随机有限集

【实例截图】

【核心代码】

Contents
Preface xxiii
Acknowledgments xxv
Chapter 1 Introduction to the Book 1
1.1 What Is the Purpose of This Book? 1
1.2 Major Challenges in Information Fusion 7
1.3 Why Random Sets—or FISST? 8
1.3.1 Why Isn’t Multitarget Filtering Straightforward? 9
1.3.2 Beyond Heuristics 10
1.3.3 How Do Single-Target and Multitarget Statistics
Differ? 11
1.3.4 How Do Conventional and Ambiguous Data Differ? 11
1.3.5 What Is Formal Bayes Modeling? 13
1.3.6 How Is Ambiguous Information Modeled? 13
1.3.7 What Is Multisource-Multitarget Formal Modeling? 14
1.4 Random Sets in Information Fusion 15
1.4.1 Statistics of Multiobject Systems 15
1.4.2 Statistics of Expert Systems 16
1.4.3 Finite Set Statistics 17
1.5 Organization of the Book 17
1.5.1 Part I: Unified Single-Target Multisource Integration 17
1.5.2 Part II: Unified Multitarget-Multisource Integration 20
1.5.3 Part III: Approximate Multitarget Filtering 21
1.5.4 Appendixes 22
vii
viii Contents
I Unified Single-Target Multisource Integration 23
Chapter 2 Single-Target Filtering 25
2.1 Introduction to the Chapter 25
2.1.1 Summary of Major Lessons Learned 26
2.1.2 Organization of the Chapter 27
2.2 The Kalman Filter 27
2.2.1 Kalman Filter Initialization 28
2.2.2 Kalman Filter Predictor 28
2.2.3 Kalman Filter Corrector 29
2.2.4 Derivation of the Kalman Filter 30
2.2.5 Measurement Fusion Using the Kalman Filter 32
2.2.6 Constant-Gain Kalman Filters 32
2.3 Bayes Formulation of the Kalman Filter 33
2.3.1 Some Mathematical Preliminaries 34
2.3.2 Bayes Formulation of the KF: Predictor 35
2.3.3 Bayes Formulation of the KF: Corrector 37
2.3.4 Bayes Formulation of the KF: Estimation 40
2.4 The Single-Target Bayes Filter 42
2.4.1 Single-Target Bayes Filter: An Illustration 43
2.4.2 Relationship Between the Bayes and Kalman Filters 45
2.4.3 Single-Target Bayes Filter: Modeling 51
2.4.4 Single-Target Bayes Filter: Formal Bayes Modeling 56
2.4.5 Single-Target Bayes Filter: Initialization 61
2.4.6 Single-Target Bayes Filter: Predictor 61
2.4.7 Single-Target Bayes Filter: Corrector 62
2.4.8 Single-Target Bayes Filter: State Estimation 63
2.4.9 Single-Target Bayes Filter: Error Estimation 64
2.4.10 Single-Target Bayes Filter: Data Fusion 67
2.4.11 Single-Target Bayes Filter: Computation 68
2.5 Single-Target Bayes Filter: Implementation 70
2.5.1 Taylor Series Approximation: The EKF 71
2.5.2 Gaussian-Mixture Approximation 72
2.5.3 Sequential Monte Carlo Approximation 79
2.6 Chapter Exercises 87
Chapter 3 General Data Modeling 89
3.1 Introduction to the Chapter 89
3.1.1 Summary of Major Lessons Learned 91
Contents ix
3.1.2 Organization of the Chapter 91
3.2 Issues in Modeling Uncertainty 92
3.3 Issues in Modeling Uncertainty in Data 94
3.4 Examples 97
3.4.1 Random, Slightly Imprecise Measurements 97
3.4.2 Imprecise, Slightly Random Measurements 101
3.4.3 Nonrandom Vague Measurements 102
3.4.4 Nonrandom Uncertain Measurements 103
3.4.5 Ambiguity Versus Randomness 106
3.5 The Core Bayesian Approach 109
3.5.1 Formal Bayes Modeling in General 109
3.5.2 The Bayes Filter in General 110
3.5.3 Bayes Combination Operators 111
3.5.4 Bayes-Invariant Measurement Conversion 113
