《工程分析的模糊集和非精确概率方法》课程教学课件（讲稿）L1_2_Intro

UncertaintyQuantificationLeibniztilMichaelBeer102UniversitatInstituteforRiskand Reliability1o0:4Hannover

Engineering & Uncertainty Analysis Part I: Engineering Mathematics − Numerical Solution of Engineering Problems Uncertainty Quantification Michael Beer Institute for Risk and Reliability

ReliabilityandRiskAnalysisLeibniz102UniversitatGENERALSITUATIONto04HannoverEndeavornumericalmodeling-physicalphenomena,structure,andenvironmentprognosis-systembehavior,hazards,safety,risk,robustness,economic and social impact,...closeto realitynumericallyefficientDeterministicmethodsdeterministicdeterministicRealitycomputationalmodelsstructural parametersUncertaintyImprecisionVagueness?Fluctuations?Ambiguity ?ConsequencesVariability?Indeterminacy?2/29

2 / 29 Reliability and Risk Analysis GENERAL SITUATION » close to reality » numerically efficient Endeavor • numerical modeling − physical phenomena, structure, and environment prognosis − system behavior, hazards, safety, risk, robustness, economic and social impact, . Deterministic methods deterministic structural parameters Reality deterministic computational models • • Uncertainty ? Variability ? Consequences ? Vagueness ? Indeterminacy ? Fluctuations ? Imprecision ? Ambiguity ?

ReliabilityandRiskAnalysisLeibniz10:2UniversitatUncertaintiesto04HannoverGeneralstatement,There is nothing so wrong with the analysisas believing the answer!"RichardP.FeynmanClassification of error-related problemsblunders/grosserrors:stronglymisleadingresultsexample:linear analysiscomputationaltargetof a stronglyresultsnonlinearprobleminaccuracy/bias:systematicdeviationfromtheactualvaluecomputationalexample: negligence of a certain minoreffect8target-resultsinanapproximationsolutionimprecision:spread around theactual valueuncertaintyscattercomputationalexample:randomeffectsinthespecificationtargetofinputquantitiessuchasamaterialstrengthresultsStatistics and Probability Theory3/29

3 / 29 Reliability and Risk Analysis Uncertainties General statement „There is nothing so wrong with the analysis as believing the answer!“ Richard P. Feynman Classification of error-related problems target computational results example: linear analysis of a strongly nonlinear problem target computational results example: negligence of a certain minor effect in an approximation solution ● inaccuracy / bias: systematic deviation from the actual value ● blunders / gross errors: strongly misleading results target computational results example: random effects in the specification of input quantities such as a material strength ● uncertainty / imprecision: scatter / spread around the actual value Statistics and Probability Theory

ReliabilityandRiskAnalysisLeibniz10:2UniversitatUncertaintiesto04HannoverExample:compressivestrengthofconcreteetakeasamplefromafreshconcretemixture.form 20specimensof egualgeometrystorethespecimensforhardening over28 days under equal conditions.testthe cylinder compressive strengthf.samplestrengthsamplestrengthX=f,[N/mm2]X=f[N/mm2]elementelement11122.3426.20Which value to use21226.6126.40for calculations?31326.6820.6141424.5925.79Howriskyis itto51523.9625.30selecta certainvalue?61630.3726.0871724.1028.2781822.6823.20Can wefindacertain91920.3924.94valuecorrespondingto201025.7425.60an acceptablerisk?4/29

4 / 29 Reliability and Risk Analysis Can we find a certain value corresponding to an acceptable risk ? Example: compressive strength of concrete ● take a sample from a fresh concrete mixture ● form 20 specimens of equal geometry ● store the specimens for hardening over 28 days under equal conditions ● test the cylinder compressive strength fc sample strength element x=fc [N/mm2] 1 22.34 2 26.40 3 20.61 4 24.59 5 25.30 6 26.08 7 24.10 8 22.68 9 24.94 10 25.60 sample strength element x=fc [N/mm2] 11 26.20 12 26.61 13 26.68 14 25.79 15 23.96 16 30.37 17 28.27 18 23.20 19 20.39 20 25.74 Which value to use for calculations ? How risky is it to select a certain value ? Uncertainties

ReliabilityandRiskAnalysisiLeibniz102UniversitatUncertaintiesto04HannoverMission of statistics and probability theory.real datasummaryofthedescriptivetypical properties of the datastatistics(e.g.,magnitudeand scatter)drawconclusionsregardingtheinferentialunderlyinggeneration scheme of thedata,modeling and guantification of uncertaintystatistics(e.g.,certain"form"of the scatter)processing oftheuncertaintycalculation of uncertain quantitiesprobabilityforthe results as a basis to derive decisionstheory(e.g.,scatter of a deflection,failureprobabilities)Stochastics.decisions(e.g.,design,strengthening)fromGreekandLatin:NOTE:Statisticscannotmakedecisions!"theart of conjecturingIthelpsustomakedecisions.5/29

