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Part 3 Sensitivity Analysis

Part 3 Sensitivity Analysis. (Design Exploration). optiSLang Sensitivity Analysis. optiSLang scans the design space and measures the sensitivity with statistical measures Results of a global sensitivity study are: Global sensitivities of the variables due to important responses

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Part 3 Sensitivity Analysis

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  1. Part 3 Sensitivity Analysis (Design Exploration)

  2. optiSLang Sensitivity Analysis • optiSLang scans the design space and measures the sensitivity with statistical measures • Results of a global sensitivity study are: • Global sensitivities of the variables due to important responses • Identification of reduced sets of important variables which have the most significant influence on the responses • Estimate the variation of responses • Estimate the solver noise • Better understanding and verification of dependences between input parameter variation and design responses Part 3: Sensitivity Analysis

  3. Input/output of models Inputs CAE, …, experiments Outputs Part 3: Sensitivity Analysis

  4. Statistical properties of input/output • Mean value • Variance • Standard deviation • Coefficient of variation Part 3: Sensitivity Analysis

  5. Methods for Sensitivity Analysis • Local methods • Local derivatives • Standardized derivatives • Global methods • Anthill plots • Coefficients of correlation (linear, quadratic) • Rank order correlation • Standardized regression coefficients • Stepwise polynomial regression • Reduced polynomial models: Coefficients of Importance • Advanced surrogate models including prediction analysis and optimal subspace detection: Coefficients of Prognosis Part 3: Sensitivity Analysis

  6. Scanning the design space with DOE Inputs Design of experiments Design evaluation Outputs • Output variability and input sensitivities • Input parameter significance and multivariate dependencies • Regression analysis • Quantification of the model predictability and noise fraction Part 3: Sensitivity Analysis

  7. Deterministic DOE schemes Full factorial Central composite D-Optimal quadr. • Simple DOE schemes can not identify multivariate dependencies • More complex schemes only efficient for small number of variables • Optimized only for polynomial regression • Not always uniformly distributed • Fixed size of samples, critical if failed designs occur Part 3: Sensitivity Analysis

  8. Latin Hypercube Sampling Standard Monte Carlo Simulation Latin Hypercube Sampling • Improved Monte Carlo Simulation • Cumulative distribution function is subdivided into N classes with same probability • Reduced number of required samples for statistical estimates • Reduced unwanted input correlations • Add optimal samples to an existing set of LHS samples (ALHS) • LHS requires N≥k+1 samples, if not possible ALHS can be used Part 3: Sensitivity Analysis

  9. DOE settings Part 3: Sensitivity Analysis

  10. Anthill plots • Two-dimensional scatter-plots of two sample vectors of any design variable or response • Reveal both linear and nonlinear dependencies • Strongly nonlinear dependency, e.g. as bifurcation may become clearly visible Part 3: Sensitivity Analysis

  11. Covariance • Covariance measures dependence between two selected parameters (design variables or responses) • Linear dependence is assumed • Covariance value depends on parameter dimensions Part 3: Sensitivity Analysis

  12. Coefficient of correlation • Defined as standardized covariance of two variables • Coefficient of correlation is always between -1 and 1 • Defines degree of linear dependence Part 3: Sensitivity Analysis

  13. Coefficient of correlation No correlation Weak correlation Strong correlation • Positive value indicates a positive relationship between two variables Xand Y, e.g. in case that the value of X increases, the value for Y increases as well • A value near zero indicates a random or nonlinear relationship between the two variables Part 3: Sensitivity Analysis

  14. Correlation matrix Output-Output Input-Output • Symmetric matrix: • One at diagonal: Input-Input Output-Input • Significant deviation from the target correlation of the input parameters indicates a possible error during the design procedure, or that the number of samples is too small Part 3: Sensitivity Analysis

  15. Example: Analytical nonlinear function • Additive linear and nonlinear terms and one coupling term • Contribution to the output variance (reference values):X1: 18.0%, X2: 30.6%, X3: 64.3%, X4: 0.7%, X5: 0.2% Part 3: Sensitivity Analysis

  16. Example: Analytical nonlinear function Part 3: Sensitivity Analysis

  17. Example: Analytical nonlinear function Part 3: Sensitivity Analysis

  18. Example: Analytical nonlinear function • Since optiSLang 3.2: Extended correlation matrix Part 3: Sensitivity Analysis

  19. Confidence interval • Correlation coefficients are statistical estimates with an accuracy depending on the number of samples N • Lower and upper bounds estimated from 95% confidence interval • Number of required samples: 50…100 if k<20, else 100…200 Part 3: Sensitivity Analysis

