Comparativestudies withrespecttoprobabilitytheory
Michael Beer 1 / 14 Comparative studies with respect to probability theory
VagueInformation:ComparativeStudyPROBABILITIESORINTERVALSOR???Simplesettlementproblem(Ang&Tang1984)StructureNmodel correction factorCccompressionindex(clay)Sande.soil initial void ratioHclaylayerthicknessP。initial stress in clayNormallyHConsolidatedApstressincreasebystructureClayP, + ApC.H文8=NlogP°1+e.RockStatistical data?Experience?Distribution type?Data quality?Physical justification?Samplesize?Bounds?Assumptions?2/14MichaelBeer
Michael Beer 2 / 14 PROBABILITIES OR INTERVALS OR ??? Vague Information: Comparative Study Simple settlement problem (Ang & Tang 1984) Sand Normally Consolidated Clay Structure Rock H N model correction factor Cc compression index (clay) eo soil initial void ratio H clay layer thickness Po initial stress in clay ∆p stress increase by structure + ∆ δ = + c o 0 o CH P p N 1e P log Statistical data ? Distribution type ? Sample size ? Experience ? Assumptions ? Data quality ? Bounds ? Physical justification ?
Vague Information:Comparative StudyPROBABILITIESORINTERVALSOR???Parameter uncertainty and imprecisionexample(Ang&Tang1984)probabilisticanalysis》performancefunctionExpectedC.O.V.G() = 8. -88.=6.35cmvalueμ1N0.1》MonteCarlo simulation,N=106Cc0.40.251.20.15eoμG= 3.7954270.05H(cm)0=0.8220.05P。(kPa)17814*00.2g(.)△p(kPa)*modifiedP, = P(G() ≤0) = 8.94 ·10-4normal distributionsComparative studyintervalsforallparametersintervalanalysisintervalsforsomeparametersimpreciseprobabilitiesMichael Beer3/14
Michael Beer 3 / 14 Parameter uncertainty and imprecision Expected value µ c.o.v. N 1 0.1 Cc 0.4 0.25 eo 1.2 0.15 H(cm) 427 0.05 Po(kPa) 178 0.05 ∆p(kPa) 14* 0.2 Comparative study normal distributions *modified • example (Ang & Tang 1984) • probabilistic analysis 0 g(.) ( ) =δ −δ G c . δ = c 6 35 cm . » performance function » Monte Carlo simulation, N = 106 ( ( ) ) − = ≤= ⋅ 4 P P G 0 8 94 10 f . . µ = σ = G 2 G 3 795 0 822 . . • intervals for all parameters interval analysis • intervals for some parameters imprecise probabilities PROBABILITIES OR INTERVALS OR ??? Vague Information: Comparative Study
Vague Information:Comparative StudyPROBABILITIESORINTERVALSOR???Parameter intervals.[Xi,Xu]=[ux-30x-μx +30xintervalforperformance function:[g,g.].no probabilistic modelPropagation ofparameterintervals·intuitive suggestion: use of uniform distributions over[xu,xuand Mcs》histogramartificial》only extremevalues usefulN=106-6.236.22][g,gu Mcs二》lowapproximationqualityof boundsexact solution:[g,g,]=[-9.66,6.24]-6.23*6.22* g(.)*average of 500 analyses》high numericalcostinterval analysis methods:global optimization9.66,6.24》highapproximationquality:[g,,guo》numericallyefficient:N=165MichaelBeer4/14
Michael Beer 4 / 14 Parameter intervals =µ −σ µ +σ i ii i xx 3 3 il iu X X X X , , Propagation of parameter intervals • no probabilistic model • interval for performance function: g gl u , • intuitive suggestion: use of uniform distributions over and MCS x x il iu , N = 106 −6.23* 6.22* g(.) * average of 500 analyses • interval analysis methods: global optimization » histogram artificial » only extreme values useful » low approximation quality of bounds, exact solution: = − l u MCS g g 6 23 6 22 , . ,. = − g g 9 66 6 24 l u , . ,. » high numerical cost » high approximation quality: = − l u GO g g 9 66 6 24 , . ,. » numerically efficient: N = 165 PROBABILITIES OR INTERVALS OR ??? Vague Information: Comparative Study
VagueInformation:ComparativeStudyPROBABILITIESORINTERVALSOR???Interval analysisProbabilisticanalysis·[g,9,]=[-9.66,6.24]-[-9.66,0] (0,6.24]· P, = P(G()≤0) = 8.94 10-4Given thatinput information isquitevaguefailuremay occur in afailure mayoccurmoderate number of casescomparablemagnitude of exceedancesignificant exceedance of.g =0of g=0rather small,strongmayoccurexceedance quite unlikelyconclusionsfocussedon differentissuesGeneralrelationshipbounding property P(Y [yu,yu) ≥P(Xe[x,xu) for general mapping X-Y》knowndistributionofX》unknowndistributionofXconclusions fromprobabilistic results maybetoointervalanalysismostlyoptimistic,worstcase(whichistooconservativeemphasizedinintervalanalysis)maybe likelyMichaelBeer5/14