3.6 Formal Modeling of Generalized Data 114
3.7 Chapter Exercise 117
Chapter 4 Random Set Uncertainty Representations 119
4.1 Introduction to the Chapter 119
4.1.1 Summary of Major Lessons Learned 119
4.1.2 Organization of the Chapter 120
4.2 Universes, Events, and the Logic of Events 120
4.3 Fuzzy Set Theory 121
4.3.1 Fuzzy Logics 122
4.3.2 Random Set Representation of Fuzzy Events 123
4.3.3 Finite-Level Fuzzy Sets 126
4.3.4 Copula Fuzzy Logics 129
4.3.5 General Random Set Representations of Fuzzy Sets 131
4.4 Generalized Fuzzy Set Theory 133
4.4.1 Random Set Representation of Generalized Fuzzy
Events 134
4.5 Dempster-Shafer Theory 134
4.5.1 Dempster’s Combination 136
4.5.2 “Zadeh’s Paradox” and Its Misinterpretation 138
4.5.3 Converting b.m.a.s to Probability Distributions 141
4.5.4 Random Set Representation of Uncertain Events 143
4.6 Fuzzy Dempster-Shafer Theory 144
4.6.1 Random Set Representation of Fuzzy DS Evidence 145
x Contents
4.7 Inference Rules 147
4.7.1 What Are Rules? 147
4.7.2 Combining Rules Using Conditional Event Algebra 148
4.7.3 Random Set Representation of First-Order Rules 150
4.7.4 Random Set Representation of Composite Rules 151
4.7.5 Random Set Representation of Second-Order Rules 152
4.8 Is Bayes Subsumed by Other Theories? 152
4.9 Chapter Exercises 154
Chapter 5 UGA Measurements 157
5.1 Introduction to the Chapter 157
5.1.1 Notation 158
5.1.2 Summary of Major Lessons Learned 159
5.1.3 Organization of the Chapter 161
5.2 What Is a UGA Measurement? 162
5.2.1 Modeling UGA Measurements 162
5.2.2 Modeling the Generation of UGA Measurements 164
5.3 Likelihoods for UGA Measurements 164
5.3.1 Special Case: Θ Is Statistical 165
5.3.2 Special Case: Θ Is Fuzzy 166
5.3.3 Special Case: Θ Is Generalized Fuzzy 169
5.3.4 Special Case: Θ Is Discrete/Dempster-Shafer 171
5.3.5 Special Case: Θ Is Fuzzy Dempster-Shafer 173
5.3.6 Special Case: Θ Is a First-Order Fuzzy Rule 174
5.3.7 Special Case: Θ Is a Composite Fuzzy Rule 179
5.3.8 Special Case: Θ Is a Second-Order Fuzzy Rule 180
5.4 Bayes Unification of UGA Fusion 181
5.4.1 Bayes Unification of UGA Fusion Using Normalized and Unnormalized Dempster’s Combinations 185
5.4.2 Bayes Unification of UGA Fusion Using Normalized and Unnormalized Fuzzy Dempster’s Combinations 186
5.4.3 Bayes Unification of UGA Fusion Using Copula
Fuzzy Conjunctions 186
5.4.4 Bayes Unification of UGA Rule-Firing 187
5.4.5 If Z0 Is Finite, Then Generalized Likelihoods Are
Strict Likelihoods 188
Contents xi
5.4.6 Bayes-Invariant Conversions Between UGA Measurements 189
5.5 Modeling Other Kinds of Uncertainty 194
5.5.1 Modeling Unknown Statistical Dependencies 195
5.5.2 Modeling Unknown Target Types 196
5.6 The Kalman Evidential Filter (KEF) 199
5.6.1 Definitions 204
5.6.2 KEF Predictor 205
5.6.3 KEF Corrector (Fuzzy DS Measurements) 205
5.6.4 KEF Corrector (Conventional Measurements) 207
5.6.5 KEF State Estimation 208
5.6.6 KEF Compared to Gaussian-Mixture and Kalman
Filters 208
5.7 Chapter Exercises 209
Chapter 6 AGA Measurements 211
6.1 Introduction to the Chapter 211
6.1.1 Summary of Major Lessons Learned 212
6.1.2 Organization of the Chapter 213
6.2 AGA Measurements Defined 213
