Soft Methods for Integrated Uncertainty Modelling

397 Pages·2006·13.631 MB·English

by  Lotfi A. Zadeh (auth.), Jonathan Lawry, Enrique Miranda, Alberto Bugarin, Shoumei Li, Maria Angeles Gil, Przemys aw Grzegorzewski, Olgierd Hyrniewicz (eds.)

JonathanLawry,EnriqueMiranda,AlbertoBugarin,ShoumeiLi,MariaAngelesGil,
Przemysa(cid:20) wGrzegorzewski,OlgierdHyrniewicz(Eds.)
SoftMethodsforIntegratedUncertaintyModelling
AdvancesinSoftComputing
Editor-in-chief
Prof.JanuszKacprzyk
SystemsResearchInstitute
PolishAcademyofSciences
ul.Newelska6
01-447Warsaw
Poland
E-mail:[e-mail]
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MiguelLÛpez-DÌaz,MarÌaÁ.Gil, ISBN3-540-33520-X
Przemysa(cid:20) wGrzegorzewski,Olgierd
JonathanLawry,EnriqueMiranda,Alberto
Hryniewicz,JonathanLawry
Bugarin,ShoumeiLi,MariaAngelesGil,
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Przemysa(cid:20) wGrzegorzewski,Olgierd
Systems,2004
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KwangH.Lee Modelling,2006
FirstCourseonFuzzyTheoryand ISBN3-540-34776-3
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Jonathan Lawry
Enrique Miranda
Alberto Bugarin
Shoumei Li
Maria Angeles Gil
Przemys(cid:20)aw Grzegorzewski
Olgierd Hyrniewicz
(Eds.)
Soft Methods for Integrated
Uncertainty Modelling
ABC
JonathanLawry ShoumeiLi
AIGroup DepartmentofAppliedMathematics
DepartmentofEngineering BeijingUniversityofTechnology
Mathematics Beijing100022,P.R.China
UniversityofBristol
Bristol,BS81TR,UK
EnriqueMiranda MariaAngelesGil
ReyJuanCarlosUniversity UniversiaddeOviedo
StatisticsandOperationsResearch Fac.Ciencias
C-Tulip·ns/n Dpto.EstadisticaeI.OyD.M.
MÛstoles28933,Spain CalleCalvoSotelos/n
33071Oviedo,Spain
AlbertoBugarin Przemysa(cid:20) wGrzegorzewski
OlgierdHyrniewicz
IntelligentSystemsGroup
DepartmentofElectronics SystemsResearchInstitute
&ComputerScience PolishAcademyofSciences
UniversityofSantiagodeCompostela Newelska6
SantiagodeCompostela,Spain 01-447Warsaw,Poland
LibraryofCongressControlNumber:2006928309
ISSNprintedition:1615-3871
ISSNelectronicedition:1860-0794
ISBN-10 3-540-34776-3SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-34776-7SpringerBerlinHeidelbergNewYork
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Preface
Theideaofsoftcomputingemergedintheearly1990sfromthefuzzysystemscom-
munity, and refers to an understanding that the uncertainty, imprecision and igno-
rance present in a problem should be explicitly represented and possibly even ex-
ploitedratherthaneithereliminatedorignoredincomputations.Forinstance,Zadeh
defined‘SoftComputing’asfollows:
Soft computing differs from conventional (hard) computing in that, unlike
hard computing, it is tolerant of imprecision, uncertainty and partial truth.
Ineffect,therolemodelforsoftcomputingisthehumanmind.
Recently soft computing has, to some extent, become synonymous with a hybrid
approach combining AI techniques including fuzzy systems, neural networks, and
biologicallyinspiredmethodssuchasgeneticalgorithms.Here,however,weadopt
a more straightforward definition consistent with the original concept. Hence, soft
methodsareunderstoodasthoseuncertaintyformalismsnotpartofmainstreamsta-
tisticsandprobabilitytheorywhichhavetypicallybeendevelopedwithintheAIand
decisionanalysiscommunity.Thesearemathematicallysounduncertaintymodelling
methodologies which are complementary to conventional statistics and probability
theory.
Inadditiontoprobabilisticfactorssuchasmeasurementerrorandotherrandom
effects,themodellingprocessoftenrequiresustomakequalitativeandsubjectjudge-
ments that cannot easily be translated into precise probability values. Such judge-
ments give rise to a number of different types of uncertainty including; fuzziness
if they are based on linguistic information; epistemic uncertainty when their relia-
bility is in question; ignorance when they are insufficient to identify or restrict key
modelling parameters; imprecision when parameters and probability distributions
canonlybeestimatedwithincertainbounds.Statisticaltheoryhasnottraditionally
been concerned with modelling uncertainty arising in this manner but soft meth-
ods, a range of powerful techniques developed within AI, attempt to address those
problemswheretheencodingofsubjectiveinformationisunavoidable.Therefore,a
morerealisticmodellingprocessprovidingdecisionmakerswithanaccuratereflec-
tionofthetruecurrentstateofourknowledge(andignorance)requiresanintegrated
VI Preface
framework incorporating both probability theory, statistics and soft methods. This
fusionmotivatesinnovativeresearchattheinterfacebetweencomputerscience(AI),
mathematicsandsystemsengineering.
