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 ~