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Generalized models for uncertainty and imprecision in engineering

Generalized models for uncertainty and imprecision in engineering. Michael Beer University of Liverpool Institute for Risk & Uncertainty. General Situation. Vagueness ?. Uncertainty ?. Imprecision ?. Indeterminacy ?. Fluctuations ?. Ambiguity ?. ?. Consequences. Variability ?.

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Generalized models for uncertainty and imprecision in engineering

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  1. Generalized modelsfor uncertainty and imprecision in engineering Michael Beer University of Liverpool Institute for Risk & Uncertainty

  2. General Situation Vagueness ? Uncertainty ? Imprecision ? Indeterminacy ? Fluctuations ? Ambiguity ? ? Consequences Variability ? 1 Introduction Endeavor numerical modeling − physical phenomena, structure, and environment · prognosis − system behavior, hazards, safety, risk, robustness, economic and social impact, ... » close to reality » numerically efficient Deterministic methods deterministic computational models deterministic structural parameters · · Reality

  3. Traditional modeling of uncertainty probability space [, ,P] xa Ek ● axiomatic definition P is a real-valued set function defined on a system of sets/events Ek, which complies with the requirements: xb » A1 − nonnegetivity: measure space [X, ,P] » A2 − normalization: » A3 − i) additivity: ii) σ-additivity: 1 Introduction Probability random variable X: mapping  X ● generation of events random elementary events (measurable function)  Ω F(x) : sample space X: universe, fundamental set 5 1.0 event of interest Ek: xa < X  xb 4 3 typically X =  2 1 0 x4=x(4) x1=x(1) x2=x(2) Borel -algebra x3=x(3) x X x5=x(5) realizations (images)

  4. Subjective probabilities 1 Introduction Combination of rare data and prior expert knowledge BAYESian approach prior distribution posterior distribution » update of expert knowledge by means of data » influence of subjective assessment (prior distribution) decays quickly with increase of sample size » questions ▪ evaluation of remaining subjective uncertainty ▪ consideration of imprecision (data and expert knowledge) ▪ very small sample size or no data

  5. Some Questions 1 Introduction of imprecise and rare data statistical analysis Is it safe ? ~ F(x) set of plausible s model reliability analysis ~ Pf Is the reliability analysis still reliable ? [Pf,l, Pf,r] imprecision reflected in Pf Effects on Pf ? Sensitivity of Pf to imprecision ?

  6. 1 Introduction Some Questions Any concern or doubt ? How precise will it be ? modeling, quantification, processing, evaluation, interpretation ?

  7. linguistic assessments expert assessment / experience imprecise measurements · · · 2 Uncertainty and Imprecision Problematic cases Summary of examples measurement / observation under dubious conditions · high plausible range x medium low imprecise sample elements · x small samples · incomplete probabilistic elicitation exercise observations which cannot be separated clearly · · conditional probabilities observed under unclear conditions vague / dubious probabilistic information · · only marginals of a joint distribution available without copula function changing environmental conditions · · mixture of information from different sources and with different characteristics classification and mathematical modeling ?

  8. 2 Uncertainty and Imprecision Classification and modeling According to sources aleatory uncertainty epistemic uncertainty · · » irreducible uncertainty » property of the system » fluctuations / variability » reducible uncertainty » property of the analyst » lack of knowledge or perception collection of all problematic cases, inconsistency of information stochastic characteristics no specific model traditional probabilistic models According to information content uncertainty imprecision · · » probabilistic information » non-probabilistic characteristics set-theoretical models traditional and subjective probabilistic models In view of the purpose of the analysis averaged results, value ranges, worst case, etc. ? ·

  9. 2 Uncertainty and Imprecision Classification and modeling Simultaneous appearance of uncertainty and imprecision information defies a pure probabilistic modeling · separate treatment of uncertainty and imprecision in one model · generalized models combining probabilistics and set theory · concepts of imprecise probabilities » interval probabilities » sets of probabilities / p-box approach » random sets » fuzzy random variables / fuzzy probabilities » evidence theory / Dempster-Shafer theory common basic feature: set of plausible probabilistic models over a range of imprecision (set of models which agree with the observations) bounds on probabilities for events of interest

  10. 3 Non-probabilistic Models Intervals Mathematical model (classical set X) · common suggestion: assignment of uniform distributions x xr xl Information content possible value range between crisp bounds · no additional information (on fluctuations etc.) · Numerical processing engineering analysis option I: enclosure schemes » narrow actual result interval from outside » high numerical efficiency » procedures restricted to specific problems · option II: global optimization » explicit search for result interval bounds » reasonable or high numerical effort » procedures applicable to a large variety of problems · meaning ? effort ? does it serve the purpose of the analysis ?

