Bayesian Estimation and Tracking: A Practical Guide

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Author(s): Anton J. Haug
Publisher: John Wiley & Sons
Year: 2012

Language: English
Pages: 397
City: Hoboken
Tags: Математика;Теория вероятностей и математическая статистика;Математическая статистика;Прикладная математическая статистика;

Bayesian Estimation and Tracking: A Practical Guide......Page 1
CONTENTS......Page 9
PREFACE......Page 17
ACKNOWLEDGMENTS......Page 19
LIST OF FIGURES......Page 21
LIST OF TABLES......Page 27
PART I: PRELIMINARIES......Page 29
1 Introduction......Page 31
1.1 Bayesian Inference......Page 32
1.2 Bayesian Hierarchy of Estimation Methods......Page 33
1.3.2 Chapter Overview and Prerequisites......Page 34
1.4 Modeling and Simulation with MATLAB®......Page 36
References......Page 37
2.1.1 Vector and Matrix Conventions and Notation......Page 39
2.1.2 Sums and Products......Page 40
2.1.3 Matrix Inversion......Page 41
2.1.4 Block Matrix Inversion......Page 42
2.1.5 Matrix Square Root......Page 43
2.2 Vector Point Generators......Page 44
2.3.1 Approximating Scalar Nonlinear Functions......Page 47
2.3.2 Approximating Multidimensional Nonlinear Functions......Page 51
2.4.1 General Definitions......Page 57
2.4.2 The Gaussian Density......Page 60
References......Page 68
3 General Concepts of Bayesian Estimation......Page 70
3.2 Point Estimators......Page 71
3.3 Introduction to Recursive Bayesian Filtering of Probability DensityFunctions......Page 74
3.4 Introduction to Recursive Bayesian Estimation of the State Mean andCovariance......Page 77
3.4.1 State Vector Prediction......Page 78
3.4.2 State Vector Update......Page 79
References......Page 83
4 Case Studies: Preliminary Discussions......Page 84
4.1 The Overall Simulation/Estimation/Evaluation Process......Page 85
4.2.1 Ship Dynamics Model......Page 86
4.2.3 Scenario Specifics......Page 87
4.3 DIFAR Buoy Signal Processing......Page 90
4.4 The DIFAR Likelihood Function......Page 95
References......Page 97
PART II: THE GAUSSIAN ASSUMPTION: A FAMILY OF KALMANFILTER ESTIMATORS......Page 99
5 The Gaussian Noise Case: Multidimensional Integration ofGaussian-Weighted Distributions......Page 101
5.1 Summary of Important Results From Chapter 3......Page 102
5.2 Derivation of the Kalman Filter Correction (Update) EquationsRevisited......Page 104
5.3 The General Bayesian Point Prediction Integrals for GaussianDensities......Page 106
5.3.1 Refining the Process Through an Affine Transformation......Page 108
5.3.2 General Methodology for Solving Gaussian-WeightedIntegrals......Page 110
References......Page 113
6.1 Linear Dynamic Models......Page 114
6.2 Linear Observation Models......Page 115
6.4 Application of the LKF to DIFAR Buoy Bearing Estimation......Page 116
References......Page 120
7.1 One-Dimensional Consideration......Page 121
7.1.1 One-Dimensional State Prediction......Page 122
7.1.2 One-Dimensional State Estimation Error VariancePrediction......Page 123
7.1.4 Transformation of One-Dimensional Prediction Equations......Page 124
7.2 Multidimensional Consideration......Page 126
7.2.1 The State Prediction Equation......Page 127
7.2.2 The State Covariance Prediction Equation......Page 128
7.2.3 Observation Prediction Equations......Page 130
7.2.4 Transformation of Multidimensional PredictionEquations......Page 131
7.2.6 Second-Order Extended Kalman Filter......Page 133
7.3 An Alternate Derivation of the Multidimensional CovariancePrediction Equations......Page 135
7.4.1 The Ship Motion Dynamics Model......Page 136
7.4.2 The DIFAR Buoy Field Observation Model......Page 137
7.4.3 Initialization for All Filters of the Kalman Filter Class......Page 139
7.4.5 The EKF Tracking Filter Results......Page 140