6.3 Likelihoods for AGA Measurements 214
6.3.1 Special Case: Θ and Σx Are Fuzzy 215
6.3.2 Special Case: Θ and Σx Are Generalized Fuzzy 219
6.3.3 Special Case: Θ and Σx Are Dempster-Shafer 219
6.3.4 Special Case: Θ and Σx Are Fuzzy DS 220
6.4 Filtering with Fuzzy AGA Measurements 221
6.5 Example: Filtering with Poor Data 222
6.5.1 A Robust-Bayes Classifier 223
6.5.2 Simulation 1: More Imprecise, More Random 225
6.5.3 Simulation 2: Less Imprecise, Less Random 225
6.5.4 Interpretation of the Results 232
6.6 Unmodeled Target Types 232
6.7 Example: Target ID Using Link INT Data 238
6.7.1 Robust-Bayes Classifier 240
6.7.2 “Pseudodata” Simulation Results 243
6.7.3 “LONEWOLF-98” Simulation Results 243
6.8 Example: Unmodeled Target Types 244
6.9 Chapter Exercises 245
xii Contents
Chapter 7 AGU Measurements 249
7.1 Introduction to the Chapter 249
7.1.1 Summary of Major Lessons Learned 250
7.1.2 Why Not Robust Statistics? 250
7.1.3 Organization of the Chapter 251
7.2 Random Set Models of UGA Measurements 252
7.2.1 Random Error Bars 252
7.2.2 Random Error Bars: Joint Likelihoods 252
7.3 Likelihoods for AGU Measurements 254
7.4 Fuzzy Models of AGU Measurements 255
7.5 Robust ATR Using SAR Data 260
7.5.1 Summary of Methodology 264
7.5.2 Experimental Ground Rules 266
7.5.3 Summary of Experimental Results 268
Chapter 8 Generalized State-Estimates 271
8.1 Introduction to the Chapter 271
8.1.1 Summary of Major Lessons Learned 273
8.1.2 Organization of the Chapter 274
8.2 What Is a Generalized State-Estimate? 274
8.3 What Is a UGA DS State-Estimate? 275
8.4 Posterior Distributions and State-Estimates 277
8.4.1 The Likelihood of a DS State-Estimate 278
8.4.2 Posterior Distribution Conditioned on a DS StateEstimate 278
8.4.3 Posterior Distributions and Pignistic Probability 279
8.5 Unification of State-Estimate Fusion Using Modified Dempster’s Combination 280
8.6 Bayes-Invariant Transformation 280
8.7 Extension to Fuzzy DS State-Estimates 281
8.8 Chapter Exercises 285
Chapter 9 Finite-Set Measurements 287
9.1 Introduction to the Chapter 287
9.1.1 Summary of Major Lessons Learned 287
9.1.2 Organization of the Chapter 288
9.2 Examples of Finite-Set Measurements 288
9.2.1 Ground-to-Air Radar Detection Measurements 288
9.2.2 Air-to-Ground Doppler Detection Measurements 291
Contents xiii
9.2.3 Extended-Target Detection Measurements 292
9.2.4 Features Extracted from Images 292
9.2.5 Human-Mediated Features 292
9.2.6 General Finite-Set Measurements 293
9.3 Modeling Finite-Set Measurements? 293
9.3.1 Formal Modeling of Finite-Set Measurements 293
9.3.2 Multiobject Integrals 297
9.3.3 Finite-Set Measurement Models 299
9.3.4 True Likelihoods for Finite-Set Measurements 302
9.3.5 Constructive Likelihood Functions 302
9.4 Chapter Exercises 303
II Unified Multitarget-Multisource Integration 305
Chapter 10 Conventional Multitarget Filtering 307
10.1 Introduction to the Chapter 307
10.1.1 Summary of Major Lessons Learned 308
10.1.2 Organization of the Chapter 311
10.2 Standard Multitarget Models 311
10.2.1 Standard Multitarget Measurement Model 311
10.2.2 Standard Multitarget Motion Model 313
10.3 Measurement-to-Track Association 315
10.3.1 Distance Between Measurements and Tracks 315
