Possibility Theory for the Design of Information Fusion Systems

294 Pages·2019·12.283 MB·English

by  Basel Solaiman, Éloi Bossé

Information Fusion and Data Science
Series Editor: Henry Leung
Basel Solaiman
Éloi Bossé
Possibility Theory
for the Design of
Information Fusion
Systems
Information Fusion and Data Science
Serieseditor
HenryLeung,UniversityofCalgary,Calgary,AB,Canada
Thisbookseriesprovidesaforumtosystematicallysummarizerecentdevelopments,
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mation fusion along with its connection to data-enabled science. Emphasis is also
placed on fundamental theories, algorithms and real-world applications of massive
dataaswellasinformationprocessing,analysis,fusionandknowledgegeneration.
The aim of this book series is to provide the most up-to-date research results and
tutorialmaterialsoncurrenttopicsinthisgrowingfieldaswellastostimulatefurther
research interest by transmitting the knowledge to the next generation of scientists
and engineers in the corresponding fields. The target audiences are graduate stu-
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to computational sciences and engineering, complex systems and artificial intelli-
gence. Formats suitable for the series are contributed volumes, monographs and
lecturenotes.
Moreinformationaboutthisseriesathttp://www.springer.com/series/15462
(cid:129)
Basel Solaiman Éloi Bossé
Possibility Theory for the
Design of Information Fusion
Systems
BaselSolaiman ÉloiBossé
ImageandInformationProcessing ImageandInformationProcessing
Department Department
IMTAtlantique IMTAtlantique
Brest,France Brest,France
ISSN2510-1528 ISSN2510-1536 (electronic)
ISBN978-3-030-32852-8 ISBN978-3-030-32853-5 (eBook)
https://doi.org/10.1007/978-3-030-32853-5
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Preface
Possibilitytheoryisamathematicaltheory,coinedbyL.A.Zadehinthelate1970s
(1978)todealwithvaguepiecesofinformationdescribedbymeansoffuzzysetsand
fuzzylogic.Thereafter,DidierDuboisandHenriPradehavebeenthemainfounders
of this theory to theextent that we have today, a crediblealternative toprobability
theory. A considerable body of literature has flourished around fuzzy sets and
possibilitytheoryconceptsinaverywiderange ofapplications, frommathematics
andlogicstoadvancedengineeringmethodologies,frommedicaldomaintofinance,
from human factors toconsumer products, and so on. There is a plethora of books
andpapersdescribingthisrichdomainofapplications.
The ambition of this book is to address a niche still uncovered by the existing
availablebooks:acomprehensiveassemblageofthebasicconcepts,themathemat-
ical developments, and the engineering methodologies to position and exploit
possibilitytheoryforthedesignofcomputer-baseddecision-supportsystems.Usu-
ally, decision-support systems comprise three main parts: analysis (analytics),syn-
thesis(informationfusion),andprescription(decideandact).Literature showsthat
possibilitytheorycanbeappliedtothethreeparts.
This book consists of nine chapters. The first three chapters discuss the funda-
mental possibilistic concepts: distribution, necessity measure, possibility measure,
jointdistribution,andtheimportantconceptofconditioning.Chapter4examinesthe
conceptofsimilaritythatplaysanessentialroleinawiderangeofapplicationfields
like patternrecognition, reasoning,data andknowledge miningbut with respectto
whatcanpossibilitytheorybringtoimplementthatcomplicatedconcept.Chapter5
addressesthelinksandtransformationsbetweentheinterrelateduncertaintymodel-
ing theories. The following next two chapters treat aspects of decision-making
through possibilistic and fuzzy integrals, fusion operators, and decision-making
criteria in the framework of possibility theory. Chapter 8 presents three low-level
complexityapplicationsofpossibilisticconcepts:(1)onpixel-basedimageclassifi-
cation,(2)onspatialunmixing,and(3)onimagesegmentation.