  11. X x k k l x k r 3 Non-probabilistic Models Fuzzy Sets Modeling of imprecision, numerical representation -level set aggregation of information · · ~ » max-min operator -discretization · ▪ min operator = t-norm ▪ max operator = t-co-norm (x) 1.0 (.) not important but analysis with various intensities of imprecision k 0.0 x set of nested intervals of various size · utilization of interval analysis for -level sets, calculation of result-bounds for each -level

  12. plausibility and belief measures for sets (events of interest) · B2 Pl(.), Bel(.) B1 1.0 0.8 0.6 0.4 0.2 0.0 A5 A5 A4 A4 A3 A3 A1 A1 A2 A2 4 Imprecise Probabilities Evidence theory (DempsterShafer) Mathematical model body of evidence with focal subsets Ai (observations), · assignment of probability weights to the Ai: basic probability assignment · (Ai sometimes interpreted as random sets) example · Pl(x) Bel(x) x Michael Beer

  13. 4 Imprecise Probabilities Evidence theory (DempsterShafer) Relationship to traditional probability subjective bounding property for given basic probability assignment · generation schemes for observations not considered · subjective character of the basic probability assignment · statistical methods (estimations and tests) not defined · covers the special case of traditional probability » focal subsets are disjoint singletons (dissonant case), as images of elementary events · Information content possible value ranges with crisp bounds for a series of observed events · bounds on subjective probability · Numerical processing direct processing of empirical information (no theoretical distribution model) · stochastic simulation (Monte Carlo, Latin Hypercube) · general applicability

  14. fuzzy Random Variables ~ ~ ~ ~ ~ x(2) x(4) x(3) x(1) x(5) original Xj » real-valued random variable X that is completely enclosed in X · ~ 4 Imprecise Probabilities Mathematical model fuzzy realizations generated by elementary events · random elementary events (x)    1.0 fuzzy numbers 5 ~  4 X:   F() 3 2 1 0.0 ~ representation of X » fuzzy set of all possible originals Xj · x X =  realizations -discretization random -level sets X ·

  15. fuzzy Probabilities ~ F(x) ~ F(xi) F(x) 1.0  = 0  = 1 ~ 0.5 F(x) with fuzzy parameters and fuzzy functional type (F(x)) 0.0 x xi 1.0 0.0 4 Imprecise Probabilities Mathematical model (cont'd) fuzzy probability · ~ » » evaluation of all random -level sets X Bj note: in evidence theory, these events represent "belief" and "plausibility" Bj X X l X r x x fuzzy probability distribution functions · » bunch of the distribution functions Fj(x) of the originals Xj of X ~ (related to p-box approach for  = 0)

  16. fuzzy Probabilities 4 Imprecise Probabilities Relationship to evidence theory focal subsets represented by fuzzy realizations · probability of the focal subsets induced by elementary events · Relationship to traditional probability traditional probabilistic generation scheme for realizations · statistical methods (estimations and tests) fully applicable to originals · fuzzy set of possible probabilistic models subjective bounding property · special case of traditional probability directly included with P = 1(.) · Information content set of probabilistic models with subjective weights for their plausibility · probability bounds for various intensities of imprecision in the model · Numerical processing ▪ sensitivities with respect to probabilistic model choice ▪ conclusions in association with the initial observation utilization of the complete framework of stochastic simulation · fuzzy / interval analysis applied to fuzzy parameters and probabilities · general applicability

  17. Fuzzy probabilities in reliability analysEs ~ ~ ~ ~ Pf Pf X X 4 Imprecise Probabilities Fuzzy parameters Failure probability structural parameters · probabilistic model parameters · acceptable Pf µ(x) µ(Pf) 1 1 acceptable parameter interval sensitivity of Pf w.r.t. x i i mapping  x Pf 0 0 coarse specifications of design parameters & probabilistic models attention to / exclude model options leading to large imprecision of Pf acceptable imprecision of parameters & probabilistic models indications to collect additional information definition of quality requirements (.) not important, but analysis with various intensities of imprecision robust design

  18. example 1  reliability analysis Loading Process dead load · horizontal load PH · vertical loads n·PV 0 and n·pV 0 · 5 Examples Reinforced concrete frame n·PV 0 n·PV 0 n·p0 6.00 m PH PH 3 4 4 ø 16 n – load factor fuzzy 2 ø 16 4 ø 16 fuzzy random PH = 10 kN PV 0 =100 kN p0 = 10 kN/m 8.00 cross section 500/350 kφ kφ 1 2 simultaneous processing of imprecision and uncertainty