10.4 Single-Hypothesis Correlation (SHC) 319
10.4.1 SHC: No Missed Detections, No False Alarms 319
10.4.2 SHC: Missed Detections and False Alarms 320
10.5 Multihypothesis Correlation (MHC) 321
10.5.1 Elements of MHC 323
10.5.2 MHC: No Missed Detections or False Alarms 326
10.5.3 MHC: False Alarms, No Missed Detections 329
10.5.4 MHC: Missed Detections and False Alarms 332
10.6 Composite-Hypothesis Correlation (CHC) 335
10.6.1 Elements of CHC 335
10.6.2 CHC: No Missed Detections or False Alarms 337
10.6.3 CHC: Probabilistic Data Association (PDA) 337
10.6.4 CHC: Missed Detections, False Alarms 338
10.7 Conventional Filtering: Limitations 338
10.7.1 Real-Time Performance 338
xiv Contents
10.7.2 Is a Hypothesis Actually a State Variable? 340
10.8 MHC with Fuzzy DS Measurements 341
Chapter 11 Multitarget Calculus 343
11.1 Introduction to the Chapter 343
11.1.1 Transform Methods in Conventional Statistics 344
11.1.2 Transform Methods in Multitarget Statistics 345
11.1.3 Summary of Major Lessons Learned 346
11.1.4 Organization of the Chapter 348
11.2 Random Finite Sets 348
11.3 Fundamental Statistical Descriptors 356
11.3.1 Multitarget Calculus—Why? 357
11.3.2 Belief-Mass Functions 359
11.3.3 Multiobject Density Functions and Set Integrals 360
11.3.4 Important Multiobject Probability Distributions 364
11.3.5 Probability-Generating Functionals (p.g.fl.s) 370
11.4 Functional Derivatives and Set Derivatives 375
11.4.1 Functional Derivatives 375
11.4.2 Set Derivatives 380
11.5 Key Multiobject-Calculus Formulas 383
11.5.1 Fundamental Theorem of Multiobject Calculus 384
11.5.2 Radon-Nikodym´ Theorems 385
11.5.3 Fundamental Convolution Formula 385
11.6 Basic Differentiation Rules 386
11.7 Chapter Exercises 394
Chapter 12 Multitarget Likelihood Functions 399
12.1 Introduction to the Chapter 399
12.1.1 Summary of Major Lessons Learned 401
12.1.2 Organization of the Chapter 402
12.2 Multitarget State and Measurement Spaces 403
12.2.1 Multitarget State Spaces 403
12.2.2 Multisensor State Spaces 406
12.2.3 Single-Sensor, Multitarget Measurement Spaces 407
12.2.4 Multisensor-Multitarget Measurement Spaces 408
12.3 The Standard Measurement Model 408
12.3.1 Measurement Equation for the Standard Model 411
12.3.2 Case I: No Target Is Present 412
12.3.3 Case II: One Target Is Present 414
Contents xv
12.3.4 Case III: No Missed Detections or False Alarms 416
12.3.5 Case IV: Missed Detections, No False Alarms 418
12.3.6 Case V: Missed Detections and False Alarms 420
12.3.7 p.g.fl.s for the Standard Measurement Model 421
12.4 Relationship with MHC 422
12.5 State-Dependent False Alarms 424
12.5.1 p.g.fl. for State-Dependent False Alarms 426
12.6 Transmission Drop-Outs 426
12.6.1 p.g.fl. for Transmission Drop-Outs 427
12.7 Extended Targets 427
12.7.1 Single Extended Target 428
12.7.2 Multiple Extended Targets 430
12.7.3 Poisson Approximation 431
12.8 Unresolved Targets 432
12.8.1 Point Target Clusters 434
12.8.2 Single-Cluster Likelihoods 435
12.8.3 Multicluster Likelihoods 442
12.8.4 Continuity of Multicluster Likelihoods 444
12.9 Multisource Measurement Models 445
12.9.1 Conventional Measurements 445
12.9.2 Generalized Measurements 447
