《工程分析的模糊集和非精确概率方法》课程教学课件（讲稿）L9_10_Quantification of heterogeneous information

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,,,,,,,,,,,,,,,,,,,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 • , , , , , , , , , , , , , , , , , , , 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 ~

QuantificationEXAMPLE2Smallsamplesize,expertknowledgemeasurementof thecompressivestrengthofconcrete》20sampleelementsforx=f[N/mm2]29.8,23.1,27.6,20.228.3,26.8,31.5,35.3,35.2,26.3,30.7,29.2,25.2,25.7,34.6,34.2,24.8,19.2,22.828.9,expertknowledge》distribution type:normal distribution》choice ofestimator? sample mean for mx·samplevarianceforox2》constructionof confidenceintervals(typeandlevel): both-sided.levels:=0.50,0.75,0.90,0.99》assignment of membership degrees to confidencelevels:point estimation-μ=1.0-=0.50-μ=0.75,=0.75-μ=0.50=0.75-=0.25,=0.99-=0.00》subseguentmodification of the initial draftof the membershipfunctionsMichael Beer6/21

Michael Beer 6 / 21 EXAMPLE 2 Quantification Small sample size, expert knowledge measurement of the compressive strength of concrete » 20 sample elements for x = fc [N/mm²] • 28.3, 26.8, 31.5, 35.3, 35.2, 26.3, 29.8, 23.1, 27.6, 20.2 30.7, 29.2, 25.2, 25.7, 34.6, 34.2, 28.9, 24.8, 19.2, 22.8 expert knowledge » distribution type ▪ normal distribution » choice of estimator ▪ sample mean for mX ▪ sample variance for σX ² » construction of confidence intervals (type and level) ▪ both-sided ▪ levels: γ = 0.50, 0.75, 0.90, 0.99 » assignment of membership degrees to confidence levels ▪ point estimation − μ = 1.0 ▪ γ = 0.50 − μ = 0.75, γ = 0.75 − μ = 0.50 γ = 0.75 − μ = 0.25, γ = 0.99 − μ = 0.00 » subsequent modification of the initial draft of the membership functions •

QuantificationEXAMPLE2Smallsamplesize,expertknowledgestatisticalestimationconfidenceexpectedvaluestandard deviationlevelmx[N/mm2]0x[N/mm2]4.7527.97point estimationinterval0.50[27.24, 28.70][4.35,5.43]0.75[26.71,29.23][4.05,5.92]estimation0.90[26.13,29.81][3.77,6.52]0.99[24.93,31.01][3.34, 7.92]construction of membership functionsμ(mx)μ(ax)1.001.00ox0.75mx0.750.500.500.250.250.000.0024.9327.9731.01mx[N/mm2]24.9327.9731.01 x[N/mm2]Michael Beer7/21

Michael Beer 7 / 21 EXAMPLE 2 Quantification Small sample size, expert knowledge • statistical estimation confidence expected value standard deviation level γ mX [N/mm²] σX [N/mm²] point estimation − 27.97 4.75 interval 0.50 [27.24, 28.70] [4.35, 5.43] estimation 0.75 [26.71, 29.23] [4.05, 5.92] 0.90 [26.13, 29.81] [3.77, 6.52] 0.99 [24.93, 31.01] [3.34, 7.92] • construction of membership functions 0.50 0.25 0.00 1.00 0.75 μ(mX) mX 24.93 27.97 31.01 [N/mm²] 0.50 0.25 0.00 1.00 0.75 μ(σX) σX 24.93 27.97 31.01 [N/mm²] mX ~ σX ~

QuantificationEXAMPLE3Inconsistentenvironmental conditions,....measurementofthecompressive strengthofconcrete》620sampleelementsforx=f.[N/mm2]》samplegeneration undervarying environmental conditions.differentmanufacturers:differentaggregates/additives(different suppliers):different hardening conditions (temperature,humidity):different motivation of personnelexpertknowledge》classify sample elements with respecttotheir attributes(conditions)》determinegroupsofsampleelementswithsameattributesquantification options》parametricquantification:distributionassumptionfromexpertknowledge》non-parametricquantification:useof empirical distributionfunctionsMichaelBeer8/21

Michael Beer 8 / 21 EXAMPLE 3 Quantification Inconsistent environmental conditions, . measurement of the compressive strength of concrete » 620 sample elements for x = fc [N/mm²] » sample generation under varying environmental conditions ▪ different manufacturers ▪ different aggregates / additives (different suppliers) ▪ different hardening conditions (temperature, humidity) ▪ different motivation of personnel • expert knowledge » classify sample elements with respect to their attributes (conditions) » determine groups of sample elements with same attributes • quantification options » parametric quantification ▪ distribution assumption from expert knowledge » non-parametric quantification ▪ use of empirical distribution functions •

QuantificationEXAMPLE3Inconsistentenvironmentalconditions,...(cont'd)parametricquantification》distributiontypeforeachgroup:normal distribution》choice of estimator and point/interval estimationfor each group:pointestimation:samplemeanfor mxsample variance for x2samplefgroupgroupsampleOxmxmx[N/mm2] [N/mm2]number size[N/mm2] [N/mm2]number size1545.3727.3555.026.428484726.64.930.14.639424.2645.929.228.3453381031.43.827.93.8575445.6116.328.329.661252483.229.427.84.7MichaelBeer9/21

Michael Beer 9 / 21 EXAMPLE 3 Quantification Inconsistent environmental conditions, . (cont’d) parametric quantification » distribution type for each group ▪ normal distribution » choice of estimator and point / interval estimation for each group ▪ point estimation ▪ sample mean for mX ▪ sample variance for σX ² • group sample mX σX number size [N/mm²] [N/mm²] 1 54 27.3 5.3 2 48 26.6 4.9 3 42 29.2 4.2 4 38 31.4 3.8 5 44 28.3 5.6 6 48 29.4 3.2 group sample mX σX number size [N/mm²] [N/mm²] 7 55 26.4 5.0 8 47 30.1 4.6 9 64 28.3 5.9 10 53 27.9 3.8 11 75 29.6 6.3 12 52 27.8 4.7

QuantificationEXAMPLE3Inconsistentenvironmental conditions,...(cont'd)parametric guantification》histogram-likerepresentationforparametersforallgroups》constructionof membership functionsfortheparameters》assessment of interaction4n4n44mx [N/mm?]3332.002211OXmxμ(mx)μ(gx)1.001.00Ox0.750.75mx0.500.5025.430.250.250x[N/mm2]0.000.00-25.43 27.91 32.002.796.962.794.886.960x [N/mm2]mx [N/mm2]MichaelBeer10 /21

Michael Beer 10 / 21 EXAMPLE 3 Quantification Inconsistent environmental conditions, . (cont’d) parametric quantification » histogram-like representation for parameters for all groups » construction of membership functions for the parameters • 0.50 0.25 0.00 1.00 0.75 μ(mX) mX [N/mm²] 25.43 27.91 32.00 mX 0.50 0.25 0.00 1.00 0.75 μ(σX) σX [N/mm²] 2.79 4.88 6.96 ~ σX ~ mX σX n 4 3 2 1 n 4 3 2 1 σX [N/mm²] 2.79 6.96 mX [N/mm²] 25.43 32.00 » assessment of interaction