References......Page 142
8 The Sigma Point Class: The Finite Difference Kalman Filter......Page 143
8.1.1 One-Dimensional Finite Difference State Prediction......Page 144
8.1.2 One-Dimensional Finite Difference State VariancePrediction......Page 145
8.1.5 Simplified One-Dimensional Finite Difference PredictionEquations......Page 146
8.2.1 Multidimensional Finite Difference State Prediction......Page 148
8.2.2 Multidimensional Finite Difference State CovariancePrediction......Page 151
8.2.3 Multidimensional Finite Difference Observation PredictionEquations......Page 152
8.3 An Alternate Derivation of the Multidimensional Finite DifferenceCovariance Prediction Equations......Page 153
References......Page 155
9.1 Introduction to Monomial Cubature Integration Rules......Page 156
9.2.1 Background......Page 158
9.2.2 The UKF Developed......Page 159
9.2.4 The UKF State Vector Covariance Prediction Equation......Page 162
9.2.7 An Alternate Version of the Unscented Kalman Filter......Page 163
9.3 Application of the UKF to the DIFAR Ship Tracking Case Study......Page 165
References......Page 166
10 The Sigma Point Class: The Spherical Simplex Kalman Filter......Page 168
10.1 One-Dimensional Spherical Simplex Sigma Points......Page 169
10.2 Two-Dimensional Spherical Simplex Sigma Points......Page 170
10.4 The Spherical Simplex Kalman Filter......Page 172
10.5 The Spherical Simplex Kalman Filter Process......Page 173
10.6 Application of the SSKF to the DIFAR Ship Tracking Case Study......Page 174
Reference......Page 175
11 The Sigma Point Class: The Gauss–Hermite Kalman Filter......Page 176
11.1 One-Dimensional Gauss–Hermite Quadrature......Page 177
11.2 One-Dimensional Gauss–Hermite Kalman Filter......Page 181
11.3 Multidimensional Gauss–Hermite Kalman Filter......Page 183
11.4 Sparse Grid Approximation for High Dimension/High PolynomialOrder......Page 188
References......Page 191
12 The Monte Carlo Kalman Filter......Page 192
Reference......Page 195
13.1 Analytical Kalman Filters......Page 196
13.2 Sigma Point Kalman Filters......Page 198
13.3 A More Practical Approach to Utilizing the Family of KalmanFilters......Page 202
References......Page 203
14.1 Error Ellipses......Page 204
14.1.1 The Canonical Ellipse......Page 205
14.1.2 Determining the Eigenvalues of P......Page 206
14.1.3 Determining the Error Ellipse Rotation Angle......Page 207
14.1.4 Determination of the Containment Area......Page 208
14.1.5 Parametric Plotting of Error Ellipse......Page 209
14.2 Root Mean Squared Errors......Page 210
14.3 Divergent Tracks......Page 211
14.4.1 The One-Dimensional Case......Page 212
14.4.3 A Recursive Approach to the CRLB......Page 214
14.4.4 The Cramer–Rao Lower Bound for Gaussian AdditiveNoise......Page 218
14.4.6 The Gaussian Cramer–Rao Lower Bound with LinearModels......Page 219
14.5 Performance of Kalman Class DIFAR Track Estimators......Page 220
References......Page 226
PART III: MONTE CARLO METHODS......Page 227
15 Introduction to Monte Carlo Methods......Page 229
15.1.2 Approximating a Density by Its MultidimensionalHistogram......Page 230
15.1.3 Kernel Density Approximation......Page 232
15.3 Summary......Page 243
References......Page 244
16.1 General Concept of Sequential Importance Sampling......Page 246
16.2.1 The Inverse Transform Method......Page 250
16.2.2 SIS Particle Filter with Resampling......Page 254
16.2.3 Regularization......Page 255
16.3 The Bootstrap Particle Filter......Page 258
16.3.1 Application of the BPF to DIFAR Buoy Tracking......Page 259
16.4 The Optimal SIS Particle Filter......Page 261
16.4.1 Gaussian Optimal SIS Particle Filter......Page 263
16.4.2 Locally Linearized Gaussian Optimal SIS Particle Filter......Page 264
16.5 The SIS Auxiliary Particle Filter......Page 266
