Quantificationofheterogeneous information
Michael Beer 1 / 21 Quantification of heterogeneous information
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 variablesMichaelBeer2/21
Michael Beer 2 / 21 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
QuantificationFUZZYPROBABILISTICMODELINGGeneralconcept.exploitationof statistical informationrealistic considerationof imprecisionno mixingbetweenstatisticalinformationandimprecisionTypical cases in engineeringimprecisesampleelementsstatistics withfuzzyvariablesutilization of fuzzy arithmetic in statistical estimations and testssmallsamplesize,expertknowledge》weakstatisticalinformationfromestimationsandtestsutilizationofstatisticalimprecisionfor the specificationof fuzzy parameters and fuzzy distribution typesinconsistentenvironmentalconditions,expertknowledge,conflictinginformation》criticalconditionsfor statisticalestimationsandtestsseparation of fuzzinessand randomness by constructingconsistentgroups (discretizedfuzziness)MichaelBeer3/21
Michael Beer 3 / 21 FUZZY PROBABILISTIC MODELING Quantification General concept Typical cases in engineering • exploitation of statistical information • realistic consideration of imprecision • no mixing between statistical information and imprecision imprecise sample elements » statistics with fuzzy variables • inconsistent environmental conditions, expert knowledge, conflicting information » critical conditions for statistical estimations and tests • utilization of fuzzy arithmetic in statistical estimations and tests separation of fuzziness and randomness by constructing consistent groups (discretized fuzziness) small sample size, expert knowledge » weak statistical information from estimations and tests • utilization of statistical imprecision for the specification of fuzzy parameters and fuzzy distribution types
QuantificationEXAMPLE1Imprecisesampleelements.measurementof the compressive strength of concrete》20sampleelementsforx=f.[N/mm2]:imprecision dueto individual careand readings inthetests:measurementsmodeled withfuzzytriangularnumbers<26.3, 28.3, 30.3>,<29.5, 31.5,33.5>,<24.8,26.8,28.8>,<33.3,35.3,37.3>,<33.2,35.2, 37.2>,<24.3, 26.3, 28.3>,<27.8,29.8,31.8>,<21.1,23.1,25.1>,<25.6, 27.6, 29.6>,<18.2, 20.2, 22.2>,<28.7, 30.7, 32.7>,<27.2, 29.2, 31.2>,<23.2, 25.2, 27.2>,<23.7, 25.7, 27.7>,<32.6, 34.6, 36.6>,<32.2, 34.2, 36.2>,<26.9,28.9,30.9>,<22.8, 24.8, 26.8>,<17.2, 19.2, 21.2>,<20.8,22.8,24.8>statisticalevaluation》distributiontype:normaldistribution(expertknowledge)》applicationofestimatorstofuzzysampleelementsinteractionC(闵）-n.ni-ibetweendependabilityproblemX and SxX= (xi...,x,)ex= (x..,x) = (x,sx)e (x,sx)Michael Beer4/21
Michael Beer 4 / 21 EXAMPLE 1 Quantification _ _ _ Imprecise sample elements measurement of the compressive strength of concrete » 20 sample elements for x = fc [N/mm²] ▪ imprecision due to individual care and readings in the tests ▪ measurements modeled with fuzzy triangular numbers • <26.3, 28.3, 30.3>, <24.8, 26.8, 28.8>, <29.5, 31.5, 33.5>, <33.3, 35.3, 37.3>, <33.2, 35.2, 37.2>, <24.3, 26.3, 28.3>, <27.8, 29.8, 31.8>, <21.1, 23.1, 25.1>, <25.6, 27.6, 29.6>, <18.2, 20.2, 22.2>, <28.7, 30.7, 32.7>, <27.2, 29.2, 31.2>, <23.2, 25.2, 27.2>, <23.7, 25.7, 27.7>, <32.6, 34.6, 36.6>, <32.2, 34.2, 36.2>, <26.9, 28.9, 30.9>, <22.8, 24.8, 26.8>, <17.2, 19.2, 21.2>, <20.8, 22.8, 24.8> statistical evaluation » distribution type: normal distribution (expert knowledge) » application of estimators to fuzzy sample elements • ~ = = ∑ n i i 1 1 x x n ( ) ( ) = = = ∑ ∑ − − 2 n n 2 2 X ii i 1 i 1 1 1 s xx n1 n x x x x x x xs xs = ∈= ⇒ ∈ ( 1n 1n ,., ) ( ,., ) ( , , X X ) ( ) ~ ~ interaction between x and sX ~ ~ ! ~ ~ ~ ~ ~ ~ ~ dependability problem
OuantificationEXAMPLE1Imprecise sample elements (cont'd)numerical evaluation of statistical estimationsSx extremeparametervaluesF(×)全Sα=0r1.0H=1μ=O,interactionSα=1F(x)betweenxandsxnegligenceSα=01文of interaction0.0Xα=01X=1XXα=0rfuzzyparametersfortheimprecisedistributionfunctionsμ(mx)μ(x)1.001.00mx0.750.750.500.500.250.250.000.0025.97227.97 29.973.223.954.755.636.540x [N/mm2]mx [N/mm2]5/21MichaelBeer
Michael Beer 5 / 21 EXAMPLE 1 Quantification extreme parameter values Imprecise sample elements (cont’d) • numerical evaluation of statistical estimations sα=0 l sα=0 r xα=0 r _ xα=0 l _ sX x _ x F(x) 1.0 0.0 xα=1 _ sα=1 F(x) ~ • fuzzy parameters for the imprecise distribution functions μ = 1 negligence of interaction μ = 0, interaction between x and sX _ 0.50 0.25 0.00 1.00 0.75 μ(mX) mX [N/mm²] 25.97 27.97 29.97 mX ~ 0.50 0.25 0.00 1.00 0.75 μ(σX) σX [N/mm²] 3.22 3.95 4.75 5.63 6.54 σX ~