Bayesian Estimation and Tracking: A Practical Guide

386 Pages·2012·10.586 MB·English

by  Anton J. Haug(auth.)

BAYESIAN ESTIMATION
AND TRACKING
BAYESIAN ESTIMATION
AND TRACKING
A Practical Guide
ANTONJ.HAUG
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LibraryofCongressCataloging-in-PublicationData:
Haug,AntonJ.,1941–
Bayesianestimationandtracking:apracticalguide/AntonJ.Haug.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-470-62170-7(hardback)
1. Bayesianstatisticaldecisiontheory. 2. Automatic
tracking–Mathematics. 3. Estimationtheory. I. Title.
QA279.5.H38 2012
519.5(cid:1)42–dc23
2011044308
PrintedintheUnitedStatesofAmerica
10987654321
DEDICATION
Tomywife,whoinspiresusalltoachieveatthehighestlimitofourabilities.
Tomychildren,whoseachievementsmakemeproudtobetheirfather.
Tomygrandchildren,whowillalwayskeepmeyoung.
Andinmemoryofmyparents,RoseandDavid,whoselovewaswithoutbound.
CONTENTS
PREFACE xv
ACKNOWLEDGMENTS xvii
LISTOFFIGURES xix
LISTOFTABLES xxv
PARTI PRELIMINARIES
1 Introduction 3
1.1 BayesianInference, 4
1.2 BayesianHierarchyofEstimationMethods, 5
1.3 ScopeofThisText, 6
1.3.1 Objective, 6
1.3.2 ChapterOverviewandPrerequisites, 6
1.4 ModelingandSimulationwithMATLAB®, 8
References, 9
2 PreliminaryMathematicalConcepts 11
2.1 AVeryBriefOverviewofMatrixLinearAlgebra, 11
2.1.1 VectorandMatrixConventionsandNotation, 11
2.1.2 SumsandProducts, 12
2.1.3 MatrixInversion, 13
2.1.4 BlockMatrixInversion, 14
2.1.5 MatrixSquareRoot, 15
vii
viii CONTENTS
2.2 VectorPointGenerators, 16
2.3 ApproximatingNonlinearMultidimensionalFunctionswith
MultidimensionalArguments, 19
2.3.1 ApproximatingScalarNonlinearFunctions, 19
2.3.2 ApproximatingMultidimensionalNonlinearFunctions, 23
2.4 OverviewofMultivariateStatistics, 29
2.4.1 GeneralDefinitions, 29
2.4.2 TheGaussianDensity, 32
References, 40
3 GeneralConceptsofBayesianEstimation 42
3.1 BayesianEstimation, 43
3.2 PointEstimators, 43
3.3 IntroductiontoRecursiveBayesianFilteringofProbabilityDensity
Functions, 46
3.4 IntroductiontoRecursiveBayesianEstimationoftheStateMeanand
Covariance, 49
3.4.1 StateVectorPrediction, 50
3.4.2 StateVectorUpdate, 51
3.5 DiscussionofGeneralEstimationMethods, 55
References, 55
4 CaseStudies:PreliminaryDiscussions 56
4.1 TheOverallSimulation/Estimation/EvaluationProcess, 57
4.2 AScenarioSimulatorforTrackingaConstantVelocityTarget
ThroughaDIFARBuoyField, 58
4.2.1 ShipDynamicsModel, 58
4.2.2 MultipleBuoyObservationModel, 59
4.2.3 ScenarioSpecifics, 59
4.3 DIFARBuoySignalProcessing, 62
4.4 TheDIFARLikelihoodFunction, 67
References, 69
PARTII THEGAUSSIANASSUMPTION:AFAMILYOFKALMAN
FILTERESTIMATORS
5 TheGaussianNoiseCase:MultidimensionalIntegrationof
Gaussian-WeightedDistributions 73
5.1 SummaryofImportantResultsFromChapter3, 74
5.2 DerivationoftheKalmanFilterCorrection(Update)Equations
Revisited, 76
5.3 TheGeneralBayesianPointPredictionIntegralsforGaussian
Densities, 78
CONTENTS ix
5.3.1 RefiningtheProcessThroughanAffineTransformation, 80
5.3.2 GeneralMethodologyforSolvingGaussian-Weighted
Integrals, 82