5 / 29 Reliability and Risk Analysis Mission of statistics and probability theory ● real data descriptive statistics inferential statistics probability theory ● summary of the typical properties of the data (e.g., magnitude and scatter) ● draw conclusions regarding the underlying generation scheme of the data, modeling and quantification of uncertainty (e.g., certain "form" of the scatter) ● processing of the uncertainty, calculation of uncertain quantities for the results as a basis to derive decisions (e.g., scatter of a deflection, failure probabilities) ● decisions (e.g., design, strengthening) NOTE: Statistics cannot make decisions ! It helps US to make decisions. Stochasticsmathematical statistics from Greek and Latin: "the art of conjecturing" Uncertainties

ReliabilityandRiskAnalvsisLeibniz102UniversitatProbability Theoryto04HannoverGeneral understanding of probability Pmeasure for the likelihood of occurrence of a specific event (event of interest)relative to the occurrence of all alternative events.basic requirements:definition of(i)aprobabilityspace[Q,,P]with》the samplespaceQ asa set comprised of all possible elementaryeventsO;eQ(spaceof elementary events),definition of therandomexperimenta"complete"system*(2)of subsetsofQcoveringALLeventswhichcanbeformulated withtheelementaryevents》a functionPwithcertainproperties whichassignsprobabilitiestotheelementsof (2)Probabilityis areal-valued set function.(i)anassociatedmeasurespace[x,G(x),P]with》a fundamentalsetX(universe)coveringALLpossibleobservationsresultingfrom the randomexperiment》a"complete"system*G(x)ofsubsetsofX》afunctionPassigningprobabilitiestoALLelementsof G(x)*(2)and G(x)must satisfy certain requirements from measure theory,theyrepresento-algebras.6/29

6 / 29 Reliability and Risk Analysis Probability Theory General understanding of probability P ● measure for the likelihood of occurrence of a specific event (event of interest) relative to the occurrence of all alternative events ● basic requirements: definition of (i) a probability space [Ω, ,P] with F » a fundamental set X (universe) covering ALL possible observations resulting from the random experiment » a "complete" system* of subsets of X » a function P assigning probabilities to ALL elements of F () F () » the sample space Ω as a set comprised of all possible elementary events ωi ∈ Ω (space of elementary events), definition of the random experiment » a "complete" system* of subsets of Ω covering ALL events, which can be formulated with the elementary events » a function P with certain properties which assigns probabilities to the elements of Probability is a real-valued set function. (ii) an associated measure space [X, ,P] with S(X) * and must satisfy certain requirements from measure theory, they represent σ-algebras. S(X) S(X) F () S(X)

ReliabilityandRiskAnalysisLeibniz10:2UniversitatProbability Theoryto04HannoverGeneral understandingof probabilityP (cont'd)random variable X:mappingQ→X.generation of eventsOEQ(measurablefunction)probability space05Q:samplespace04[2, 3(2) ,P]X:universe,fundamental set0-random elementarytypicallyX=R02events01event of interestEk:X≤XkmeasurespaceX4=X(04)(x|x≤xk) eG(X)[X,G(X),P]XKX1=x(01)BorelrealizationsX2=X(02)g-algebraX3=x(03)(images)XEXX5=X(05)example》elementary events o:testthecompressivestrengthof concretespecimens》realizationsx:measured valuesf,ofthecompressive strength》event of interestEk:f,takes on valuesleadingtostructural failureP(Ek)=Pf:failure probabilityFrequently,the distinction between Q and x is not considered explicitly.7/29

7 / 29 Reliability and Risk Analysis Probability Theory General understanding of probability P (cont’d) ● generation of events {xx x ≤ ∈ k } S(X) ● example » elementary events ωi : test the compressive strength of concrete specimens » realizations xi : measured values fc of the compressive strength » event of interest Ek: fc takes on values leading to structural failure P(Ek) = Pf : failure probability ω1 ω2 ω3 ω4 ω5 realizations (images) x ∈ X ω ∈ Ω random elementary events x1=x(ω1) x4=x(ω4) x2=x(ω2) x5=x(ω5) x3=x(ω3) xk event of interest Ek: X ≤ xk probability space [Ω, ,P] F () Borel σ-algebra measure space [X, ,P] S(X) Ω: sample space X: universe, fundamental set typically X =  (measurable function) random variable X: mapping Ω → X Frequently, the distinction between Ω and X is not considered explicitly

