Sensitivity Analysis with Correlated Variables

Transcription

1 Sensitivity Analysis with Correlated Variables st Workshop on Nonlinear Analysis of Shell Structures INTALES GmbH Engineering Solutions University of Innsbruck, Faculty of Civil Engineering University of Innsbruck, Faculty of Mathematics, Informatics and Physics Natters/Tyrol, 23/4/2 Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 / 2

2 ARIANE 5 Launcher Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 2 / 2

3 Introduction Tasks Sensitivity analysis of limit state Further understanding of local failure modes Foundations for decisions on robust design Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 3 / 2

4 Introduction Tasks Sensitivity analysis of limit state Further understanding of local failure modes Foundations for decisions on robust design Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 3 / 2

5 Introduction DATA, INPUT OUTPUT Buckling load: depending on loads, material, geometry (currently 7 3 input variables) Flight critical behavior: investigation of (currently) 37 output variables Challenges: excessive computational cost IO-map continuous but non-differentiable Scarce data: flight scenarios supplied by the architect Statistical distribution of input parameters unknown (nonparametric methods required) Method of choice: Monte Carlo simulation, statistical indicators of dependence Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 4 / 2

6 Introduction Activities Monte Carlo simulation Latin hypercube sampling Correlation control Copula correlated input sample Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 5 / 2

7 Introduction Activities Monte Carlo simulation Latin hypercube sampling Correlation control Copula correlated input sample Descriptive statistics Sensitivity indicators Bootstrap confidence intervals Partial coefficients of determination Nonparametric tolerance intervals Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 5 / 2

8 Introduction Activities Monte Carlo simulation Latin hypercube sampling Correlation control Copula correlated input sample Descriptive statistics Sensitivity indicators Bootstrap confidence intervals Partial coefficients of determination Nonparametric tolerance intervals Application on ARIANE-frontskirt flight scenarios: assessment of sensitivity of structural behavior with respect to input variables Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 5 / 2

9 Monte Carlo Simulation Concept Situation: numerical model is given by an Input-Output map Y = g(x,..., X d ) Method: generate sample of input (X,..., X d ) and compute corresponding output Y Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 6 / 2

10 Monte Carlo Simulation Concept Situation: numerical model is given by an Input-Output map Y = g(x,..., X d ) Method: generate sample of input (X,..., X d ) and compute corresponding output Y Input Sampling Question: how to choose the sample (x i,..., x id ) N i= Consistency with desired F X of (X,..., X d ) Regular coverage of the sample space Small sample size N Small variance (accuracy of the estimator) Solution: Latin hypercube sampling Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 6 / 2

11 Correlation Control Concept Situation: random variables (X,..., X d ) with marginal distributions F Xj Goal: multidimensional sample X with desired correlation structure given by a correlation matrix C Method: restricted pairing Generate sample X according to marginal distributions F Xj 2 Cleverly rearrange X Remarks: Especially for small samples correlation control proved important 2 Iterating the process improves results Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 7 / 2

12 Correlation Control Comparison Study Goal: uncorrelated input sample correlation without control No Correlation Control Inp = Inp = 2 TOL correlation after control Correlation Control Inp = Inp = 2 TOL ratio sample size : nr. inputs ratio sample size : nr. inputs Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 8 / 2

13 Copulas Concept Goal: construction of joint distribution H of two random variables X, Y from marginal distributions F, G Method: copula C function such that H = C (F, G) Archimedean copula: copula C given by generator φ v Sample via Archimedean Copula u v Asymmetric Sample via Archimax ,5 u Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 9 / 2

14 Preliminary Analysis Scatter Plots Input vs. elastic strain energy density: ring 2 min (left) max (right) X X 2 X 3 X 4 X X 2 X 3 X 4 X 5 X 6 X 7 X 8 X 5 X 6 X 7 X 8 X 9 X X X 2 X 9 X X X 2 X 3 X 4 X 5 X 6 X 3 X 4 X 5 X 6 X 7 X 7 Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 / 2

15 Statistical Indicators Disadvantage: scatterplots do not eliminate the influence of scale and of interaction of input variables. Remedy: statistical indicators that remove these effects. Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 / 2

16 Statistical Indicators Disadvantage: scatterplots do not eliminate the influence of scale and of interaction of input variables. Remedy: statistical indicators that remove these effects. Indicators in use (X i vs. Y ): CC Pearson correlation coefficient; PCC partial correlation coefficient; SRC standardized regression coefficient; RCC Spearman rank correlation coefficient; PRCC partial rank correlation coefficient; SRRC standardized rank regression coefficient. Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 / 2

17 CCs, RCCs, PCCs, PRCCs, SRCs, SRRCs Input vs. elastic strain energy density: ring 2 min (left) max (right) CC RCC CC RCC PCC PRCC PCC PRCC SRC SRRC SRC SRRC Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 2 / 2