ThebookisconcludedbyChapter9ontheuseofpossibilitytheoryinthedesign
of information fusion systems in today’s ever-increasing complexity of our real
v
vi Preface
world. Information overload and complexity are core problems to most organiza-
tionsoftoday.Theadvancesinnetworkingcapabilitieshavecreatedtheconditions
of complexity by enabling richer, real-time interactions between and among indi-
viduals, objects, systems, and organizations. Fusion of Information and Analytics
Technologies(FIAT)arekeyenablersforthedesignofcurrentandfuturedecision-
support systems to support prognosis, diagnosis, and prescriptive tasks in such
complex environments. Hundreds of methods and technologies exist, and several
bookshavebeendedicatedtoeitheranalyticsorinformationfusionsofar.Thisbook
presentstheoverallpictureinwhichpossibilitytheorycanbeofanyuse.
Brest,France BaselSolaiman
Brest,France ÉloiBossé
Contents
1 IntroductiontoPossibilityTheory. . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 InformationConcept. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 InformationElementDefinition. . . . . . . . . . . . . . . . . . . . 3
1.2.2 IntrinsicInformationImperfectionTypes. . . . . . . . . . . . . . 7
1.3 PossibilisticInformationConcept. . . . . . . . . . . . . . . . . . . . . .. . . 10
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 FundamentalPossibilisticConcepts. . . . . . . . . . . . . . . .. . . . . . . . . . 13
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 PossibilityDistributionsConcept. . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 DefiningaPossibilityDistribution. . . . . . . . . . . . . . . . . . 14
2.2.2 PossibilityDistributionModels. . . . . . . . . . . . . . . . . . . . . 16
2.2.3 PossibilityDistributionsDiscounting. . . . . . . . . . . . . . . . . 21
2.2.4 PossibilisticExtensionPrinciple. . . . . . . . . . . . . . . . . . . . 22
2.2.5 SpecificityConceptandMinimalSpecificity
Principle(MSP). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 PossibilityandNecessityMeasures. . . . . . . . . . . . . . . . . . . . . . . 25
2.3.1 PossibilityMeasure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.2 NecessityMeasure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.3 DualityRelevantPropertiesofPossibility
andNecessityMeasures. . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.4 RelativePossibilityandNecessityMeasures
ofAmbiguousEvents. . .. . . . . .. . . . . .. . . . . .. . . . .. . 33
2.3.5 ImportantPropertiesofPossibility/Necessity
DegreesofMatching. . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
vii
viii Contents
2.4 SubnormalPossibilityDistributions. . . . . . . . . . . . . . . . . . . . . . . 38
2.4.1 PossibilityDistributionsNormalizationMethods. . . . . . . . 40
2.4.2 Dubois’sAlternativeNecessityMeasure. . . . . . . . . . . . . . 42
2.4.3 NormalVersusSubnormalDistributionsProperties. . . . . . 44
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3 JointPossibilityDistributionsandConditioning. . . . . . . . . . . . . . . . 47
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 JointandMarginalPossibilityDistributions. . . . . . . . . . . . . . . . . 49
3.3 CylindricalExtensionofNon-interactivePossibilistic
Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3.1 ProjectionsofaCylindricalExtension. . . . . . . . . . . . . . . . 55
3.3.2 JointPossibilityandNecessityMeasures. . . . . . . . . . . . . . 56
3.4 ConditioningUndertheKnowledgeoftheJointPossibility
Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.4.1 Zadeh’sConditioningRule. . . . . . . . . . . . . . . . . . . . . . . . 61
3.4.2 Hisdal’sConditioningRule. . . . . . . . . . . . . . . . . . . . . . . 63
3.4.3 Dempster’sConditioningRule. . . . . . . . . . . . . . . . . . . . . 66
3.4.4 Nguyen’sConditioningRule. . . . . . . . . . . . . . . . . . . . . . 67
3.4.5 CausalLinkConditioning.. . . .. . . . .. . . . .. . . .. . . . .. 70
3.5 ConditioningandBeliefRevision. . . . . . . . . . . . . . . . . . . . . . . . 72
3.5.1 CrispEvent-BasedPossibilisticRevision. . . . . . . . . . . . . . 72
3.5.2 UnreliableCrispEvent-BasedPossibilisticRevision. . . . . . 76