Michael Beer 5 / 14 Probabilistic analysis • failure may occur in a moderate number of cases Interval analysis • =− =− ∪ ( g g 9 66 6 24 9 66 0 0 6 24 l u , . ,. . , ,. failure may occur magnitude of exceedance of g = 0 rather small, strong exceedance quite unlikely significant exceedance of g = 0 may occur comparable conclusions focussed on different issues Given that input information is quite vague » known distribution of X General relationship • bounding property for general mapping X PY y y PX x x ( ∈ ≥∈ l u , , ) ( l u ) →Y conclusions from interval analysis mostly too conservative ( ( ) ) − = ≤= ⋅ 4 P P G 0 8 94 10 f . . » unknown distribution of X probabilistic results may be too optimistic, worst case (which is emphasized in interval analysis) maybe likely PROBABILITIES OR INTERVALS OR ??? Vague Information: Comparative Study
Vague Information:Comparative StudyPROBABILITIESORINTERVALSOR???ProbabilisticanalysisIntervalanalysisnormaldistributionsforall X[xuxu]=[μx -30xμx +30x for all X(initialexampledescription)P(X, [xu,u],i= 1,.,6) = 0.98391→[9,9]=-9.66,6.24histogram for G(.)P(G(-) =[g),g.) ≥0.98391P(G() =[g,9,) =0.999931》estimation of intervals [gip,gup]largedifferenceduetolow0.98391With P(G() [gip,gup)probabilitydensityforsmallg(),=but critical forfailurefromhistogram- both-sided[1.15,5.53[gip, gup Jentral.left-sided w.r.t.lowerbound g0-1010 g(.)-9.66,5.39[g, 9up eft =moderate differenceintervalresultisconservativedue to high probabilitydifferences controlled by distributionof G(.)densityatupperboundsMichael Beer6/14
Michael Beer 6 / 14 Probabilistic analysis • Interval analysis • P X x x i 1 6 0 98391 ( i il iu ∈ == , , ,., . ) normal distributions for all Xi (initial example description) =µ −σ µ +σ i ii i xx 3 3 il iu X X X X , , for all Xi P G . g g 0 98391 ( ( ) ∈ ≥ l u , . ) histogram for G(.) P G . g g 0 99993 ( ( ) ∈ = l u , . ) » estimation of intervals [glP,guP] with from histogram P G . g g 0 98391 ( ( ) ∈ = lP uP , . ) differences controlled by distribution of G(.) interval result is conservative −10 0 10 g(.) = − g g 9 66 6 24 l u , . ,. ▪ both-sided = lP uP central g g 1 15 5 53 , . ,. ▪ left-sided w.r.t. lower bound gl = − l uP left g g 9 66 5 39 , . ,. large difference due to low probability density for small g(.), but critical for failure moderate difference due to high probability density at upper bounds PROBABILITIES OR INTERVALS OR ??? Vague Information: Comparative Study
Vague Information:Comparative StudyPROBABILITIESORINTERVALSOR???Interval analysisProbabilistic approximation[g,9]-[-9.66,6.24].usingestimatedmomentsofG(.)Hg=3.795,=0.822P(G()=[g,g.)≥0.98391》Chebyshev'sinequalitywith0.98391P(G() [9ip, 9up he[9ip, 9up heby[-3.35,10.940-1010 g(.)interval analysisintervalresultshiftedtowardsfailure domain,even moreChebyshevconservativethanChebyshevrecall histogram for G(.)interval result reflectstendencyfor uniform Xof thedistribution of G(.)to left-skewnessfor right-skewed distribution of G()Chebychev'sinequalitymayleadtothemoreconservativeresult-1010 g(.)Michael Beer7/14
Michael Beer 7 / 14 Probabilistic approximation • Interval analysis using estimated moments of G(.) • P G . g g 0 98391 ( ( ) ∈ ≥ l u , . ) = − g g 9 66 6 24 l u , . ,. µ= σ= 2 G G 3 795 0 822 ., . P G . g g 0 98391 ( ( ) ∈ > lP uP , . Cheby ) » Chebyshev’s inequality with = − lP uP Cheby g g 3 35 10 94 , .,. interval result shifted towards failure domain, even more conservative than Chebyshev interval result reflects tendency of the distribution of G(.) to left-skewness −10 0 10 g(.) interval analysis Chebyshev for right-skewed distribution of G(.), Chebychev‘s inequality may lead to the more conservative result −10 10 g(.) recall histogram for G(.) for uniform Xi PROBABILITIES OR INTERVALS OR ??? Vague Information: Comparative Study
