Sequential change detection and hypothesis testing: general non-i.i.d. stochastic models and asymptotically optimal rules

321 Pages·2020·8.862 MB·English

by  Tartakovsky, Alexander

Sequential Change Detection
and Hypothesis Testing

Sequential Change Detection
and Hypothesis Testing
General Non-i.i.d. Stochastic Models and
Asymptotically Optimal Rules
Alexander G. Tartakovsky
Moscow,Russia andLosAngeles,USA
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IN MEMORY OF MY FATHER
GEORGIY P. TARTAKOVSKY
AS WELL AS
TO MY WIFE MARINA AND MY SON DANIEL

Contents
Preface xi
NotationandSymbols xiii
Introduction xvii
1 SequentialHypothesisTestinginMultipleDataStreams 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 SequentialMultistreamHypothesisTestingProblem . . . . . . . . . . . . . . . . 2
1.3 GeneralizedLikelihoodRatioandMixtureSequentialTests . . . . . . . . . . . . 3
1.4 AsymptoticOperatingCharacteristicsintheGeneralNon-i.i.d.Case . . . . . . . . 6
1.4.1 ProbabilitiesofErrorsintheGeneralNon-i.i.d.Case . . . . . . . . . . . . 6
1.4.1.1 UpperBoundsontheErrorProbabilities . . . . . . . . . . . . . 6
1.4.1.2 ErrorExponents . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.1.3 MonteCarloImportanceSampling . . . . . . . . . . . . . . . . 9
1.4.1.4 AsymptoticOptimalityofGSLRTandMSLRT . . . . . . . . . 12
1.4.2 TheCaseofIndependentDataStreams . . . . . . . . . . . . . . . . . . . 15
1.4.2.1 AsymptoticOptimalityintheCaseofIndependentStreams . . . 16
1.4.2.2 ScalabilityintheCaseofIndependentStreams . . . . . . . . . . 16
1.4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.4 MonteCarloSimulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.5 HigherOrderApproximationsandOptimalityinthei.i.d.Case . . . . . . . . . . . 26
1.5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.5.2 AsymptoticApproximationsfortheProbabilitiesofErrors . . . . . . . . . 29
1.5.3 Third-OrderAsymptoticApproximationsfortheESS . . . . . . . . . . . . 31
1.5.3.1 Asymptotic Approximations for the ESS Under Hypothesis HB
andUnderHypothesisH intheAsymmetricCase . . . . . . . . 31
0
1.5.3.2 AsymptoticApproximationsfortheESSUnderHypothesisH in
0
theGeneralCase . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.5.4 High-OrderAsymptoticOptimality . . . . . . . . . . . . . . . . . . . . . 52
1.5.4.1 UniformAsymptoticOptimality . . . . . . . . . . . . . . . . . 52
1.5.4.2 Bayesian-typeAsymptoticOptimality . . . . . . . . . . . . . . 53
1.5.4.3 Asymptotic Minimax Properties with Respect to Kullback–
LeiblerInformation . . . . . . . . . . . . . . . . . . . . . . . . 57
1.5.4.4 FurtherOptimizationandMCSimulations . . . . . . . . . . . . 58
2 Sequential Detection of Changes: Changepoint Models, Performance Metrics and
OptimalityCriteria 63
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.2 ChangepointModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.2.1 ModelsforObservedProcesses . . . . . . . . . . . . . . . . . . . . . . . 64
2.2.1.1 ASingleStreamScenario . . . . . . . . . . . . . . . . . . . . . 64
2.2.1.2 AMultistreamScenario . . . . . . . . . . . . . . . . . . . . . . 65
vii
viii Contents
2.2.2 ModelsfortheChangePoint . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.2.2.1 TypesofChanges . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.2.2.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.3 OptimalityCriteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.3.2 MeasuresoftheFalseAlarmRisk . . . . . . . . . . . . . . . . . . . . . . 69
2.3.2.1 AverageRunLengthtoFalseAlarm . . . . . . . . . . . . . . . 70
2.3.2.2 WeightedProbabilityofFalseAlarm . . . . . . . . . . . . . . . 71
2.3.2.3 GlobalProbabilityofFalseAlarm . . . . . . . . . . . . . . . . 71
2.3.2.4 LocalProbabilitiesofFalseAlarm . . . . . . . . . . . . . . . . 72
2.3.3 AnExpectedDelaytoDetectioninaGeneralCase . . . . . . . . . . . . . 74
2.3.4 BayesianCriteriawithRespecttotheExpectedDelaytoDetection . . . . . 75
2.3.5 MinimaxCriteriawithRespecttotheExpectedDelaytoDetection . . . . . 77
2.3.6 PointwiseUniformOptimalityCriterion . . . . . . . . . . . . . . . . . . . 79
2.3.7 CriteriaMaximizingProbabilityofDetection . . . . . . . . . . . . . . . . 79
2.3.8 AsymptoticOptimalityCriteria . . . . . . . . . . . . . . . . . . . . . . . 81
3 BayesianQuickestChangeDetectioninaSinglePopulation 83
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.2 TheShiryaevandShiryaev–RobertsMixtureRules . . . . . . . . . . . . . . . . . 85
3.3 AsymptoticProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.4 AsymptoticOptimalityoftheMixtureShiryaevRule . . . . . . . . . . . . . . . . 88
3.4.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.4.2 Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.4.3 AsymptoticOptimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.5 AsymptoticPerformanceoftheMixtureShiryaev–RobertsRule . . . . . . . . . . 99
3.6 AsymptoticOptimalitywithRespecttotheIntegratedRisk . . . . . . . . . . . . . 104
