Chapter 3 122
5.2 Derivation of the Kalman Filter Correction (Update) Equations Revisted 124
5.3 The General Bayesian Point Prediction Integrals for Gaussian Densities 128
6. The Linear Class of Kalman Filters 141
6.1 Linear Dynamic Models 142
6.2 Linear Observation Models 143
6.3 The Linear Kalman Filter 144
6.4 Application of the LKF to DIFAR Buoy Bearing Estimation 146
7. The Analytical Linearization Class of Kalman Filters: The Extended Kalman Filter 153
7.1 One-Dimensional Consideration 154
7.2 Multidimensional Consideration 159
7.3 An Alternate Derivation of the Multidimensional Covariance Prediction Equations 172
7.4 Application of the EKF to the DIFAR Ship Tracking Case Study 174
8. The Sigma Point Class: The Finite Difference Kalman Filter 187
8.1 One-Dimensional Finite Difference Kalman Filter 189
8.2 Multidimensional Finite Difference Kalman Filters 195
8.3 An Alternate Derivation of the Multidimensional Finite Difference Covariance Prediction Equations 201
9. The Sigma Point Class: The Unscented Kalman Filter 207
9.1 Introduction to Monomial Cubature Integration Rules 207
9.2 The Unscented Kalman Filter 211
9.3 Applications of the UKF to the DIFAR Ship Tracking Case Study 221
10. The Sigma Point Class: The Spherical Simplex Kalman Filter 227
10.1 One-Dimensional Spherical Simplex Sigma Points 228
10.2 Two-Dimensional Spherical Simplex Sigma Points 229
10.3 Higher-Dimensional Spherical Simplex Sigma Points 233
10.4 The Spherical Simplex Kalman Filter 233
10.5 The Spherical Simplex Kalman Filter Process 236
10.6 Application of the SSKF to the DIFAR Ship Tracking Case Study 236
11. The Sigma Point Class: The Gauss-Hermite Kalman Filter 241
11.1 One-Dimensional Gauss-Hermite Quadrature 242
11.2 One-Dimensional Gauss-Hermite Kalman Filter 248
11.3 Multidimensional Gauss-Hermite Kalman Filter 251
11.4 Sparse Grid Approximation for High Dimension/High Polynomial Order 257
11.5 Application of the GHKF to the DIFAR Ship Tracking Case Study 261
12. The Monte Carlo Kalman Filter 265
12.1 The Monte Carlo Kalman Filter 268
13. Summary of Gaussian Kalman Filters 273
13.1 Analytical Kalman Filters 274
13.2 Sigma-Point Kalman Filters 276
13.3 A More Practical Approach to Utilizing the Family of Kalman Filters 284
14. Performance Measures for the Family of Kalman Filters 289
14.1 Error Ellipses 290
14.2 Root Mean Squared Errors 299
14.3 Divergent Tracks 301
14.4 Cramer-Rao Lower Bound 302
14.5 Performance of Kalman Class DIFAR Track Estimators 315
Part III. Monte Carlo Methods
15. Introduction to Monte Carlo Methods 323
15.1 Approximating a Density From a Set of Monte Carlo Samples 325
15.2 General Concepts Importance Sampling 340
15.3 Summary 347
16. Sequential Importance Sampling Particle Filters 351
16.1 General Concept of Sequential Importance Sampling 351
16.2 Resampling and Regularization (Move) for SIS Particle Filters 357
16.3 The Bootstrap Particle Filter 372
16.4 The Optimal SIS Particle Filter 378
16.5 The SIS Auxiliary Particle Filter 385
16.6 Approximations to the SIS Auxiliary Particle Filter 393
16.7 Reducing the Computational Load Through Rao-Blackwellization 396
17. The Generalized Sequential Monte Carlo Particle Filter 403
17.1 The Gaussian Particle Filter 404
17.2 The Combination Particle Filter 406
17.3 Performance Comparison of all DIFAR Tracking Filters 411
Part IV Additional Case Studies
18. A Spherical Constant Velocity Model for Target Tracking in Three Dimensions 421
18.1 Tracking a Target in Cartesian Coordinates 426
18.2 Tracking a Target in Spherical Coordinates 433
18.3 Implementation of Cartesian and Spherical Tracking Filters 443
18.4 Performance Comparison for Various Estimation Methods 453
18.5 Some Observations and Future Considerations 469
19. Tracking a Falling Rigid Body Using Photogrammetry 497
19.1 Introduction 497
19.2 The Process (Dynamic) Model for Rigid Body Motion 502
19.3 Components of the Observation Model 513
19.4 Estimation Methods 517
19.5 The Generation of Synthetic Data 529
19.6 Performance Comparison Analysis 538
20. Sensor Fusion using Photogrammetric and Inertial Measurements 559
20.1 Introduction 559
20.2 The Process (Dynamic) Model for Rigid Body Motion 562
20.3 The Sensor Fusion Observational Model563
20.4 The Generation of Synthetic Data 569
20.5 Estimation Methods 572
20.6 Performance Comparison Analysis 577
20.7 Conclusions 585
20.8 Future Work 586
References 589