This edited volume is the proceedings of the 2006 International Workshop on
SoftMethodsinProbabilityandStatistics(SMPS2006)hostedbytheArtificialIn-
telligenceGroupattheUniversityofBristol,between5-7September2006.Thisis
thethirdofaseriesofbiennialmeetingsorganizedin2002bytheSystemsResearch
Institute from the Polish Academy of Sciences in Warsaw, and in 2004 by the De-
partmentofStatisticsandOperationalResearchattheUniversityofOviedoinSpain.
Theseconferencesprovideaforumfordiscussionandresearchintothefusionofsoft
methods with probability and statistics, with the ultimate goal of integrated uncer-
taintymodellingincomplexsystemsinvolvinghumanfactors.
The papers in the volume are organized into a number of key themes each ad-
dressing a different aspect of the integration of soft methods with probability and
statistics.Theseareidentifiedbothasbeinglongstandingfoundationalproblems,as
wellaspromisingavenuesofresearchwiththepotentialofprovidingsignificantad-
vancesinthemodellingandrepresentationofknowledgeanduncertainty.Alsovital
tothedevelopmentofanyacademicdisciplineistheidentificationandexplorationof
challengingnewapplicationareas.Itisonlythroughtheapplicationofexistingtools
andmethodologiestotheanalysisofuncertaintyinlarge-scalecomplexsystemsthat
fundamentalresearchissuescanbeidentifiedandnewcapabilitiesdeveloped.
Part I presents abstracts of four keynote presentations by Lotfi Zadeh, Gert de
Cooman,JimHallandVladikKreinovich.Prof.Zadeh’stalkprovidesdetailsonthe
latestdevelopmentsinhistheoryofgeneraliseduncertainty.Prof.deCooman’stalk
describesatheoryoflinguisticprobabilitiesbasedonimpreciseprobabilities.Prof.
Hall gives an overview of the application of soft methods in Earth Systems Engi-
neering. Prof. Kreinovich describes algorithms for statistical data processing under
interval uncertainty and investigates their complexity. Part II on Soft Methods in
StatisticsandRandomInformationSystemspresentscurrentresearchleadingtothe
developmentofnewstatisticaltoolsincorporatingfuzziness.PartIIIonProbability
ofImprecisely-ValuedRandomElementsWithApplicationsfocussesonaspectsof
probabilitytheoryincorporatingimprecision.PartIVonApplicationsandModelling
ofImpreciseOperatorsconsidershowlinguisticquantifierscanbeusedtodescribe
uncertainty. Part V on Imprecise Probability theory concerns the uncertainty mea-
surescorrespondingtoupperandlowerprobabilitiesandprevisions.PartVIonPos-
sibility,EvidenceandIntervalMethodscontainspapersonpossibilityandevidence
theoryaswellasintervalmethods.Finally,partVIIpresentsarangeofchallenging
applicationsrequiringtheintegrationofuncertainty,fuzzinessandimprecision.
Bristol, JonathanLawry
May2006
Contents
PartI KeynotePapers
GeneralizedTheoryofUncertainty(GTU)–PrincipalConcepts
andIdeas
LotfiA.Zadeh .................................................... 3
ReasoningwithVagueProbabilityAssessments
GertdeCooman .................................................. 5
SoftMethodsinEarthSystemsEngineering
JimW.Hall ...................................................... 7
StatisticalDataProcessingunderIntervalUncertainty:Algorithmsand
ComputationalComplexity
VladikKreinovich ................................................. 11
PartII SoftMethodsinStatisticsandRandomInformationSystems
OnTestingFuzzyIndependence
OlgierdHryniewicz................................................ 29
VarianceDecompositionofFuzzyRandomVariables
AndreasWünsche,WolfgangNäther................................... 37
FuzzyHistogramsandDensityEstimation
KevinLoquin,OlivierStrauss........................................ 45
GradedStochasticDominanceasaToolforRankingtheElements
ofaPoset
KarelDeLoof,HansDeMeyer,BernardDeBaets ....................... 53
VIII Contents
OnNeyman-PearsonLemmaforCrisp,RandomandFuzzyHypotheses
AdelMohammadpour,AliMohammad-Djafari .......................... 61
FuzzyProbabilityDistributionsInducedbyFuzzyRandomVectors
WolfgangTrutschnig............................................... 71
OntheIdentifiabilityofTSKAdditiveFuzzyRule-BasedModels
JoséLuisAznarteM.,JoséManuelBenítez ............................. 79
AnAsymptoticTestforSymmetryofRandomVariablesBased
onFuzzyTools
González-Rodríguez.G.,Colubi,A.,D’UrsoP.,Giordani,P. ............... 87
ExploratoryAnalysisofRandomVariablesBasedonFuzzifications
Colubi,A.,González-Rodríguez.G.,Lubiano,M.A.,Montenegro,M.......... 95
AMethodtoSimulateFuzzyRandomVariables