18 CCs, RCCs, PCCs, PRCCs, SRCs, SRRCs Input vs. elastic strain energy density: ring 2 min (left) max (right) CC RCC CC RCC PCC PRCC PCC PRCC SRC SRRC SRC SRRC Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 3 / 2

19 Bootstrap Confidence Interval Motivation Partial Rank Correlation Coefficient.5 PRCC Inp. Nr. Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 4 / 2

20 Bootstrap Confidence Interval Motivation Partial Rank Correlation Coefficient.5 PRCC Inp. Nr. Question Which input parameters have a significant influence? Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 4 / 2

21 Bootstrap Confidence Interval Concept Goal: measure of uncertainty/accuracy of computed sensitivity parameter Idea: 95 % - Confidence Interval Method: compute the sample distribution via resampling and then pick inner 95 % quantile Decision: significant input parameters: not contained in confidence interval Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 5 / 2

22 Bootstrap Confidence Interval Concept Goal: measure of uncertainty/accuracy of computed sensitivity parameter Idea: 95 % - Confidence Interval Method: compute the sample distribution via resampling and then pick inner 95 % quantile Decision: significant input parameters: not contained in confidence interval Resampling: generate bootstrap sample (M times) by drawing from existing simulated sample (size N) new samples of size N with replacement Advantage: no further expensive FE-computations needed Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 5 / 2

23 Bootstrap Confidence Interval Confidence Intervals Partial Rank Correlation Coefficient Confidence Interval for PRCC.5.5 PRCC CI Inp. Nr. 5 5 Inp. Nr. Interpretation Significant input parameters: not contained in confidence interval Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 6 / 2

24 Significance and Ranking Most influential Input Parameter Situation: correlation coefficients together with information about significance Ranking: sort significant input parameter by absolute correlation value (descending) Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 7 / 2

25 Significance and Ranking Most influential Input Parameter Situation: correlation coefficients together with information about significance Ranking: sort significant input parameter by absolute correlation value (descending) Inp. Nr. PRCC Rank PRCC.5 Partial Rank Correlation Coefficient Inp. Nr. Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 7 / 2

26 Partial Coefficient of Determination Method Goal: proportion of coefficient of determination R 2 caused by X j, j =,..., d Method: linear regression Idea: Compute change of R 2 when adding X j to the regression given that X,..., X j are already in the model Average over all possible permutations of (X,..., X d ) Interpretation: proportion of the variation of the output Y explained by the input factor X j Generalization: non-linear form functions, e.g. X 2, X 3,... Limitation: permutation matrix large for many input parameters Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 8 / 2

27 Partial Coefficient of Determination Application Data: 7 input loads, max load proportionality factor LPF as output Form functions: linear (left), quadratic (right) Idea: apply sensitivity analysis first, compute the PCD with significant input factors Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 9 / 2

28 Partial Coefficient of Determination Application Data: 7 input loads, max load proportionality factor LPF as output Form functions: linear (left), quadratic (right) Idea: apply sensitivity analysis first, compute the PCD with significant input factors PCD of max LPF PCD of max LPF X :.4 % X 3:.6 % X 9:.5 % X 3: 85.4 % Res:.87 % X :.68 % X 3:.3 % X 9:.83 % X 3: % X 2 :.64 % X 2 3:.32 % X 2 9:.9 % X 2 3: 42.2 % Res:.2 % Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 9 / 2

29 Nonparametric Tolerance Intervals Goal: assessing the range of the output Method: nonparametric tolerance intervals Idea: Compute interval [x l, x u ] such that at least a proportion p of the possible values lies in this interval Confidence level 9%, 95%,... Question: which proportion of possible values is captured by the computed interval [x min, x max ]? Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 2 / 2

30 Nonparametric Tolerance Intervals Goal: assessing the range of the output Method: nonparametric tolerance intervals Idea: Compute interval [x l, x u ] such that at least a proportion p of the possible values lies in this interval Confidence level 9%, 95%,... Question: which proportion of possible values is captured by the computed interval [x min, x max ]? Example: N = Confidence Level 9% 95% 99% 99.9% Proportion p 97.75% 97.5% 95.5% 93.33% Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 2 / 2

31 Software Software package Input sampling: LHS, Copula with correlation control Descriptive statistics: mean, variance, mode,... 6 sensitivity indicators: CC, PCC, SRC, RCC, PRCC, SRRC Bootstrap confidence intervals: significance Partial coefficients of determination PCD Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 2 / 2

32 Software Software package Input sampling: LHS, Copula with correlation control Descriptive statistics: mean, variance, mode,... 6 sensitivity indicators: CC, PCC, SRC, RCC, PRCC, SRRC Bootstrap confidence intervals: significance Partial coefficients of determination PCD Graphics Scatterplots, histograms, boxplots Bar charts for sensitivity indicators Error bars for confidence intervals Pie charts for PCDs Graphical representation of the correlation matrix Helene Roth (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 23/4/2 2 / 2