What is what:intervals,fuzzysets,random variables,impreciseprobabilitiesepistemicuncertainty
Michael Beer 1 / 15 What is what: intervals, fuzzy sets, random variables, imprecise probabilities epistemic uncertainty
VagueandimpreciseinformationSOMEOUESTIONSHow precisewill it be?Anyconcernordoubt?modeling,quantification,processing,evaluation,interpretation?2/15MichaelBeer
Michael Beer 2 / 15 How precise will it be ? modeling, quantification, processing, evaluation, interpretation ? Any concern or doubt ? SOME QUESTIONS Vague and imprecise information
VagueandimpreciseinformationCHALLENGEstatistical analysis ofIs it safe ?impreciseand raredataF(x)set of plausiblemodelsreliabilityanalysis2p'一一Is the[Pr,I, Pr,]imprecisionreliabilityanalysisreflectedinPstill reliable?EffectsonP?SensitivityofP,toimprecision?3/15MichaelBeer
Michael Beer 3 / 15 CHALLENGE of imprecise and rare data Is the reliability analysis still reliable ? statistical analysis Is it safe ? Effects on Pf ? reliability analysis F(x) model Pf ~ ~ set of plausible s [Pf,l, Pf,r] imprecision reflected in Pf Sensitivity of Pf to imprecision ? epistemic uncertainty Vague and imprecise information
Vagueand impreciseinformationCHALLENGINGCASESSummaryofexamplesimprecisemeasurementsexpertassessment/experiencemeasurement/observationlinguisticassessmentsunderdubious conditionshighplausiblerangemediumXlowimprecisesampleelementsX.smallsamples.incompleteprobabilisticobservationswhichcannotbeelicitationexerciseseparated clearlyconditional probabilities observedvague/dubiousprobabilisticinformationunder unclear conditions.only marginals of a joint distributionchanging environmental conditionsavailablewithoutcopulamixtureof informationfrom different sourcesand with different characteristicsclassificationand mathematical modeling?Michael Beer4/15
Michael Beer 4 / 15 CHALLENGING CASES Summary of examples mixture of information from different sources and with different characteristics classification and mathematical modeling ? • imprecise sample elements • small samples • changing environmental conditions incomplete probabilistic elicitation exercise • vague / dubious probabilistic information • • imprecise measurements x plausible range measurement / observation under dubious conditions • observations which cannot be separated clearly • conditional probabilities observed under unclear conditions • only marginals of a joint distribution available without copula • low medium high • linguistic assessments x • expert assessment / experience Vague and imprecise information
VagueandimpreciseinformationCLASSIFICATIONANDMODELINGAccording to sourcesaleatory uncertainty·epistemic uncertainty》irreducible uncertainty》reducible uncertainty》propertyofthesystem》propertyoftheanalyst》fluctuations/variability》lackofknowledgeorperceptioncollection of all problematic cases,stochasticcharacteristicsinconsistencyofinformationtraditionalno specific modelprobabilistic modelsAccording to information contentuncertaintyimprecision》probabilisticinformation》non-probabilisticcharacteristicstraditionaland subjectiveset-theoreticalmodelsprobabilistic modelsInviewof thepurpose of theanalysisaveraged results, value ranges, worst case, etc.?MichaelBeer5/15
Michael Beer 5 / 15 CLASSIFICATION AND MODELING » reducible uncertainty » property of the analyst » lack of knowledge or perception According to sources • aleatory uncertainty » irreducible uncertainty » property of the system » fluctuations / variability stochastic characteristics • epistemic uncertainty collection of all problematic cases, inconsistency of information » non-probabilistic characteristics According to information content • uncertainty » probabilistic information traditional and subjective probabilistic models • imprecision set-theoretical models traditional no specific model probabilistic models In view of the purpose of the analysis • averaged results, value ranges, worst case, etc. ? Vague and imprecise information
Imprecision and uncertaintyEPISTEMICUNCERTAINTIESProbabilisticmodeling-uncertainty.subjectiveprobabilities,beliefSet-theoretical modeling -imprecision·Intervals,fuzzysets,etc.》thevariablemaytakeonanyvaluebetweenboundsbutthere is no basis to assume probabilities to theoptions》thevariablehas aparticularrealvalue,but that valueis unknown except that it is between bounds》thevariablemay takea singlevalue ormultiplevalues in some range,but it is not known whichis thecase》thevariableis set-valuedExamples:》poor data or linguisticexpressions indicating a value range or bounds》designparameters in the early design stage,specified onlyroughlyand underlielaterchangesasthedesignmatures》physicalinequalitiesMichael Beer6/15
Michael Beer 6 / 15 • subjective probabilities, belief Probabilistic modeling − uncertainty EPISTEMIC UNCERTAINTIES Imprecision and uncertainty • Intervals, fuzzy sets, etc. Set-theoretical modeling − imprecision » the variable may take on any value between bounds, but there is no basis to assume probabilities to the options » the variable has a particular real value, but that value is unknown except that it is between bounds » the variable may take a single value or multiple values in some range, but it is not known which is the case » the variable is set-valued Examples: » poor data or linguistic expressions indicating a value range or bounds » design parameters in the early design stage, specified only roughly and underlie later changes as the design matures » physical inequalities