3.6 ConditioningandPossibilisticMedicalDiagnosis. . . . . . . . . . . . . 78
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4 PossibilisticSimilarityMeasures. . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2 TaxonomyofSimilarityMeasures. . . . . . . . . . . . . . . . . . . . . . . . 86
4.2.1 Metric-BasedSimilarityMeasures. . . . . . . . . . . . . . . . .. . 87
4.2.2 Set-BasedSimilarityMeasures. . . . . . . . . . . . . . . . . . . . . 94
4.3 FuzzySetsTheoryandSimilarityMeasures. . . . . . . . . . . . . . . . . 98
4.3.1 Metric-BasedSimilarityMeasuresofFuzzySets. . . . . . . . 102
4.3.2 Set-BasedSimilarityMeasuresofFuzzySets. . . . . . . . . . . 104
4.3.3 Implication-BasedSimilarityMeasuresofFuzzySets. . . . . 111
4.4 PossibilityDistributionsSimilarityMeasures. . . . . . . . . . . . . . . . 116
4.4.1 Definition,PossibilisticSimilarityMeasures. . . . . . . . . . . 118
4.4.2 Metric-BasedPossibilisticSimilarityMeasures. . . . . . . . . 121
4.4.3 Set-BasedPossibilisticSimilarityMeasures. . . . . . . . . . . . 122
4.4.4 Informational-BasedPossibilisticSimilarity
Measures. . .. . . .. . . . .. . . .. . . .. . . .. . . .. . . .. . . .. 124
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Contents ix
5 TheInterrelatedUncertaintyModelingTheories. . . . . . . . . . . . . . . 137
5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.2 TheMonotoneMeasuresTheory. . . . . . . . . . . . . . . . . . . . . . . . . 140
5.2.1 SugenoMonotoneMeasureDefinition. . . . . . . . . . . . . . . 140
5.2.2 DistinguishedClassesofMonotoneMeasures. . . . . . . . . . 142
5.3 UncertaintyTheoriesintheFrameworkofMonotone
MeasuresTheory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.4 Evidence-PossibilityTransformations. . . . . . . . . . . . . . . . . . . . . . 148
5.4.1 Transformingab.p.aintoaPossibilityDistribution. . . . . . 149
5.4.2 TransformingaPossibilityDistributionintoab.p.a. . . . . . 150
5.5 Probability-PossibilityTransformations. . . . . . . . . . . . . . . . . . . . 151
5.5.1 Probability-PossibilityConsistencyConcepts. . . . . . . . . . . 152
5.5.2 Probability-PossibilityTransformationMethods. . . . . . . . . 155
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6 PossibilityIntegral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.2 AggregationFunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.3 MonotoneMeasuresandFuzzyIntegrals. . . . . . . . . . . . . . . . . . . 169
6.3.1 MonotoneMeasuresDefinition. . . . . . . . . . . . . . . . . . . . . 170
6.3.2 SpecialMonotoneMeasures. . . . . . . . . . . . . . . . . . . . . . . 172
6.4 DiscreteChoquetIntegral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.4.1 ImportantPropertiesoftheDiscreteChoquet
Integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.4.2 DiscreteChoquetIntegralforSomeTypes
ofMonotoneMeasures. . . . . . . .. . . . . . . . .. . . . . . . . .. 179
6.5 DiscreteSugenoIntegral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6.5.1 ImportantPropertiesoftheDiscreteSugenoIntegral. . . . . 186
6.5.2 DiscreteSugenoIntegralforSomeMonotone
Measures. . .. . . .. . . . .. . . .. . . .. . . .. . . .. . . .. . . .. 188
6.5.3 TwofoldIntegral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
6.6 PossibilityIntegral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
6.6.1 PossibilisticChoquetIntegral. . . . . . . . . . . . . . . . . . . . . . 190
6.6.2 PossibilisticSugenoIntegral. . . . . . . . . . . . . . . . . . . . . . . 192
6.6.3 SubnormalPossibilisticSugenoIntegral. . . . . . . . . . . . . . 196
6.7 ApplicationofthePossibilityIntegraltoPatternRecognition. . . . . 198
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
7 FusionOperatorsandDecision-MakingCriteria
intheFrameworkofPossibilityTheory. . . . . . . . . . . . . . . . . . . . . . 205
7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
7.2 PossibilityDistributionsFusion. . . . . . . . . . . . . . . . . . . . . . . . . . 205
7.2.1 ConjunctivePossibilityDistributionsFusion. . . . . . . . . . . 206
7.2.2 DisjunctivePossibilityDistributionsFusion. . . . . . . . . . . . 208
7.2.3 Trade-OffPossibilityDistributionsFusion. . . . . . . . . . . . . 209