  19. example 1  reliability analysis ~ b (u) ~ b1 1.0 ~ b2 0.5 ~ ~ b2 < 3.8 = req_b b1 ≥ 3.8 = req_b 0.0 6.43 7.24 7.63 n 5 Examples Structural response Safety Level fuzzy failure load (limit state) fuzzy reliability index (fuzzy-FORM) · · traditional, crisp safety level m(b) req_b = 3.8 1.0 Pf = () 0.4 0.0 2.54 3.94 5.21 5.21 7.56 6.59 b worst and best case results for various intensities of imprecision maximum intensity of imprecision to meet requirements required reduction of input imprecision

  20. example 2  Reliability Analysis 5 Examples Fixed jacket platform imprecision in the models for » wave, drag and ice loads » wind load » corrosion » joints of tubular members » foundation » possible damage ·

  21. example 2  Reliability Analysis 5 Examples Probabilistic model (After R.E. Melchers) (time-dependent corrosion depth, uniform) · » c(t,E)  corrosion depth » f(t,E)  mean value function » b(t,E)  bias function » (t,E)  uncertainty function (zero mean Gaussian white noise) » E  collection of environmental and material parameters Modeling ? b(t,E) f(t,E)

  22. example 2  Reliability Analysis ~ X 5 Examples Models for the bias factor model variants Bounded random variable  beta distribution · Case II Case I with X representing a random bias b(t,E) Reliability analysis interval and fuzzy set · select one value b(t,E) = x · µ(x) calculate Pf via Monte Carlo simulation 1 · » beta distribution pdf for Pf ("overall" sensitivity of Pf) i » interval interval for Pf (bounds) » fuzzy set fuzzy set for Pf 0 (set of intervals with various intensities of imprecision "incremental" sensitivities of Pf) x

  23. example 2  Reliability Analysis Beta (q=r=1) 9.60 9.73 Beta (q=r=2) 9.49 Interval Pf (×10-7) 7.0 8.0 9.0 10.0 5 Examples Fixed jacket platform dimensions » height: 142 m » top: 27 X 54 m » bottom: 56 X 70 m loads, environment » T = 15C, t = 5 a » random: wave height, current, yield stress, and corrosion depth c(t,E) » beta distribution / interval for b(.)  [0.8,1.6] · · Reliability analysis Monte Carlo simulation with importance sampling and response surface approximation · Failure probability Nb = 2000 Nb = 114 beta, CI 9.60 beta, CII 9.73 9.49 interval b(.) = 1.0 NPf = 5000 7.0 8.0 9.0 Pf [107]

  24. example 3  Structuralanalysis p 1.0 1.2 1.5 m  [N/mm²] 22 29 E[v] 1 0 v [mm] 17.58 17.89 18.34 5 Examples (DFG SFB 528) Textile reinforced roof structure fuzzy random field for material strength stochastic finite elements · · plane 2 v plane 1 F(v) 1  = 0  = 1 v [mm] 0 16 14 20 18

  25. example 4  Collapse Simulation σ σ2 σ1 fuzzy probability distribution function F(t) for failure time · ~ εu ε ε1 ε2 ~ 1 P(t<τ) ~ ~ ~ f(σ1) f(ε2) f(εu) (P(t<τ)) fuzzy random function X(σ1,ε2,εu) ~ 0 t t = τ, critical failure time 5 Examples (DFG Research Unit 500) Controlled demolition of a store house fuzzy random variables · » material behavior fuzzy stochastic structural analysis · » Monte Carlo simulation with response surface approximation

  26. Resumé Generalized models for uncertainty and imprecision in engineering Concepts of imprecise probabilities high degree of flexibility in uncertainty quantification · » modeling adjustable to the particular situation » inclusion of vague and imprecise information » utilization of traditional stochastic methods and techniques comprehensive reflection of imprecision in the computational results; bounds on probability · » aspects in safety and design ▪ identification of sensitivities, hazards and robust design ▪ dealing with coarse specifications in early design stages » economic aspects ▪ reduction of required information: only as much and precise as necessary ▪ re-assessment of existing, damaged structures: quick first assessment and justification of further detailed investigation appropriate model choice in each particular case depending on the available information AND the purpose of the analysis application of different uncertainty models in parallel

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