Table Of ContentNETWORKED
MULTISENSOR DECISION AND ESTIMATION FUSION
Based on Advanced Mathematical Methods
NETWORKED
MULTISENSOR DECISION AND ESTIMATION FUSION
Based on Advanced Mathematical Methods
Yunmin Zhu
Jie Zhou
Xiaojing Shen
Enbin Song
Yingting Luo
Boca Raton London New York
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Library of Congress Cataloging-in-Publication Data
Networked multisensor decision and estimation fusion : based on advanced mathematical methods /
Yunmin Zhu … [et al.].
p. cm.
Summary: “Multisource information fusion has become a crucial technique in areas such as sensor
networks, space technology, air traffic control, military engineering, communications, industrial
control, agriculture, and environmental engineering. Exploring recent signficant results, this
book presents essential mathematical descriptions and methods for multisensory decision and
estimation fusion. It covers general adapted methods and systematic results, includes computer
experiments to support the theoretical results, and fixes several popular but incorrect results in the
field”-- Provided by publisher.
Includes bibliographical references and index.
ISBN 978-1-4398-7452-3 (hardback)
1. Sensor networks. 2. Multisensor data fusion--Mathematics. I. Zhu, Yunmin, 1944-
TK7872.D48N47 2012
681’.2--dc23 2012015651
Visit the Taylor & Francis Web site at
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and the CRC Press Web site at
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Contents
Preface ........................................................................ xiii
Acknowledgment ............................................................. xix
1 Introduction............................................................... 1
1.1 FundamentalProblems ................................................ 2
1.2 CoreofFundamentalTheoryandGeneralMathematicalIdeas...... 3
1.3 ClassicalStatisticalDecision........................................... 4
1.3.1 BayesDecision ................................................. 5
1.3.2 Neyman–PearsonDecision .................................... 8
1.3.2.1 Neyman–PearsonCriterion .......................... 8
1.3.3 MinimaxDecision ............................................. 10
1.4 LinearEstimationandKalmanFiltering .............................. 11
1.5 BasicsofConvexOptimization........................................ 17
1.5.1 ConvexOptimization.......................................... 17
1.5.1.1 BasicTerminologyofOptimization.................. 17
1.5.2 Duality ......................................................... 22
1.5.3 Relaxation ...................................................... 24
1.5.3.1 S-ProcedureRelaxation............................... 24
1.5.3.2 SDPRelaxation....................................... 26
2 ParallelStatisticalBinaryDecisionFusion .............................. 29
2.1 OptimalSensorRulesforBinaryDecisionGivenFusionRule....... 30
2.1.1 FormulationforBayesBinaryDecision ....................... 30
2.1.2 FormulationofFusionRulesviaPolynomials
ofSensorRules................................................. 31
2.1.3 Fixed-PointTypeNecessaryConditionfortheOptimal
SensorRules ................................................... 33
2.1.4 FiniteConvergenceoftheDiscretizedAlgorithm............. 37
2.2 UnifiedFusionRule ................................................... 45
2.2.1 ExpressionoftheUnifiedFusionRule........................ 45
2.2.2 NumericalExamples........................................... 48
2.2.2.1 TwoSensors ......................................... 48
v
vi ■ Contents
2.2.2.2 ThreeSensors........................................ 50
2.2.2.3 FourSensors ......................................... 52
2.3 ExtensiontoNeyman–PearsonDecision.............................. 53
2.3.1 AlgorithmSearchingforOptimalSensorRules............... 56
2.3.2 NumericalExamples........................................... 57
3 GeneralNetworkStatisticalDecisionFusion............................ 59
3.1 ElementaryNetworkStructures....................................... 60
3.1.1 ParallelNetwork............................................... 60
3.1.2 TandemNetwork.............................................. 62
3.1.3 Hybrid(Tree)Network........................................ 64
3.2 FormulationofFusionRuleviaPolynomialsofSensorRules ........ 64
3.3 Fixed-PointTypeNecessaryConditionforOptimal
SensorRules ........................................................... 69
3.4 IterativeAlgorithmandConvergence ................................. 71
3.5 UnifiedFusionRule ................................................... 74