12.10 A Model for Bearing-Only Measurements 448
12.10.1 Multitarget Measurement Model 450
12.10.2 Belief-Mass Function 451
12.10.3 Multitarget Likelihood Function 452
12.11 A Model for Data-Cluster Extraction 452
12.11.1 Finite-Mixture Models 453
12.11.2 A Likelihood for Finite-Mixture Modeling 456
12.11.3 Extraction of Soft Data Classes 457
12.12 Chapter Exercises 458
Chapter 13 Multitarget Markov Densities 461
13.1 Introduction to the Chapter 461
13.1.1 Summary of Major Lessons Learned 465
13.1.2 Organization of the Chapter 466
13.2 “Standard” Multitarget Motion Model 466
13.2.1 Case I: At Most One Target Is Present 469
13.2.2 Case II: No Target Death or Birth 470
xvi Contents
13.2.3 Case III: Target Death, No Birth 471
13.2.4 Case IV: Target Death and Birth 471
13.2.5 Case V: Target Death and Birth with Spawning 472
13.2.6 p.g.fl.s for the Standard Motion Model 473
13.3 Extended Targets 474
13.4 Unresolved Targets 475
13.4.1 Intuitive Dynamic Behavior of Point Clusters 475
13.4.2 Markov Densities for Single Point Clusters 476
13.4.3 Markov Densities for Multiple Point Clusters 477
13.5 Coordinated Multitarget Motion 478
13.5.1 Simple Virtual Leader-Follower 478
13.5.2 General Virtual Leader-Follower 481
13.6 Chapter Exercises 482
Chapter 14 The Multitarget Bayes Filter 483
14.1 Introduction to the Chapter 483
14.1.1 Summary of Major Lessons Learned 484
14.1.2 Organization of the Chapter 486
14.2 Multitarget Bayes Filter: Initialization 486
14.2.1 Initialization: Multitarget Poisson Process 486
14.2.2 Initialization: Target Number Known 487
14.3 Multitarget Bayes Filter: Predictor 487
14.3.1 Predictor: No Target Birth or Death 489
14.4 Multitarget Bayes Filter: Corrector 490
14.4.1 Conventional Measurements 490
14.4.2 Generalized Measurements 493
14.4.3 Unified Multitarget-Multisource Integration 493
14.5 Multitarget Bayes Filter: State Estimation 494
14.5.1 The Failure of the Classical State Estimators 494
14.5.2 Marginal Multitarget (MaM) Estimator 497
14.5.3 Joint Multitarget (JoM) Estimator 498
14.5.4 JoM and MaM Estimators Compared 501
14.5.5 Computational Issues 504
14.5.6 State Estimation and Track Labeling 505
14.6 Multitarget Bayes Filter: Error Estimation 509
14.6.1 Target Number RMS Deviation 509
14.6.2 Track Covariances 509
14.6.3 Global Mean Deviation 510
Contents xvii
14.6.4 Information Measures of Multitarget Dispersion 512
14.7 The JoTT Filter 514
14.7.1 JoTT Filter: Models 516
14.7.2 JoTT Filter: Initialization 518
14.7.3 JoTT Filter: Predictor 519
14.7.4 JoTT Filter: Corrector 520
14.7.5 JoTT Filter: Estimation 520
14.7.6 JoTT Filter: Error Estimation 523
14.7.7 SMC Implementation of JoTT Filter 523
14.8 The p.g.fl. Multitarget Bayes Filter 528
14.8.1 The p.g.fl. Multitarget Predictor 528
14.8.2 The p.g.fl. Multitarget Corrector 530
14.9 Target Prioritization 531
14.9.1 Tactical Importance Functions (TIFs) 533
14.9.2 The p.g.fl. for a TIF 533
14.9.3 The Multitarget Posterior for a TIF 535
14.10 Chapter Exercises 537
III Approximate Multitarget Filtering 539
Chapter 15 Multitarget Particle Approximation 541
15.1 Introduction to the Chapter 541
15.1.1 Summary of Major Lessons Learned 542