  20. Simple polynomial regression • Approximation of one variable Y in terms of a single variable X • Minimization of squared error sum for given set of samples • Least squares solution Part 3: Sensitivity Analysis

  21. Quadratic correlation • Quadratic regression of variable Y on variable X by a least-squares fit of the sample values • Correlation between approximated and exact values of Y Part 3: Sensitivity Analysis

  22. Quadratic correlation matrix • Unsymmetric matrix: • One at diagonal: Output-Output Input-Output Input-Input Output-Input • A value near zero means that there is a random or highly nonlinear relationship between the two variables Part 3: Sensitivity Analysis

  23. Spearman’s rank correlation • It assesses how well an arbitrary monotonic function could describe the relationship between two variables without making any assumptions about the regression function of the variables • Data are converted to ranks before calculating the coefficient • Insensitive towards outliers Part 3: Sensitivity Analysis

  24. Multiple polynomial regression • Set of input variables • Definition of polynomial basis • Approximation function • Least squares solution • Number of samples, linear: • Quadratic: Part 3: Sensitivity Analysis

  25. Coefficient of Determination (CoD) • Fraction of explained variation of an approximated variable • Total variation • Explained variation • Unexplained variation • Quality measure of approximation quality of a simple or multiple polynomial regression Part 3: Sensitivity Analysis

  26. Coefficient of Determination (CoD) • Adjusted Coefficient of Determination to penalize over-fitting • One dimensional CoDs are equivalent to squared linear or quadratic correlation coefficients of a certain response Yb with respect to a single variable Xa • Linear CoDs of single variables sum up to CoD of multi-dimensional linear polynomial if inputs are uncorrelated Part 3: Sensitivity Analysis

  27. Example: Analytical nonlinear function Part 3: Sensitivity Analysis

  28. Coefficient of Importance (CoI) • Explained variation of a response Yb due to a single variable Xaincluding its coupling terms • Reduced polynomial basis • Coefficient of Importance as reduction of CoD by removing Xa from the regression model • Sum of single CoIs larger than full CoD indicates important coupling terms • For linear basis and independent inputs Part 3: Sensitivity Analysis

  29. Example: Analytical nonlinear function all inputs,linear all inputs, quadratic 3 inputs, quadratic(N=100, p=6) (N=100, p=21) (N=100, p=10) • Calculated CoIs are more precise for a smaller number of regression coefficients (at least N>2p samples are recommended) Part 3: Sensitivity Analysis

  30. Significance filter • In large dimensions, the necessary number of solver runs for sensitivity analysis increases • But in reality, often only a small number of variables is important • Therefore, optiSLang includes filter technology to estimate significant correlation between inputs and outputs • Significance level for linear and quadratic correlation coefficient from Input- Input correlation errors Part 3: Sensitivity Analysis

  31. Limitations of CoD/CoI • CoD/CoI is only based on how good regression model fits through the sample points, but not on how good is the prediction quality • Approximation quality is too optimistic for small number of samples • For interpolation models with perfect fit, CoD is equal to one • Better approximation models are required for highly nonlinear problems, but CoD/CoI works only with polynomials Part 3: Sensitivity Analysis

  32. Moving Least Squares approximation • Local polynomial regression with position-dependent coefficients • Distance depending weighting of support points • Smoothing of approximation is controlled by distance radius D • Choice of D directly influences CoD Part 3: Sensitivity Analysis

  33. Moving Least Squares interpolation • Regularized weighting function • Interpolation condition is almost fulfilled • Almost independent of influence radius D • Noise is fully interpolated and can not be filtered • CoD is equal to one Part 3: Sensitivity Analysis

  34. Box-Cox transformation • Flexible transformation to transform nonlinear problems to almost linear • Family of power transformations • Geometric mean • Determination of optimal l by minimization of approximation errors • If optimal l is equal one, transformation is not active • Requires scaling of response Part 3: Sensitivity Analysis

  35. Coefficient of Prognosis (CoP) • Fraction of explained variation of prediction • optiSLang 3.1: CoP is estimated by approximation error of an additional test data set which is not used to build the model • Problems: • Test data set not used for approximation • For small number of samples: CoP estimate may not be reliable • optiSLang 3.2: Estimation of CoP by cross validation using a partitioning of all available samples: • Due to reverse checking every sample point is used for approximation and prediction, CoP estimate is more reliable • All samples are used for final approximation model Part 3: Sensitivity Analysis

  36. Coefficient of Prognosis (CoP) • Sample splitting with different splitting rates may lead to large variation of CoP estimate while approximation function is almost unchanged COP =0.59 COP =0.46 COP =0.67 COP =0.54 COP =0.69 COP =0.76 Part 3: Sensitivity Analysis