3.5.1 UnifiedFusionRuleforParallelNetworks.................... 75
3.5.2 UnifiedFusionRuleforTandemand
HybridNetworks.............................................. 78
3.5.3 NumericalExamples........................................... 79
3.5.3.1 Three-SensorSystem................................. 80
3.5.3.2 Four-SensorSystem.................................. 82
3.6 OptimalDecisionFusionwithGivenSensorRules................... 84
3.6.1 ProblemFormulation.......................................... 85
3.6.2 ComputationofLikelihoodRatios............................ 87
3.6.3 LocallyOptimalSensorDecisionRules
withCommunicationsamongSensors ........................ 88
3.6.4 NumericalExamples........................................... 90
3.6.4.1 Two-SensorNeyman–Pearson
DecisionSystem ..................................... 91
3.6.4.2 Three-SensorBayesianDecisionSystem ............ 91
3.7 SimultaneousSearchforOptimalSensorRulesand
FusionRule............................................................ 96
3.7.1 ProblemFormulation.......................................... 96
3.7.2 NecessaryConditionsforOptimalSensorRules
andanOptimalFusionRule .................................. 99
3.7.3 IterativeAlgorithmandItsConvergence...................... 103
3.7.4 ExtensionstoMultiple-BitCompressionandNetwork
DecisionSystems .............................................. 110
3.7.4.1 ExtensionstotheMultiple-BitCompression........ 110
3.7.4.2 ExtensionstoHybridParallelDecisionSystem
andTreeNetworkDecisionSystem................. 112
3.7.5 NumericalExamples........................................... 116
Contents ■ vii
3.7.5.1 TwoExamplesforAlgorithm3.2.................... 116
3.7.5.2 AnExampleforAlgorithm3.3 ...................... 119
3.8 PerformanceAnalysisofCommunicationDirection
forTwo-SensorTandemBinaryDecisionSystem .................... 120
3.8.1 ProblemFormulation.......................................... 122
3.8.1.1 SystemModel........................................ 122
3.8.1.2 BayesDecisionRegionofSensor2.................. 122
3.8.1.3 BayesDecisionRegionofSensor1
(FusionCenter)...................................... 127
3.8.2 BayesCostFunction........................................... 128
3.8.3 Results ......................................................... 129
3.8.4 NumericalExamples........................................... 140
3.9 NetworkDecisionSystemswithChannelErrors ..................... 143
3.9.1 SomeFormulationsaboutChannelError..................... 144
3.9.2 NecessaryConditionforOptimalSensorRulesGiven
aFusionRule .................................................. 145
3.9.3 SpecialCase:MutuallyIndependentSensor
Observations................................................... 149
3.9.4 UnifiedFusionRulesforNetworkDecisionSystems......... 151
3.9.4.1 NetworkDecisionStructureswith
ChannelErrors....................................... 151
3.9.4.2 UnifiedFusionRuleinParallelBayesianBinary
DecisionSystem ..................................... 154
3.9.4.3 UnifiedFusionrulesforGeneralNetwork
DecisionSystemswithChannelErrors.............. 155
3.9.5 NumericalExamples........................................... 157
3.9.5.1 ParallelBayesianBinaryDecisionSystem........... 157
3.9.5.2 Three-SensorDecisionSystem ...................... 159
4 SomeUncertainDecisionCombinations ................................ 163
4.1 RepresentationofUncertainties....................................... 164
4.2 DempsterCombinationRuleBasedonRandom
SetFormulation ....................................................... 165
4.2.1 Dempster’sCombinationRule................................ 167
4.2.2 MutualConversionoftheBasicProbabilityAssignment
andtheRandomSet........................................... 167
4.2.3 CombinationRulesoftheDempster–ShaferEvidencesvia
RandomSetFormulation...................................... 168
4.2.4 AllPossibleRandomSetCombinationRules................. 169
4.2.5 CorrelatedSensorBasicProbabilisticAssignments............ 171
4.2.6 OptimalBayesianCombinationRule......................... 172
4.2.7 ExamplesofOptimalCombinationRule ..................... 174
viii ■ Contents
4.3 FuzzySetCombinationRuleBasedonRandomSet
Formulation ........................................................... 177
4.3.1 MutualConversionoftheFuzzySetandthe
RandomSet.................................................... 178
4.3.2 SomePopularCombinationRulesofFuzzySets.............. 179
4.3.3 GeneralCombinationRules................................... 181
4.3.3.1 UsingtheOperationsofSetsOnly.................. 182
4.3.3.2 UsingtheMoreGeneralCorrelationofthe