Non-probabilisticModelsINTERVALSMathematical model.X=[X,x,]=(×eX=R|X,≤x≤X,(classical setX)common suggestion:+XrXassignmentofuniform distributionsInformation contentpossible value rangebetweencrisp boundsno additional information (on fluctuations etc.)Numerical processingengineering analysisoption I:enclosure schemes》 narrowactual result interval from outside》highnumericalefficiency》proceduresrestrictedtospecificproblemsoptionII:globaloptimizationmeaning? explicitsearchforresultintervalboundseffort ?》reasonableorhighnumerical effortdoes it serve the purpose》proceduresapplicableof the analysis?to a largevariety of problemsMichaelBeer7/15
Michael Beer 7 / 15 INTERVALS x (classical set X) Mathematical model X , = = ∈ = ≤≤ x x x x x x l r { X l r } common suggestion: assignment of uniform distributions • xl xr Information content • possible value range between crisp bounds • no additional information (on fluctuations etc.) Numerical processing • option I: enclosure schemes » narrow actual result interval from outside » high numerical efficiency » procedures restricted to specific problems meaning ? effort ? does it serve the purpose of the analysis ? • option II: global optimization » explicit search for result interval bounds » reasonable or high numerical effort » procedures applicable to a large variety of problems engineering analysis Non-probabilistic Models
Non-probabilisticModelsFUZZYSETSModelingofimprecision,numericalrepresentation. α-level set X, =( xeX I μ(x)≥α.aggregation of information. α-discretization X=((X.μ(X.)》max-minoperator.minoperator=t-normμ(x)maxoperator=t-co-norm1.0μ()notimportantbut analysis withalkvariousintensitiesof imprecision0.0setofnestedintervalsofvarioussizeutilizationof interval analysisforα-level sets,calculationofresult-boundsforeachα-level8/15MichaelBeer
Michael Beer 8 / 15 FUZZY SETS Xαk µ(x) 1.0 0.0 αk xαk l xαk r x » max-min operator ▪ min operator = t-norm ▪ max operator = t-co-norm Modeling of imprecision, numerical representation α-level set Xα = ∈ µ ≥α { x (x) X } ~ α-discretization XX X = µ {( α α , ( ))} • set of nested intervals of various size • utilization of interval analysis for α-level sets, calculation of result-bounds for each α-level • aggregation of information • µ(.) not important but analysis with various intensities of imprecision Non-probabilistic Models
Imprecision and uncertaintyCLASSIFICATIONANDMODELINGSimultaneousappearance of uncertainty and imprecision.informationproblematicforapureprobabilisticmodelingseparatetreatment of uncertaintyand imprecision in one modelgeneralized modelscombining probabilitytheoryand settheoryconceptsofimpreciseprobabilities》intervalprobabilities》setsofprobabilities/p-boxapproach》randomsets》fuzzyrandomvariables/fuzzyprobabilities》evidencetheory/Dempster-Shafertheorycommonbasicfeature:setof plausibleprobabilistic modelsoverarangeof imprecision(setof modelswhichagreewiththeobservations)boundsonprobabilitiesfor events of interestMichael Beer9/15
Michael Beer 9 / 15 CLASSIFICATION AND MODELING Simultaneous appearance of uncertainty and imprecision separate treatment of uncertainty and imprecision in one model • generalized models combining probability theory and set theory information problematic for a pure probabilistic modeling common basic feature: set of plausible probabilistic models over a range of imprecision » interval probabilities » sets of probabilities / p-box approach » random sets » fuzzy random variables / fuzzy probabilities » evidence theory / Dempster-Shafer theory concepts of imprecise probabilities • • bounds on probabilities for events of interest (set of models which agree with the observations) Imprecision and uncertainty
Imprecisionand uncertaintyIMPRECISEPROBABILITIESConceptualCategorization.coarselyobservedevents》coarseobservationsatphenomenologicallevelwithcomplexbackgrounde.g."severeshearcracksina wall"》probabilities assignedto entire sets,which representtheobservationsboundsforasetof distributionfunctionsevidencetheorydistributionbounds》onlyboundsavailableforparameters,typeorcurveofadistributione.g.as a result of conflicting informationfrom statistics》intervalsasdistributionparameters,typesandcdfdescriptionsset of distribution functionsinterval probabilitiesimprecisesampleelements》outcomesfrom random experimentappearblurrede.g.as linguistic variables》sampleelementsdescribedwithfuzzysetsfuzzyset ofdistributionfunctionsfuzzyrandom variablesMichaelBeer10/15
Michael Beer 10 / 15 coarsely observed events » coarse observations at phenomenological level with complex background e.g. “severe shear cracks in a wall” » probabilities assigned to entire sets, which represent the observations • Conceptual Categorization bounds for a set of distribution functions evidence theory IMPRECISE PROBABILITIES distribution bounds » only bounds available for parameters, type or curve of a distribution e.g. as a result of conflicting information from statistics » intervals as distribution parameters, types and cdf descriptions • set of distribution functions interval probabilities imprecise sample elements » outcomes from random experiment appear blurred e.g. as linguistic variables » sample elements described with fuzzy sets • fuzzy set of distribution functions fuzzy random variables Imprecision and uncertainty