Aleatory or epistemic? Does it matter?

Transcription

1 Worksho on isk Accetance and isk Communication tanord University, March 6-7, 007 Aleatory or eistemic? Does it matter? Armen Der Kiureghian University o Caliornia, Berkeley Ove Ditlevsen Technical University o Denmark

2 An old story, retold Deinitions: Aleatory: From Latin alea rolling o dice; uncertainty that arises rom intrinsic randomness o a henomenon. Eistemic: From Greek επιστημη (eisteme) knowledge; uncertainty that arises rom lack o knowledge (or data). Generally acceted criterion: I the uncertainty can be reduced by increasing knowledge or gathering o more data eistemic I the uncertainty cannot be reduced by increasing knowledge or gathering o more data aleatory

3 Element o risk analysis X ( X, K, X n ) vector o basic random variables X ( x, Θ ) distribution o X y i g i ( x, Θ g ), i, K, m redictive hysical models Θ arameters in robabilistic model Θ g arameters in hysical models

4 ources o uncertainty Uncertainty inherent in the basic random variables X Uncertain error in the orm o the robabilistic model X (x,θ ) Uncertain errors in the hysical models y i g i (x,θ g ), i,, m tatistical uncertainty in estimation o distribution arameters Θ tatistical uncertainty in estimation o hysical model arameters Θ g Uncertain measurement errors Uncertainty inherent in derived variables Y Human error

5 Nature o uncertainties: Uncertainty in basic random variables X Can be eistemic or aleatory: e.g., uncertainty inherent in the strength o concrete existing building eistemic uture building aleatory There is a undamental dierence in reliability analysis o existing and uture structures.

6 Nature o uncertainties: Physical model uncertainty y g( x, z) exact but unknown relation y gˆ( x, Θ g ) + ε redictive model, ε N(0,σ ε ) Two comonents: Error due to missing variables z (can be aleatory or eistemic) Error in model orm ĝ (mostly eistemic)

7 Nature o uncertainties: Probabilistic model uncertainty X ( x, Θ ) itted to observed data or based on a riori assumtions mall robabilities are sensitive to tails o distributions uncertain error o eistemic tye, diicult to assess tandard goodness-o-it tests do not guarantee it in the tail (Ditlevsen 994). Need or standardization o distribution models in codes Caution must be exercised in erormance-based engineering alications, which rely on absolute robability estimates

8 Nature o uncertainties: Parameter uncertainty Θ, Θ g estimated based on statistical analysis o observed data statistical uncertainty Θ ( θ) osterior distribution All statistical uncertainty is eistemic in nature.

9 Nature o uncertainties: emarks The choice between basic and derived variables deends on the available data and models. As scientiic knowledge advances, more uncertainties that aeared to be aleatory become eistemic in nature. Why then dierentiate? Dierentiating aleatory and eistemic uncertainties hels us identiy areas, where uncertainty can be reduced and models can be imroved in near term. It also hels us to more accurately ormulate risk and reliability roblems.

10 Inluence o uncertainties Proer understanding o the nature o uncertainties is essential or ormulation o risk and reliability roblems. Two demonstrative examles: tatistical deendence among system comonents due to statistical uncertainty. tatistical deendence among successive events in time due to non-ergodic uncertainties.

11 Examle : ystem reliability k-out-o-n system with statistically indeendent and identically distributed comonents: g(x) x x limit-state or tyical comonent X i N(M i,σ i ), σ i known, Μ i N( x,σ n), i, B β(m,m ) P Φ [β(m,m )] i i / Bayesian uncertain comonent reliability index Bayesian uncertain comonent ailure robability 3 β( β, 3, 0 ) B (β) β( β, 3, 30 ) (, 3, 0) P (, ( ) 3, 30) 00 n 0 n β β βin ( μβ) : μβ

12 Examle : ystem reliability k-out-o-n system with statistically indeendent and identically distributed comonents: Predictive comonent ailure robability: + + (μ,μ ) M (μ ) M (μ )dμ dμ 0 ( )d βtilde( 3, n).9 tilde( 3, n) β βti n n n n

13 Examle : ystem reliability k-out-o-n system with statistically indeendent and identically distributed comonents: Predictive system ailure robability: s N 0 j N k + N! j!( N j)! j ( ) N j P ( ) d Ptilde ( 3, n,, ) Pin ( 3,, ).5 Ptilde ( 3, n,, ) Pin ( 3,, ).4 Ptilde ( 3, n, 3, 3) s Pin ( 3, 3, 3).3 s, n Ptilde ( 3, n, 4, 4) Pin ( 3, 4, 4). Ptilde ( 3, n, 5, 5) Pin ( 3, 5, 5). μβ : k N -5 eries systems μ 3 B Ptilde ( μβ, n,, ) Pin ( μβ,, ) Ptilde ( μβ, n,, ) Pin ( μβ,, ) s Ptilde ( μβ, n, 3, ) Pin ( μβ, 3, ) s, n Ptilde ( μβ, n, 4, ) Pin ( μβ, 4, ) Ptilde ( μβ, n, 5, ) Pin ( μβ, 5, ) /3 / / Parallel systems μ B k / N / n n n n /4

14 Examle : ystem reliability k-out-o-n system with statistically indeendent and identically distributed comonents: Predictive system ailure robability: s N 0 j N k + N! j!( N j)! j ( ) N j P ( ) d Ptilde ( 3, 0, 5, K) Pin ( s 3, 5, K) Ptilde ( 3, 30, 5, K) s, n Pin ( 3, 5, K) N μ n 0 B n K k

15 Examle : Time-variant reliability tructure subject to reeated earthquake loads modeled as Poisson events with mean rate ν er year: Predictive system ailure robability: g( r, s,ε) ln r ε ε LN (λ LN (λ,ζ,ζ,ζ ) ) ) structure caacity N(0,σ N(0,σ Θ (ν, λ ), λ + ε,ζ ln s + ε earthquake load at each occurrence,σ,σ ) caacity model error load model error limit-state model model arameters

16 Examle : Time-variant reliability tructure subject to reeated earthquake loads modeled as Poisson events with mean rate ν er year: θ Φ ζ λ + σ λ + ζ + σ Θ ( θ) dθ redictive ailure robability at each occurrence P, Psn ex ( μ t) ν redictive ailure robability i events are assumed to be Poisson (the conventional aroach) P ln r e λ ex ν + Φ t ( r) ε ( e) Θ( θ)dr dedθ redictive ailure robability, correct solution r, e, θ ζ σ +

17 Examle : Time-variant reliability tructure subject to reeated earthquake loads modeled as Poisson events with mean rate ν er year: Assumed arameter values: ν ζ LN(mean μ 0.94, λ ν,c.o.v. 0.5) N(0,ζ / n), n 0 n P, Psn ζ 0.47, λ N(,ζ / n), 0.05 σ σ P μ ν t

18 ummary and conclusions The nature o uncertainties (aleatoric or eistemic) deends on circumstances and modeling assumtions. Perhas in the inal analysis, all uncertainties are eistemic. However, characterization as aleatory or eistemic hels us in identiying uncertainties that can be reduced in short term by imroving our models and by collecting data. Characterization o uncertainties is also imortant or roer ormulation o risk and reliability roblems. In articular, eistemic uncertainties can introduce correlation between the comonents o a system, and both tyes o uncertainties can introduce deendence among successive occurrences o events in time.