Engineering analyseswithimpreciseprobabilities,applications
Michael Beer 1 / 23 Engineering analyses with imprecise probabilities, applications
ImpreciseProbabilitiesFUZZYPROBABILITIESINRELIABILITYANALYSISFuzzy parametersFailure probability.structural parametersprobabilisticmodelparametersacceptablePfμ(x)μ(P.)acceptable11PparameterintervalXSensitivityof Pw.r.t.x(probabilisticmodelchoice)mappinga;aαiX→PX00coarse specificationsofdesignparameters & probabilisticmodelsattentionto/excludemodeloptionsleadingtolargeimprecisionofPacceptableimprecisionof parameters & probabilisticmodelsindicationstocollectadditionalinformationdefinitionofqualityrequirementsμ)notimportant,butrobustdesignanalysis with variousintensities of imprecision2/23MichaelBeer
Michael Beer 2 / 23 FUZZY PROBABILITIES IN RELIABILITY ANALYSIS • structural parameters Fuzzy parameters Failure probability X ~ acceptable Pf Pf 0 αi µ(x) 1 x Pf ~ • probabilistic model parameters acceptable parameter interval Sensitivity of Pf w.r.t. x (probabilistic model choice) 0 αi µ(Pf) 1 mapping X ~ Pf → ~ coarse specifications of design parameters & probabilistic models attention to / exclude model options leading to large imprecision of Pf acceptable imprecision of parameters & probabilistic models indications to collect additional information definition of quality requirements robust design µ(.) not important, but analysis with various intensities of imprecision Imprecise Probabilities
ImpreciseProbabilitiesIMPLEMENTINGEPISTEMICUINCERTAINTYSubjectiveprobabilitiesBayesian updateprior distribution[f(x[0)] 9。 (0)[f (x, 0) . g (0) deposterior distribution》approachresultfrom"insidetheepistemicuncertaintyImpreciseprobabilities个F(x).set-theoreticalmodelsforindeterminacy/imprecision(imprecise data,vague conditions orcopulas etc.)F(x)e.g.,set-valuedparametersF(×) = ([F(x),F(×)) Vx)in probabilistic modelsX》approachresultfrom"outsidetheepistemicuncertaintychoicedepending onavailableinformationand purpose of the analysisapproaches not competing but complementaryandcanbecombined(e.g.setof priors,updatewithimprecisedata)Michael Beer3/23
Michael Beer 3 / 23 IMPLEMENTING EPISTEMIC UNCERTAINTY ( ) ( ) ( ) ( ) ( ) Θ = Θ Θ = ∏ θ⋅ θ θ = ∫ ∏ θ⋅ θθ 1 n n i i 1 X , ,X 1 n n i i 1 fx g f x , ,x fx g d posterior distribution prior distribution » approach result from “inside the epistemic uncertainty” Subjective probabilities • Bayesian update Imprecise probabilities • set-theoretical models for indeterminacy / imprecision choice depending on available information and purpose of the analysis » approach result from “outside the epistemic uncertainty” F x F x ,F x x ( ) = { li ui i ( ) ( ) ∀ } e.g., set-valued parameters in probabilistic models F(x) F(x) ~ x (imprecise data, vague conditions or copulas etc.) approaches not competing but complementary and can be combined (e.g. set of priors, update with imprecise data) Imprecise Probabilities
ConceptualcomparisonRELIABILITYANALYSIS-EXAMPLE1Fixedjacketplatformimprecisioninthemodelsfor》wave,dragandiceloadswindload》corrosion》jointsoftubularmembers》foundation》possibledamage4/23MichaelBeer
Michael Beer 4 / 23 RELIABILITY ANALYSIS − EXAMPLE 1 Fixed jacket platform imprecision in the models for » wave, drag and ice loads » wind load » corrosion » joints of tubular members » foundation » possible damage • Conceptual comparison
Conceptual comparisonRELIABILITYANALYSIS-EXAMPLE1Probabilisticmodel(AfterR.E.Melchers)c(t,E) =b(t,E)·f(t,E)+(t,E)(time-dependentcorrosiondepth,uniform)》c(t,E)-corrosiondepth》f(t,E)-meanvaluefunction》b(t,E)-biasfunction》e(t,E)-uncertainty function(zero mean Gaussian white noise)>E-collection of environmental and material parameterscorrosionaModeling ?rstransition2Lvg1.0b(t,E)f(t,E)180APCsC0.ta2dimensionless exposure period t/t,timephase23MichaelBeer5/23
Michael Beer 5 / 23 Probabilistic model (After R.E. Melchers) » c(t,E) − corrosion depth » f(t,E) − mean value function » b(t,E) − bias function » ε(t,E) − uncertainty function (zero mean Gaussian white noise) » E − collection of environmental and material parameters • ctE btE f tE tE ( , , , ) = ⋅ +ε ( ) ( ) ( ) ca cs ro ta ra rs 1 2 3 4 AP corrosion phase time transition (t,E) f(t,E) b(t,E) Modeling ? (time-dependent corrosion depth, uniform) RELIABILITY ANALYSIS − EXAMPLE 1 Conceptual comparison