15.1.2 Organization of the Chapter 543
15.2 The Multitarget Filter: Computation 543
15.2.1 Fixed-Grid Approximation 544
15.2.2 SMC Approximation 545
15.2.3 When Is the Multitarget Filter Appropriate? 546
15.2.4 Implementations of the Multitarget Filter 547
15.3 Multitarget Particle Systems 551
15.4 M-SMC Filter Initialization 554
15.4.1 Target Number is Known 554
15.4.2 Null Multitarget Prior 555
15.4.3 Poisson Multitarget Prior 555
15.5 M-SMC Filter Predictor 556
15.5.1 Persisting and Disappearing Targets 557
15.5.2 Appearing Targets 558
15.6 M-SMC Filter Corrector 560
xviii Contents
15.7 M-SMC Filter State and Error Estimation 561
15.7.1 PHD-Based State and Error Estimation 561
15.7.2 Global Mean Deviation 562
15.7.3 Track Labeling for the Multitarget SMC Filter 563
Chapter 16 Multitarget-Moment Approximation 565
16.1 Introduction to the Chapter 565
16.1.1 Single-Target Moment-Statistic Filters 566
16.1.2 First-Order Multitarget-Moment Filtering 568
16.1.3 Second-Order Multitarget-Moment Filtering 572
16.1.4 Summary of Major Lessons Learned 574
16.1.5 Organization of the Chapter 575
16.2 The Probability Hypothesis Density (PHD) 576
16.2.1 First-Order Multitarget Moments 576
16.2.2 PHD as a Continuous Fuzzy Membership Function 579
16.2.3 PHDs and Multitarget Calculus 580
16.2.4 Examples of PHDs 583
16.2.5 Higher-Order Multitarget Moments 586
16.3 The PHD Filter 587
16.3.1 PHD Filter Initialization 587
16.3.2 PHD Filter Predictor 587
16.3.3 PHD Filter Corrector 590
16.3.4 PHD Filter State and Error Estimation 595
16.3.5 Target ID and the PHD Filter 599
16.4 Physical Interpretation of PHD Filter 599
16.4.1 Physical Interpretation of PHD Predictor 600
16.4.2 Physical Interpretation of PHD Corrector 603
16.5 Implementing the PHD Filter 609
16.5.1 Survey of PHD Filter Implementations 610
16.5.2 SMC-PHD Approximation 615
16.5.3 GM-PHD Approximation 623
16.6 Limitations of the PHD Filter 631
16.7 The Cardinalized PHD (CPHD) Filter 632
16.7.1 CPHD Filter Initialization 633
16.7.2 CPHD Filter Predictor 634
16.7.3 CPHD Filter Single-Sensor Corrector 636
16.7.4 CPHD Filter State and Error Estimation 639
16.7.5 Computational Complexity of the CPHD Filter 640
Contents xix
16.7.6 CPHD and JoTT Filters Compared 641
16.8 Physical Interpretation of CPHD Filter 642
16.9 Implementing the CPHD Filter 642
16.9.1 Survey of CPHD Filter Implementations 643
16.9.2 Particle Approximation (SMC-CPHD) 644
16.9.3 Gaussian-Mixture Approximation (GM-CPHD) 646
16.10 Deriving the PHD and CPHD Filters 649
16.10.1 Derivation of PHD and CPHD Predictors 650
16.10.2 Derivation of PHD and CPHD Correctors 651
16.11 Partial Second-Order Filters? 652
16.12 Chapter Exercise 653
Chapter 17 Multi-Bernoulli Approximation 655
17.1 Introduction to the Chapter 655
17.1.1 p.g.fl.-Based Multitarget Approximation 655
17.1.2 Why Multitarget Multi-Bernoulli Processes? 657
17.1.3 The Multitarget Multi-Bernoulli Filter 657
17.1.4 The Para-Gaussian Filter 658
17.1.5 Summary of Major Lessons Learned 659
17.1.6 Organization of the Chapter 660
17.2 Multitarget Multi-Bernoulli Filter 660
17.2.1 MeMBer Filter Initialization 661
17.2.2 MeMBer Filter Predictor 661
17.2.3 MeMBer Filter Corrector 662