3.7 TheCaseofaSimplePost-ChangeHypothesis . . . . . . . . . . . . . . . . . . . 108
3.8 TheCaseofIndependentObservations . . . . . . . . . . . . . . . . . . . . . . . 110
3.9 Thei.i.d.Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
3.10 Window-LimitedChangeDetectionRules . . . . . . . . . . . . . . . . . . . . . . 116
3.11 SufficientConditionsofAsymptoticOptimalityforMarkovProcesses . . . . . . . 122
3.12 AsymptoticOptimalityforHiddenMarkovModels . . . . . . . . . . . . . . . . . 129
3.12.1 MarkovRandomWalkRepresentationoftheLLRforHHM . . . . . . . . 130
3.12.2 AsymptoticOptimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.12.3 HigherOrderAsymptoticApproximationsfortheAverageDetectionDelay
andPFA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
3.12.4 TheCaseofConditionallyIndependentObservations . . . . . . . . . . . . 140
3.13 AdditionalExamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
3.14 ConcludingRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4 NearlyOptimalPointwiseandMinimaxChangeDetectioninaSinglePopulation 149
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
4.2 NearlyOptimalPointwiseandMinimaxChangeDetection . . . . . . . . . . . . . 149
4.2.1 ProblemSetup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
4.2.2 AsymptoticOptimalityoftheMSRDetectionRule . . . . . . . . . . . . . 150
4.2.2.1 TheNon-i.i.d.Case . . . . . . . . . . . . . . . . . . . . . . . . 150
4.2.2.2 TheCaseofLLRwithIndependentIncrements. . . . . . . . . . 156
4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
4.4 MonteCarloSimulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Contents ix
5 ChangeDetectionRulesOptimalfortheMaximalDetectionProbabilityCriterion 163
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.2 Shewhart’sRule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5.2.1 OptimalitywithRespecttotheExpectedDetectionDelay . . . . . . . . . 164
5.2.2 MaximalAverageProbabilityofDetection:theBayesianApproach . . . . 165
5.2.3 MaximinFrequentistCriteria. . . . . . . . . . . . . . . . . . . . . . . . . 172
5.3 BayesianandMaximinSequentialDetectioninWindowswithArbitrarilySize . . 176
5.3.1 BayesOptimalChangeDetectionRule . . . . . . . . . . . . . . . . . . . 176
5.3.2 MaximinOptimalChangeDetectionRule . . . . . . . . . . . . . . . . . . 179
5.4 ConcludingRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6 QuickestChangeDetectioninMultipleStreams 181
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.2 AMultistreamModelandChangeDetectionRules . . . . . . . . . . . . . . . . . 182
6.2.1 TheGeneralMultistreamModel . . . . . . . . . . . . . . . . . . . . . . . 182
6.2.2 Double-MixtureChangeDetectionRules . . . . . . . . . . . . . . . . . . 183
6.2.2.1 TheGeneralCase . . . . . . . . . . . . . . . . . . . . . . . . . 183
6.2.2.2 IndependentStreams . . . . . . . . . . . . . . . . . . . . . . . 185
6.3 AsymptoticOptimalityProblemsandAssumptions . . . . . . . . . . . . . . . . . 186
6.4 AsymptoticLowerBoundsforMomentsoftheDetectionDelayandAverageRisk
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
6.5 AsymptoticOptimalityofDouble-MixtureDetectionRules . . . . . . . . . . . . 189
6.5.1 AsymptoticOptimalityoftheDouble-MixtureRuleTp,W . . . . . . . . . . 189
A
6.5.2 AsymptoticOptimalityoftheDouble-MixtureRuleT(cid:101)p,W . . . . . . . . . . 193
A
6.6 AsymptoticOptimalitywithRespecttotheAverageRisk . . . . . . . . . . . . . . 197
6.7 AsymptoticOptimalityforaPutativeValueofthePost-ChangeParameter . . . . . 199
6.8 AsymptoticOptimalityintheCaseofIndependentStreams . . . . . . . . . . . . . 200
6.9 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
6.10 DiscussionandRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
7 JointChangepointDetectionandIdentification 207
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
7.2 TheModelandtheDetection–IdentificationRule . . . . . . . . . . . . . . . . . . 208
7.3 TheOptimizationProblemandAssumptions . . . . . . . . . . . . . . . . . . . . 209
7.4 UpperBoundsonProbabilitiesofFalseAlarmandMisidentification . . . . . . . . 211
7.5 LowerBoundsontheMomentsoftheDetectionDelay . . . . . . . . . . . . . . . 213
7.6 AsymptoticOptimalityoftheDetection–IdentificationRuleδ . . . . . . . . . . . 217
A
7.7 AnExample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
7.8 ConcludingRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
8 Applications 227
8.1 ApplicationtoObjectTrackManagementinSonarSystems . . . . . . . . . . . . 227
8.2 ApplicationtoDetectionofTracesofSpaceObjects . . . . . . . . . . . . . . . . 230
8.3 ApplicationtoDetectionofUnauthorizedBreak-insinComputerNetworks . . . . 235
AppendixA:UsefulAuxiliaryResults 239
AppendixB:StochasticConvergence 243
B.1 StandardModesofConvergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
B.2 Completeandr-QuickConvergence . . . . . . . . . . . . . . . . . . . . . . . . . 245