ReliabilityandRiskAnalysisLeibniz102UniversitatProbabilityTheoryto04HannoverFourviewsonprobabilitytheory.classicalprobability(1689/1713JacobBernoulli,1812Pierre-SimonLaplace)》complete systemoverview:ALLpossibleresultsofarandomexperimentareknownand independent of oneanotheruniversexis a finite countable setH(Ek)numberof favorablerealizations(iftherealizations》Pareequiprobable)H(X)number of possible realizations》characteristicfeature:a priori determination ofP(calculationofPwithoutorbeforeanexperiment)》example:ideal game of diceX = (1,2,3,4,5,6) = H(X)= 6H(E)30.5P(E)6H(x)E=(1,3,5) = H(E) =38/29

8 / 29 Reliability and Risk Analysis Four views on probability theory ● classical probability (1689/1713 Jacob Bernoulli, 1812 Pierre-Simon Laplace) » complete system overview: ALL possible results of a random experiment are known and independent of one another » universe X is a finite countable set ( ) ( ) number of favorable realizations H Ek P number of possible realizations H = = X » example: ideal game of dice ( ) ( ) ( ) k k H E 3 P E 0 5 H 6 = = = . E 135 HE 3 k k = ⇒= { , , } ( ) X X X = ⇒= {123456 H 6 , } ( ) » characteristic feature: a priori determination of P (calculation of P without or before an experiment) Probability Theory (if the realizations are equiprobable)

ReliabilityandRiskAnalvsisLeibniz102UniversitatProbabilityTheorytoo:4HannoverFourviewsonprobabilitytheory (cont'd)statistical(frequentist)probability(1689/1713JacobBernoulli1919RichardvonMises)》limitedsystemoverview:onlysomeofthepossibleresultsofarandomexperimentareknowntheyareindependentofoneanother(constantboundaryconditions)universeX isan infiniteand/or uncountable set》P=lim(h,(E)h,(Ek):relative frequency of occurrence of Ekremark:this limit does actually not exist (occasional appearance of outliers)but helpsto understandthe statistical probability》characteristic feature:a posteriori determination of P(calculationofPrequiresexperiments)》example:testofconcretecompressivestrengthX=22.34,26.40,20.61,24.59,25.30,26.08,24.10,22.68,24.94,25.60H, (Ek)4n=10, E=(×x≤24.5) = H, (E)= 4P(E)0.410n9/29

9 / 29 Reliability and Risk Analysis Four views on probability theory (cont’d) ● statistical (frequentist) probability (1689/1713 Jacob Bernoulli, 1919 Richard von Mises) » limited system overview: only some of the possible results of a random experiment are known, they are independent of one another (constant boundary conditions) » universe X is an infinite and/or uncountable set ( n k ( )) n P hE lim →∞ = » example: test of concrete compressive strength ( ) n k ( ) k H E 4 P E 0 4 n 10 { } ( ) ≈ == . E x x 24 5 H E 4 k n k =≤ ⇒ = . X = {22.34,26.40,20.61,24.59,25.30,26.08,24.10,22.68,24.94,25.60} » characteristic feature: a posteriori determination of P (calculation of P requires experiments) hn(Ek): relative frequency of occurrence of Ek n=10, Probability Theory remark: this limit does actually not exist (occasional appearance of outliers) but helps to understand the statistical probability

ReliabilityandRiskAnalysisLeibniz102UniversitatProbabilityTheoryto04HannoverFour views on probability theory (cont'd)subjectiveprobability(1937BrunodeFinetti,1954Leonard JimmieSavage)》subjectivesystemperception:subjectiveassessmentoftheprobabilityofoccurrenceofparticulareventsbased on expertknowledge and experiencegeneration scheme of the realizations not considered》characteristicfeature:validity restricted to themodel perceptionand boundary conditions of the expert》example:a doctorsassessmentofthesuccess rate inthetreatmentofpatients with a certain medication in a local clinicremark:Acombinationofstatistical and subjectiveprobability canbepursuedwithBAYESiantheory.First,asubjectiveassessmentismadefortheprobability of particular events.Then,the subjective model isupdated/adjustedbasedon observations.This resultsinasubjectiveprobabilisticmodel undertheconditionof theobservations.Withan increasing numberof observations thecharacteristicsof themodel becomemore and morestatistical.10/29

10 / 29 Reliability and Risk Analysis Four views on probability theory (cont’d) ● subjective probability (1937 Bruno de Finetti, 1954 Leonard Jimmie Savage) » subjective system perception: subjective assessment of the probability of occurrence of particular events based on expert knowledge and experience generation scheme of the realizations not considered » example: a doctors assessment of the success rate in the treatment of patients with a certain medication in a local clinic » characteristic feature: validity restricted to the model perception and boundary conditions of the expert Probability Theory remark: A combination of statistical and subjective probability can be pursued with BAYESian theory. First, a subjective assessment is made for the probability of particular events. Then, the subjective model is updated/adjusted based on observations. This results in a subjective probabilistic model under the condition of the observations. With an increasing number of observations the characteristics of the model become more and more statistical