17.2.4 MeMBer Filter Pruning and Merging 665
17.2.5 MeMBer Filter State and Error Estimation 666
17.2.6 Relationship with the Moreland-Challa Filter 667
17.3 Para-Gaussian Filter 668
17.3.1 Para-Gaussian Filter Initialization 669
17.3.2 Para-Gaussian Filter Predictor 669
17.3.3 Para-Gaussian Filter Corrector 671
17.3.4 Para-Gaussian Filter Pruning and Merging 673
17.3.5 Para-Gaussian Filter State and Error Estimation 675
17.4 MeMBer Filter Derivation 675
17.4.1 Derivation of the MeMBer Filter Predictor 675
17.4.2 Derivation of the MeMBer Filter Corrector 677
17.5 Chapter Exercise 682
Appendix A Glossary of Notation 683
xx Contents
A.1 Transparent Notational System 683
A.2 General Mathematics 684
A.3 Set Theory 685
A.4 Fuzzy Logic and Dempster-Shafer Theory 686
A.5 Probability and Statistics 687
A.6 Random Sets 689
A.7 Multitarget Calculus 690
A.8 Finite-Set Statistics 691
A.9 Generalized Measurements 692
Appendix B Dirac Delta Functions 693
Appendix C Gradient Derivatives 695
C.1 Relationship with Partial Derivatives 696
C.2 Multidimensional Taylor Series 696
C.3 Multidimensional Extrema 696
Appendix D Fundamental Gaussian Identity 699
Appendix E Finite Point Processes 705
E.1 Mathematical Representations of Multiplicity 705
E.2 Random Point Processes 707
E.3 Point Processes Versus Random Finite Sets 708
Appendix F FISST and Probability Theory 711
F.1 Multiobject Probability Theory 711
F.2 Belief-Mass Functions Versus Probability Measures 713
F.3 Set Integrals Versus Measure Theoretic Integrals 714
F.4 Set Derivatives Versus Radon-Nikod ´ym Derivatives 715
Appendix G Mathematical Proofs 717
G.1 Likelihoods for First-Order Fuzzy Rules 717
G.2 Likelihoods for Composite Rules 718
G.3 Likelihoods for Second-Order Fuzzy Rules 720
G.4 Unification of DS Combinations 721
G.5 Unification of Rule-Firing 722
G.6 Generalized Likelihoods: Z0 Is Finite 723
G.7 NOTA for Fuzzy DS Measurements 724
G.8 KEF Predictor 726
Contents xxi
G.9 KEF Corrector (Fuzzy DS Measurements) 729
G.10 Likelihoods for AGA Fuzzy Measurements 732
G.11 Likelihoods for AGA Generalized Fuzzy Measurements 733
G.12 Likelihoods for AGA Fuzzy DS Measurements 734
G.13 Interval Argsup Formula 735
G.14 Consonance of the Random State Set Γˆz 736
G.15 Sufficient Statistics and Modified Combination 737
G.16 Transformation Invariance 738
G.17 MHT Hypothesis Probabilities 739
G.18 Likelihood for Standard Measurement Model 742
G.19 p.g.fl. for Standard Measurement Model 745
G.20 Multisensor Multitarget Likelihoods 747
G.21 Continuity of Likelihoods for Unresolved Targets 749
G.22 Association for Fuzzy Dempster-Shafer 751
G.23 JoTT Filter Predictor 753
G.24 JoTT Filter Corrector 755
G.25 p.g.fl. Form of the Multitarget Corrector 757
G.26 Induced Particle Approximation of PHD 758
G.27 PHD Counting Property 760
G.28 GM-PHD Filter Predictor 761
G.29 GM-PHD Filter Corrector 763
G.30 Exact PHD Corrector 765
G.31 GM-CPHD Filter Predictor 767
G.32 GM-CPHD Filter Corrector 768
G.33 MeMBer Filter Target Number 771
G.34 Para-Gaussian Filter Predictor 773
G.35 Para-Gaussian Filter Corrector 774
Appendix H Solutions to Exercises 777
References 821
About the Author 837
Index 839