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