  37. Coefficient of Prognosis (CoP) • Cross validation CoP estimate is much more reliable then by using sample splitting COP =0.59 COP =0.59 COP =0.61 COP =0.62 COP =0.59 COP =0.58 Part 3: Sensitivity Analysis

  38. Coefficient of Prognosis (CoP) • CoP increases with increasing number of samples • CoP works for interpolation and approximation models • With MLS, continuous functions also including coupling terms can be represented with a certain number of samples • Prediction quality is better if unimportant variables are removed from the approximation model Part 3: Sensitivity Analysis

  39. Meta-model of Optimal Prognosis (MOP) • Approximation of solver output by fast surrogate model • Reduction of input space to get best compromise between available information (samples) and model representation (number of input variables) • Advanced filter technology to obtain candidates of optimal subspace (significance and CoI filters) • Determination of most appropriate approximation model (polynomials with linear or quadratic basis, MLS, …, Box-Cox) • Assessment of approximation quality (CoP) • MOP solves three important tasks: • Best variable subspace • Best meta-model • Determination of prediction quality Part 3: Sensitivity Analysis

  40. Meta-model of Optimal Prognosis (MOP) • 31 possible subspaces from 5 variables for each meta-model type =155 model evaluations • 5 possible subspaces with filter • Filter technology dramatically reduces the number of investigated subspaces to improve efficiency • DCoP measure quantifies accepted reduction in prediction quality to obtain a smaller subspace and/or simpler meta-model Part 3: Sensitivity Analysis

  41. MOP Settings Part 3: Sensitivity Analysis

  42. CoP for single variable sensitivity • Variance based sensitivity indices: fraction of variance explained by a single variable • Total indices include coupling terms • Determination of conditional variances on optimal meta-model and scaling with CoP • Sum of single CoPs larger as total CoP indicates coupling terms

  43. Example: Analytical nonlinear function CoD (quad. Polynomial, 5 inputs) CoP (MOP: MLS with 3 inputs) • Prediction quality is almost perfect with MOP on 100 LHS samples • Optimal subspace contains only X1, X2 and X3 • Highly nonlinear function of X3 and coupling term X1X2 are represented by approximation and sensitivity measures Part 3: Sensitivity Analysis

  44. Example: Analytical nonlinear function • MOP/CoP close to reference values (detects optimal subspace automatically, represents nonlinear and coupling terms) Part 3: Sensitivity Analysis

  45. CoI versus CoP • Application: 9 Inputs, 200 Sample, passive safety application • MoP can be used for visualization • global prognosis quality and local prognosis quality can be evaluated! Part 3: Sensitivity Analysis

  46. Example: Low-dim. instability problem CoD/CoI/CoP - Get ready for productive use. 1 4 3 optiSLang Version 3 (CoI find most important variable) 2 1 optiSLang Version 3.1 (CoP quantify nonlinearity) optiSLang Version 3.1 CoP: 0.73 MoP: MLS-Approximation Sample Split 70/30 optiSLang Version 2 (CoD shows no importance) Part 3: Sensitivity Analysis

  47. How to verify the CoP/MoP • Compare CoI/CoD and CoP (Explainability should continuously improve) • Check plausibility using 2D/3D visualization • If plausibility is not verified: • Sample set are to small or to much clustered • Add samples and repeat MOP/CoP calculation Part 3: Sensitivity Analysis

  48. Strategy “No Run too Much” Using advanced LHS sampling, filter technology, CoD/CoI/CoP we can start to check after 50 runs ⇒ can we explain the variation ⇒ which input scatter is important ⇒ how large is the amount of unexplainable scatter (potentially noise, extraction problems or high dimensional non-linearity) Part 3: Sensitivity Analysis “no run too much” a must for practical use!

  49. Summary • Sensitivity analysis gives advanced information about design parameters and helps to simplify the optimization problem • One-dimensional linear/quadratic dependencies can be captured by correlation coefficients /1D CoDs • Linear coupling terms can be identified with multiple CoD/CoIs but generally require a reduction of the design space • MOP serves optimal meta-model in best subspace • CoP gives reliable quality estimate (explained variation) for polynomials and advanced meta-models • Small CoP indicates insufficient number of samples or unexplainable solver behavior/problems • Single variance-based CoPs can represent highly nonlinear coupled and uncoupled dependencies • Strategy “No run too much”: Reliable sensitivity estimateswith 50…100 if k<20, else 100…200 Part 3: Sensitivity Analysis

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