References, 85
6 TheLinearClassofKalmanFilters 86
6.1 LinearDynamicModels, 86
6.2 LinearObservationModels, 87
6.3 TheLinearKalmanFilter, 88
6.4 ApplicationoftheLKFtoDIFARBuoyBearingEstimation, 88
References, 92
7 TheAnalyticalLinearizationClassofKalmanFilters:
TheExtendedKalmanFilter 93
7.1 One-DimensionalConsideration, 93
7.1.1 One-DimensionalStatePrediction, 94
7.1.2 One-DimensionalStateEstimationErrorVariance
Prediction, 95
7.1.3 One-DimensionalObservationPredictionEquations, 96
7.1.4 TransformationofOne-DimensionalPredictionEquations, 96
7.1.5 TheOne-DimensionalLinearizedEKFProcess, 98
7.2 MultidimensionalConsideration, 98
7.2.1 TheStatePredictionEquation, 99
7.2.2 TheStateCovariancePredictionEquation, 100
7.2.3 ObservationPredictionEquations, 102
7.2.4 TransformationofMultidimensionalPrediction
Equations, 103
7.2.5 The Linearized Multidimensional Extended Kalman Filter
Process, 105
7.2.6 Second-OrderExtendedKalmanFilter, 105
7.3 AnAlternateDerivationoftheMultidimensionalCovariance
PredictionEquations, 107
7.4 ApplicationoftheEKFtotheDIFARShipTrackingCaseStudy, 108
7.4.1 TheShipMotionDynamicsModel, 108
7.4.2 TheDIFARBuoyFieldObservationModel, 109
7.4.3 InitializationforAllFiltersoftheKalmanFilterClass, 111
7.4.4 ChoosingaValuefortheAccelerationNoise, 112
7.4.5 TheEKFTrackingFilterResults, 112
References, 114
8 TheSigmaPointClass:TheFiniteDifferenceKalmanFilter 115
8.1 One-DimensionalFiniteDifferenceKalmanFilter, 116
8.1.1 One-DimensionalFiniteDifferenceStatePrediction, 116
x CONTENTS
8.1.2 One-DimensionalFiniteDifferenceStateVariance
Prediction, 117
8.1.3 One-Dimensional Finite Difference Observation Prediction
Equations, 118
8.1.4 TheOne-DimensionalFiniteDifferenceKalmanFilter
Process, 118
8.1.5 Simplified One-Dimensional Finite Difference Prediction
Equations, 118
8.2 MultidimensionalFiniteDifferenceKalmanFilters, 120
8.2.1 MultidimensionalFiniteDifferenceStatePrediction, 120
8.2.2 MultidimensionalFiniteDifferenceStateCovariance
Prediction, 123
8.2.3 Multidimensional Finite Difference Observation Prediction
Equations, 124
8.2.4 TheMultidimensionalFiniteDifferenceKalmanFilter
Process, 125
8.3 An Alternate Derivation of the Multidimensional Finite Difference
CovariancePredictionEquations, 125
References, 127
9 TheSigmaPointClass:TheUnscentedKalmanFilter 128
9.1 IntroductiontoMonomialCubatureIntegrationRules, 128
9.2 TheUnscentedKalmanFilter, 130
9.2.1 Background, 130
9.2.2 TheUKFDeveloped, 131
9.2.3 TheUKFStateVectorPredictionEquation, 134
9.2.4 TheUKFStateVectorCovariancePredictionEquation, 134
9.2.5 TheUKFObservationPredictionEquations, 135
9.2.6 TheUnscentedKalmanFilterProcess, 135
9.2.7 AnAlternateVersionoftheUnscentedKalmanFilter, 135
9.3 ApplicationoftheUKFtotheDIFARShipTrackingCaseStudy, 137
References, 138
10 TheSigmaPointClass:TheSphericalSimplexKalmanFilter 140
10.1 One-DimensionalSphericalSimplexSigmaPoints, 141
10.2 Two-DimensionalSphericalSimplexSigmaPoints, 142
10.3 HigherDimensionalSphericalSimplexSigmaPoints, 144
10.4 TheSphericalSimplexKalmanFilter, 144