RandomVariables ................................... 183
4.3.4 Relationshipbetweenthet-NormandTwo-Dimensional
DistributionFunction ......................................... 184
4.3.5 Examples....................................................... 186
4.4 HybridCombinationRuleBasedonRandomSet
Formulation ........................................................... 188
5 ConvexLinearEstimationFusion........................................ 191
5.1 LMSEEstimationFusion.............................................. 192
5.1.1 FormulationofLMSEFusion................................. 192
5.1.2 OptimalFusionWeights ...................................... 195
5.2 EfficientIterativeAlgorithmforOptimalFusion..................... 200
5.2.1 AppropriateWeightingMatrix................................ 201
5.2.2 IterativeFormulaofOptimalWeightingMatrix.............. 204
5.2.3 IterativeAlgorithmforOptimalEstimationFusion........... 205
5.2.4 Examples....................................................... 210
5.3 RecursionofEstimationErrorCovarianceinDynamicSystems ..... 212
5.4 OptimalDimensionalityCompressionforSensorDatain
EstimationFusion ..................................................... 214
5.4.1 ProblemFormulation.......................................... 215
5.4.2 Preliminary..................................................... 216
5.4.3 AnalyticSolutionforSingle-SensorCase...................... 218
5.4.4 SearchforOptimalSolutionintheMultisensorCase ........ 220
5.4.4.1 ExistenceoftheOptimalSolution................... 220
5.4.4.2 OptimalSolutionataSensorWhileOtherSensor
CompressionMatricesAreGiven ................... 221
5.4.5 NumericalExample............................................ 223
5.5 QuantizationofSensorData .......................................... 224
5.5.1 ProblemFormulation.......................................... 227
5.5.2 NecessaryConditionsforOptimalSensorQuantization
RulesandOptimalLinearEstimationFusion................. 229
5.5.3 Gauss–SeidelIterativeAlgorithmforOptimalSensor
QuantizationRulesandLinearEstimationFusion............ 235
5.5.4 NumericalExamples........................................... 237
Contents ■ ix
6 KalmanFilteringFusion.................................................. 241
6.1 DistributedKalmanFilteringFusionwithCross-Correlated
SensorNoises .......................................................... 243
6.1.1 ProblemFormulation.......................................... 244
6.1.2 DistributedKalmanFilteringFusionwithoutFeedback...... 246
6.1.3 OptimalityofKalmanFilteringFusionwithFeedback....... 249
6.1.3.1 GlobalOptimalityoftheFeedback
FilteringFusion...................................... 250
6.1.3.2 LocalEstimateErrors................................ 251
6.1.3.3 TheAdvantagesoftheFeedback .................... 252
6.2 DistributedKalmanFilteringFusionwithSingularCovariances
ofFilteringErrorandMeasurementNoises........................... 254
6.2.1 EquivalenceFusionAlgorithm ................................ 255
6.2.2 LMSEFusionAlgorithm ...................................... 255
6.2.3 NumericalExamples........................................... 257
6.3 OptimalKalmanFilteringTrajectoryUpdatewithUnideal
SensorMessages ....................................................... 261
6.3.1 OptimalLocal-ProcessorTrajectoryUpdatewithUnideal
Measurements.................................................. 262
6.3.1.1 OptimalLocal-ProcessorTrajectoryUpdatewith
AdditionofOOSMs................................. 263
6.3.1.2 OptimalLocal-ProcessorTrajectoryUpdatewith
RemovalofEarlierMeasurement.................... 267
6.3.1.3 OptimalLocal-ProcessorTrajectoryUpdatewith
SequentiallyProcessingUnidealMeasurements..... 268
6.3.1.4 NumericalExamples................................. 269
6.3.2 OptimalDistributedFusionTrajectoryUpdate
withLocal-ProcessorUnidealUpdates........................ 271
6.3.2.1 OptimalDistributedFusionTrajectoryUpdate
withAdditionofLocalOOSMUpdate............. 272
6.3.2.2 OptimalDistributedStateTrajectoryUpdate
withRemovalofEarlierLocalEstimate............. 274
6.3.2.3 OptimalDistributedFusionTrajectoryUpdate
withSequentialProcessingofLocal
UnidealUpdates..................................... 275
6.4 RandomParameterMatricesKalmanFilteringFusion ............... 276
6.4.1 RandomParameterMatricesKalmanFiltering ............... 276
6.4.2 RandomParameterMatricesKalmanFiltering
withMultisensorFusion....................................... 278
6.4.3 SomeApplications............................................. 281
6.4.3.1 ApplicationtoDynamicProcesswith
FalseAlarm........................................... 281