VagueInformation:ComparativeStudyINTERVALORMOMENTS?Generalremarksinterval analysis headsforthe extremeevents,whilsta probabilisticanalysisyields probabilitiesforevents.for a defined confidence level P(X, [xu,xuD, interval analysis is moreconservative and independent of distributions of the XconservatismofintervalanalysisiscomparabletoChebyshev'sinequality.differencebetweeninterval resultsandprobabilisticresultsiscontrolledbythe distribution of theresponseforadefinedconfidencelevel,interval boundsmaybeeasiertospecifyorto control than momentsintervalanalysis can behelpful》toidentifylow-probability-but-high-consequenceeventsforriskanalysis》incaseof sensitivityof P,w.r.t.distributionassumption andveryvagueinformationforthisassumption》 ifthefirst 2 moments cannot be identified withsufficient confidenceWhat to chose in"intermediate"cases?MichaelBeer8/14
Michael Beer 8 / 14 INTERVAL OR MOMENTS ? General remarks • interval analysis heads for the extreme events, whilst a probabilistic analysis yields probabilities for events » to identify low-probability-but-high-consequence events for risk analysis • for a defined confidence level , interval analysis is more conservative and independent of distributions of the Xi PX x x ( i il iu ∈ , ) • difference between interval results and probabilistic results is controlled by the distribution of the response • conservatism of interval analysis is comparable to Chebyshev‘s inequality interval analysis can be helpful » in case of sensitivity of Pf w.r.t. distribution assumption and very vague information for this assumption » if the first 2 moments cannot be identified with sufficient confidence • for a defined confidence level, interval bounds maybe easier to specify or to control than moments What to chose in “intermediate” cases ? Vague Information: Comparative Study
Vague Information:Comparative StudyINTERVALORMOMENTS?Idea:informationcontentcompare interval representation andmoment representationof uncertainty by means of information content:Whichrepresentation tells usmore?assumethat a variable X is represented alternatively(i)bythefirst two moments μxand ox?(ii) by an interval [xi, xu] for a given confidence P(X e [x,xu])applymaximumentropyprincipletobothrepresentations;calculate the least information of the representationwithoutmakinganyadditionalassumptionschose the more informative representation;exploitavailableinformationtomaximumextent(notincontradictionwithmaximumentropyprinciple)Relating intervals and momentsanalog to the concept of confidenceintervals[X,x] =[ux-k.ox/μx+k.ox]Michael Beer9/14
Michael Beer 9 / 14 Idea: information content • compare interval representation and moment representation of uncertainty by means of information content: Which representation tells us more ? PX x x ( ∈ l u , ) • assume that a variable X is represented alternatively (i) by the first two moments μX and σX 2 (ii) by an interval [xl , xu] for a given confidence • apply maximum entropy principle to both representations; calculate the least information of the representation without making any additional assumptions • chose the more informative representation; exploit available information to maximum extent (not in contradiction with maximum entropy principle) • = µ − ⋅σ µ + ⋅σ lu X XX X xx k k , , analog to the concept of confidence intervals Relating intervals and moments INTERVAL OR MOMENTS ? Vague Information: Comparative Study
VagueInformation:ComparativeStudyENTROPY-BASEDCOMPARISONShannon's entropycontinuous entropy(f) =-Jf(x) · log2 (f(x)dx》modification for comparison (ease of derivation)In(f (x)log, (f (x)S(f) = -[f(x) · In(f(×)dxS.(f)In(2)In(2)Intervalrepresentationmaximumentropyprinciple1uniformdistributionf(xX. - X,7dx= In(×u-x,)InXrelating to momentsXu-x, =2k0xnint = In(2.k ·ox) = In(ox) +In(2.k)7m.inMichael Beer10/14
Michael Beer 10 / 14 ENTROPY-BASED COMPARISON Shannon‘s entropy • continuous entropy S f f x f x dx ( ) =− ⋅ ∫ ( ) log2 ( ( )) Interval representation • maximum entropy principle =− ⋅ ∫ − − u l x m int x ul ul 1 1 S dx xx xx , ln ( ( )) ( ( )) ( ) = 2 f x f x 2 ln log ln » modification for comparison (ease of derivation) ( ) ( ) m = ⋅ =− ⋅ ( ) ∫ ( ) ( ( )) 1 S f S f f x f x dx 2 ln ln uniform distribution ( ) = − u l 1 f x x x S 2k m int , = ⋅ ⋅σ = σ + ⋅ ln( X ) ln ln ( X ) (2 k) • relating to moments = − ln(x x u l) x x 2k u l − = ⋅ ⋅σX Vague Information: Comparative Study