10.5 TheSphericalSimplexKalmanFilterProcess, 145
10.6 ApplicationoftheSSKFtotheDIFARShipTrackingCaseStudy, 146
Reference, 147
CONTENTS xi
11 TheSigmaPointClass:TheGauss–HermiteKalmanFilter 148
11.1 One-DimensionalGauss–HermiteQuadrature, 149
11.2 One-DimensionalGauss–HermiteKalmanFilter, 153
11.3 MultidimensionalGauss–HermiteKalmanFilter, 155
11.4 Sparse Grid Approximation for High Dimension/High Polynomial
Order, 160
11.5 ApplicationoftheGHKFtotheDIFARShipTrackingCaseStudy, 163
References, 163
12 TheMonteCarloKalmanFilter 164
12.1 TheMonteCarloKalmanFilter, 167
Reference, 167
13 SummaryofGaussianKalmanFilters 168
13.1 AnalyticalKalmanFilters, 168
13.2 SigmaPointKalmanFilters, 170
13.3 AMorePracticalApproachtoUtilizingtheFamilyofKalman
Filters, 174
References, 175
14 PerformanceMeasuresfortheFamilyofKalmanFilters 176
14.1 ErrorEllipses, 176
14.1.1 TheCanonicalEllipse, 177
14.1.2 DeterminingtheEigenvaluesofP, 178
14.1.3 DeterminingtheErrorEllipseRotationAngle, 179
14.1.4 DeterminationoftheContainmentArea, 180
14.1.5 ParametricPlottingofErrorEllipse, 181
14.1.6 ErrorEllipseExample, 182
14.2 RootMeanSquaredErrors, 182
14.3 DivergentTracks, 183
14.4 Cramer–RaoLowerBound, 184
14.4.1 TheOne-DimensionalCase, 184
14.4.2 TheMultidimensionalCase, 186
14.4.3 ARecursiveApproachtotheCRLB, 186
14.4.4 The Cramer–Rao Lower Bound for Gaussian Additive
Noise, 190
14.4.5 TheGaussianCramer–RaoLowerBoundwithZeroProcess
Noise, 191
14.4.6 TheGaussianCramer–RaoLowerBoundwithLinear
Models, 191
xii CONTENTS
14.5 PerformanceofKalmanClassDIFARTrackEstimators, 192
References, 198
PARTIII MONTECARLOMETHODS
15 IntroductiontoMonteCarloMethods 201
15.1 ApproximatingaDensityFromaSetofMonteCarloSamples, 202
15.1.1 GeneratingSamplesfromaTwo-DimensionalGaussian
MixtureDensity, 202
15.1.2 ApproximatingaDensitybyItsMultidimensional
Histogram, 202
15.1.3 KernelDensityApproximation, 204
15.2 GeneralConceptsImportanceSampling, 210
15.3 Summary, 215
References, 216
16 SequentialImportanceSamplingParticleFilters 218
16.1 GeneralConceptofSequentialImportanceSampling, 218
16.2 ResamplingandRegularization(Move)forSISParticleFilters, 222
16.2.1 TheInverseTransformMethod, 222
16.2.2 SISParticleFilterwithResampling, 226
16.2.3 Regularization, 227
16.3 TheBootstrapParticleFilter, 230
16.3.1 ApplicationoftheBPFtoDIFARBuoyTracking, 231
16.4 TheOptimalSISParticleFilter, 233
16.4.1 GaussianOptimalSISParticleFilter, 235
16.4.2 LocallyLinearizedGaussianOptimalSISParticleFilter, 236
16.5 TheSISAuxiliaryParticleFilter, 238
16.5.1 ApplicationoftheAPFtoDIFARBuoyTracking, 242
16.6 ApproximationstotheSISAuxiliaryParticleFilter, 243
16.6.1 TheExtendedKalmanParticleFilter, 243
16.6.2 TheUnscentedParticleFilter, 243
16.7 ReducingtheComputationalLoadThrough
Rao-Blackwellization, 245
References, 245
17 TheGeneralizedMonteCarloParticleFilter 247
17.1 TheGaussianParticleFilter, 248
17.2 TheCombinationParticleFilter, 250
17.2.1 ApplicationoftheCPF–UKFtoDIFARBuoyTracking, 252
17.3 PerformanceComparisonofAllDIFARTrackingFilters, 253
References, 255