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“High-Confidence” Fragility Functions

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“High-Confidence” Subjective Fragility Function Estimation Suppose 19 experts give 19 opinions on fragility median A m and (logarithmic) standard deviation For each (discrete) value of strength y, find maximum of 19 cdfs and connect with a curve F max (y) – P[Strength < y|Expert A m and ], (or A m and some percentile) – Assume each maximum represents the upper 95% confidence limit (“High-Confidence”) – Fit a lognormal distribution to curve to represent a 95% “High- Confidence” fragility function using (weighted) least squares P[F(y) ≤ F max (y) for all y] 0.95! Functional confidence curve is not French curve linking 95-th percentiles at several strength values

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By Weighted Least Squares

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What if there aren’t 19 experts? Bootstrap Correl(X1,X2) estimate requires at least one subjective opinion of distribution of X1|X2 Use least squares to combine experts’ subjective distribution information – Sum of squared errors indicted magnitude of experts’ deviation from lognormal distribution

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Bootstrap 10 experts

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NoFail.xlsm Spreadsheets Spreadsheet NameContents FreqMle of lognormal fragility parameters from 19 non-failure responses, assuming 20-th would be failure BayesMoM estimates of lognormal fragility parameters of a-posteriori distribution of P[Failure|Eq] from 19 non-failure responses assuming non-informative prior BayesCorrSame as Bayes, including estimate of fragility correlations Imagine inspections after earthquakes indicate component responses and failure or non-failure

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Freq Spreadsheet Input iid responses for which no failures occurred – 19 responses were simulated for example and convenient interpretation of mean and standard deviation estimates as “High-Confidence” – Assume ln[Stress]-ln[Strength} ~ N[muX-muY, Sqrt(sigmaX^2+sigmaY^2)] Use Solver to maximize log likelihood of P[Non- failure|Response] subject to constraint – Either constrain CV or P[Failure] Output is ln(Am) and

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Bayes Spreadsheets Bayes estimate of reliability r = P{ln[Stress]- ln{strength] > 0] – Non-informative prior distribution of r – Same inputs as Freq: responses and non-failures – Use MoM to find ln(Am) and to match posterior E[r] = n/n+1 and Var[r] = n/((n+1)^2*(n+2)) – Ditto to find correlation from third moment of a- posteriori distribution of r Bayes posterior P[ESEL component life > 72 hours|Eq and plant test data]

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Parameter Estimates from 19 Non-Failure Responses Given 19 earthquake responses with ln(Median) = 0.5 and = 0.1 and reliability P[ln(Stress) < ln(strength)] ~ 95% Bayes non-informative prior on reliability P[Response posterior distribution Use Method of Moments to estimate parameters for a-posteriori distribution of reliability ParameterAssume 20-th is failure BayesBayes Correlation (ln(strength)) 0.7814040.7813890.950047891 ln(strength) 0.1481320.1481230.226336118 ln(strength) N/A 0.752266708

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What is the correlation of fragilities? See SubjFrag.xlsx:SubjCorr and NoFail.xlsm:BayesCorr spreadsheets to estimate correlations from subjective opinions on Y1|Y2 or from no-failure response observation HCLPF ignores fragility correlation Risk doesn’t ignore it

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What if multiple, co-located components? Could assume responses are same; simplifies computations In series? Parallel? RBD? Fault tree? Using event trees, some people argue that HCLPF for one component is representative of all like, co-located components. – If all like, co-located components are all in the same safety system and not in any others

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What if like-components are dependent? Fragilities could be dependent too! But not necessarily all fail if one fails – True, P[Response > strength] may be same for all like, co-located components – But what is P[g(Stress, strength) = failure] for system structure function g(.,.)?

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References NAP, “Review of Recommendations for Probabilistic Seismic Hazard Analysis: Guidance on Uncertainty and the Use of Experts (1997) / Treatment of Uncertainty,” National Academies Press, http://en.wikipedia.org/wiki/Quantification_o f_margins_and_uncertainties http://en.wikipedia.org/wiki/Quantification_o f_margins_and_uncertainties

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