16.5.1 Application of the APF to DIFAR Buoy Tracking......Page 270
16.6.2 The Unscented Particle Filter......Page 271
References......Page 273
17 The Generalized Monte Carlo Particle Filter......Page 275
17.1 The Gaussian Particle Filter......Page 276
17.2 The Combination Particle Filter......Page 278
17.2.1 Application of the CPF–UKF to DIFAR Buoy Tracking......Page 280
17.3 Performance Comparison of All DIFAR Tracking Filters......Page 281
References......Page 283
PART IV: ADDITIONAL CASE STUDIES......Page 285
18 A Spherical Constant Velocity Model for Target Trackingin Three Dimensions......Page 287
18.1 Tracking a Target in Cartesian Coordinates......Page 289
18.1.1 Object Dynamic Motion Model......Page 290
18.1.2 Sensor Data Model......Page 291
18.1.3 GaussianTracking Algorithms for a Cartesian StateVector......Page 292
18.2 Tracking a Target in Spherical Coordinates......Page 293
18.2.1 State Vector Position and Velocity Components in SphericalCoordinates......Page 294
18.2.2 Spherical State Vector Dynamic Equation......Page 295
18.2.4 GaussianTracking Algorithms for a Spherical StateVector......Page 298
18.3.1 Setting Values for q......Page 301
18.3.2 Simulating Radar Observation Data......Page 302
18.3.3 Filter Initialization......Page 304
18.4.1 Characteristics of the Trajectories Used for PerformanceAnalysis......Page 306
18.4.2 Filter Performance Comparisons......Page 310
18.5 Some Observations and Future Considerations......Page 321
18.A.1 General Velocity Components for Constant Turn RateMotion......Page 322
18.A.2 General Position Components for Constant Turn RateMotion......Page 325
18.A.4 Turn Rate Setting Based on a Desired Turn Acceleration......Page 327
APPENDIX 18.B: Three-Dimensional Coordinate Transformations......Page 329
18.B.1 Cartesian-to-Spherical Transformation......Page 330
18.B.2 Spherical-to-Cartesian Transformation......Page 333
References......Page 334
19.1 Introduction......Page 336
19.2.1 Dynamic Transition of the Translational Motion of a RigidBody......Page 339
19.2.2 Dynamic Transition of the Rotational Motion of a RigidBody......Page 341
19.2.3 Combined Dynamic Process Model......Page 344
19.2.4 The Dynamic Process Noise Models......Page 345
19.3 Components of the Observation Model......Page 346
19.4.1 A Nonlinear Least Squares Estimation Method......Page 349
19.4.2 An Unscented Kalman Filter Method......Page 351
19.4.3 Estimation Using the Unscented Combination ParticleFilter......Page 353
19.4.4 Initializing the Estimator......Page 354
19.5.2 Synthetic Trajectory......Page 356
19.5.4 Synthetic Measurements......Page 361
19.6 Performance Comparison Analysis......Page 362
19.6.1 Filter Performance Comparison Methodology......Page 363
19.6.2 Filter Comparison Results......Page 366
19.6.3 Conclusions and Future Considerations......Page 369
19.A.1 Conversions Between Rotation Representations......Page 370
19.A.2 Representation of Orientation and Rotation......Page 371
19.A.3 Point Rotations and Frame Rotations......Page 372
References......Page 373
20.1 Introduction......Page 374
20.2 The Process (Dynamic) Model for Rigid Body Motion......Page 375
20.3.1 The Inertial Measurement Unit Component of theObservation Model......Page 376
20.3.2 The Photogrammetric Component of the ObservationModel......Page 378
20.3.3 The Combined Sensor Fusion Observation Model......Page 379
20.4.3 Synthetic Measurements......Page 380
20.5.1 Initial Value Problem Solver for IMU Data......Page 382
20.6 Performance Comparison Analysis......Page 385
20.6.1 Filter Performance Comparison Methodology......Page 387
20.6.2 Filter Comparison Results......Page 388
20.7 Conclusions......Page 389
20.8 Future Work......Page 390
References......Page 392
Index......Page 395