González-Rodríguez.G.,Colubi,A.,Gil,M.A.,Coppi,R. ..................103
Friedman’sTestforAmbiguousandMissingData
EdytaMrówka,PrzemysławGrzegorzewski.............................111
PartIII ProbabilityofImprecisely-ValuedRandomElements
withApplications
Measure-FreeMartingaleswithApplicationtoClassicalMartingales
S.F.Cullender,W.-C.Kuo,C.C.A.LabuschagneandB.A.Watson............121
ANoteonRandomUpperSemicontinuousFunctions
HungT.Nguyen,YukioOgura,SantiTasenaandHienTran ................129
OptionalSamplingTheoremandRepresentationofSet-ValuedAmart
ShoumeiLi,LiGuan...............................................137
OnaChoquetTheoremforRandomUpperSemicontinuousFunctions
YukioOgura .....................................................145
AGeneralLawofLargeNumbers,withApplications
PedroTerán,IlyaMolchanov ........................................153
PartIV ApplicationsandModellingofImpreciseOperators
FuzzyProductionPlanningModelforAutomobileSeatAssembling
J.Mula,R.Poler,J.P.Garcia-Sabater .................................163
Contents IX
OptimalSelectionofProportionalBoundingQuantifiersinLinguistic
DataSummarization
IngoGlöckner ....................................................173
ALinguisticQuantifierBasedAggregation foraHumanConsistent
SummarizationofTimeSeries
JanuszKacprzyk,AnnaWilbik,SławomirZadroz˙ny.......................183
EfficientEvaluationofSimilarityQuantifiedExpressions
intheTemporalDomain
F.Díaz-Hermida,P.Cariñena,A.Bugarín .............................191
PartV ImpreciseProbabilityTheory
ConditionalLowerPrevisionsforUnboundedRandomQuantities
MatthiasC.M.Troffaes ............................................201
ExtremeLowerProbabilities
ErikQuaeghebeur,GertdeCooman ..................................211
EquivalenceBetweenBayesianandCredalNetsonanUpdatingProblem
AlessandroAntonucci,MarcoZaffalon ................................223
VaryingParameterinClassificationBasedonImpreciseProbabilities
JoaquínAbellán,SerafínMoral,ManuelGómezandAndrésMasegosa.......231
ComparingProportionsDatawithFewSuccesses
F.P.A.Coolen,P.Coolen-Schrijner....................................241
AUnifiedViewofSomeRepresentationsofImpreciseProbabilities
S.Destercke,D.Dubois ............................................249
PartVI Possibility,EvidenceandIntervalMethods
EstimatinganUncertainProbabilityDensity
YakovBen-Haim ..................................................261
TheoryofEvidencewithImperfectInformation
J.Recasens ......................................................267
ConditionalIF-probability
KatarínaLendelová ...............................................275
OnTwoWaysfortheProbabilityTheoryonIF-sets
BeloslavRiecˇan...................................................285
X Contents
AStratificationofPossibilisticPartialExplanations
SaraBoutouhami,AichaMokhtari....................................291
FiniteDiscreteTimeMarkovChainswithIntervalProbabilities
DamjanŠkulj.....................................................299
EvidenceandCompositionality
WagnerBorges,JulioMichaelStern...................................307
HighLevelFuzzyLabelsforVagueConcepts
ZengchangQinandJonathanLawry ..................................317
PartVII IntegratedUncertaintyModellinginApplications
PossibilisticChannelsforDNAWordDesign
LucaBortolussi,AndreaSgarro ......................................327
TransformationofPossibilityFunctionsinaClimateModel
ofIntermediateComplexity
HermannHeld,ThomasSchneidervonDeimling ........................337
FuzzyLogicforStochasticModeling
ÖzerCiftciogluandI.SevilSariyildiz .................................347
ACUSUMControlChartforFuzzyQualityData
DabuxilatuWang..................................................357
AFuzzySynset-BasedHiddenMarkovModel
forAutomaticTextSegmentation
VietHa-Thuc,Quang-AnhNguyen-Van,TruHoangCaoandJonathanLawry..365
ApplyingFuzzyMeasuresforConsideringInteractionEffects
inFineRootDispersalModels
WolfgangNäther,KonradWälder ....................................373
Scoring Feature Subsets for Separation Power in Supervised Bayes
Classification
TatjanaPavlenko,HakanFridén .....................................383
IntervalRandomVariablesandTheirApplicationinQueueingSystems
withLong–TailedServiceTimes
BartłomiejJacekKubica,KrzysztofMalinowski .........................393
OnlineLearningforFuzzyBayesianPrediction
N.J.Randon,J.Lawry,I.D.Cluckie ...................................405
Index ............................................................. 413