Download presentation

Presentation is loading. Please wait.

Published byNestor Winders Modified over 2 years ago

1
“High-Confidence” Fragility Functions

2
“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

3
By Weighted Least Squares

4
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

5
Bootstrap 10 experts

6
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

7
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

8
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]

9
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

10
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

11
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

12
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(.,.)?

13
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

Similar presentations

OK

CSC321: Lecture 8: The Bayesian way to fit models Geoffrey Hinton.

CSC321: Lecture 8: The Bayesian way to fit models Geoffrey Hinton.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on next generation 2-stroke engine or 4-stroke Ppt on strings in c language Ppt on channels of distribution of pepsi Ppt on diversity in living organisms of class 9 Ppt on chromosomes and chromatin definition Ppt on forex market Ppt on fire extinguisher types list Ppt on standing order act test Ppt on product advertising campaign Ppt on job rotation program