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Andrea Saltelli, Jessica Cariboni and Francesca Campolongo

Accelerating factors screening. Andrea Saltelli, Jessica Cariboni and Francesca Campolongo European Commission, Joint Research Centre SAMO 2007 Budapest. Sensitivity analysis at the Joint Research Centre of Ispra.

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Andrea Saltelli, Jessica Cariboni and Francesca Campolongo

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  1. Accelerating factors screening Andrea Saltelli, Jessica Cariboni and Francesca Campolongo European Commission, Joint Research Centre SAMO 2007 Budapest

  2. Sensitivity analysis at the Joint Research Centre of Ispra • Sensitivity analysis web at JRC (software, tutorials,..) http://sensitivity-analysis.jrc.cec.eu.int/ • New book on SA with exercises for students - at Wiley for review - Please flag errors! • Summer school in 2008 – date to be decided

  3. Where do we stands in terms of good practices for global SA : Screening: Morris – Campolongo – EE (1991-2007) Quantitative: Sobol’, plus several investigators, 1990-2007

  4. Screening: Morris – Campolongo – EE (1991-2007) • Good but not so efficient • Quantitative: Sobol’, Saltelli (1993-2002) • Efficient for Si (Mara’ + Tarantola [scrambled FAST], Ratto + Young [SDR] + proximities [Marco’s presentation of yesterday]) • Not so efficient for STi (Saltelli 2002)

  5. Where to start? From the best available practice in screening: The method ofElementary Effects (Morris 1991) The EE method can be seen as an extension of a derivative-based analysis. Max Morris, Department of Statistics Iowa State University

  6. Model The method ofElementary Effects Elementary Effectfor the ith input factor in a point Xo

  7. r elem. effects EE1i EE2i … EEri are computed at X1 , … , Xr and then averaged. Average of EEi’s (xi) Standard deviation of the EEi’s s(xi) Factors can be screened on the (xi) s(xi) plane Using EE method: The EEi is still a local measure Solution: take the average of several EE

  8. A graphical representation of results

  9. 0 0 1/3 1/3 2/3 2/3 1 1 Using the EE method Each input varies across p possible values (levels – quantiles usually) within its range of variation xi U(0,1)p = 4  p1 = 0 p2= 1/3 p3= 2/3 p4= 1 The optimal choice for  is  = p / 2 (p -1) Sampling the levels uniformly Grid in 2D

  10. x2 C’ B’ C A’ A B x1 Improving the EE (Campolongo et al., ….. 2007) - Taking the modulus of  (xi), *(xi) Instead of using the couple of  (xi) and s(xi) • Maximizing the spread of the trajectories in the input space • Application to groups of factors

  11. A comparison with variance-based methods: Is*(xi) related to either Si or STi? Empirical evidence: the g-function of Sobol’ a=99a=9a=0.9 STi available analytically

  12. A comparison with variance-based Empirical evidence: the g-function  *(xi) is a good proxy for STi

  13. A trajectory of the EE design Implementing the EE method Original implementation estimate r EE’s per input. r trajectories of (k+1) sample points are generated, each providing one EE per input Total cost = r (k + 1) r is in the range 4 -10 Each trajectory gives k effect EE at the cost of (k + 1) simulations. Efficiency =k/(k+1)~1

  14. Conclusion: the EE is a useful method Is its efficiency k/(k+1) ~ 1 good? We can compare with the Saltelli 2002 method to implement the calculation of the first order and total order sensitivity indices:

  15. Saltelli 2002 With: One of this plus … … one of this plus … plus K of these One can compute all first and total effects for k factors

  16. Saltelli 2002 Total: N(K+2) runs To obtain N*2*kelementary effects (for Si or STi) Efficiency=2k/(k+2)~2 Better that the EE method. One of this One of this K of these

  17. Conclusion: the efficiency of EE might have scope for improvement. The better efficiency of the global method (Saltelli 2002) against the screening method (EE) is due to the fact that two effects (one of the first order and one of the total order) are computed from each row of Ai. Can we do the same with EE?

  18. Saltelli 2002 From To … is one step in the non-Xi direction (all moves but Xi)

  19. Saltelli 2002 From To … is one step in the Xi direction (Xi moves and X~i does not)

  20. How about alternating steps along the Xi’s axes with steps along the along the X~i’s also for an EE-line screening method? How can we combine steps along Xi’s axes with steps along the X~i’s?

  21. Beyond Elementary Effects Method Can we generate efficiently exploration trajectories in the hyperspace of the input factors where steps in the Xi and X~i directions are nicely arranged, e.g. in a square?

  22. Beyond Elementary Effects Method

  23. Our thesis is that (1) Both |y1-y3| and |y2-y4| tells me about the first order effect of X1

  24. … and that : (2) ||y1-y4|-|y1-y2||, ||y2-y3|-|y2-y1||, ||y3-y2|-|y3-y4||,||y4-y1|-|y4-y3||, all tell me about the total order effect of a factor.

  25. Before trying to substantiate our thesis we give a look at how these squares could be built efficiently Four runs, six factors

  26. We call these four runs ‘base runs’ Four runs, six factors, six steps along the X~i directions

  27. Base runs Clones For each step in the X~i direction we add two in the Xi direction

  28. Base runs Clones Let’s count: Run 3 is a step away from run 1 in the X1 direction. Run 4 is a step away from run 2 in the X1 direction. Run 2 was already a step away from run 1 in the X~1 direction Run 4 is also a step away from run 3 in the X~1 direction … the square is closed.

  29. Beyond Elementary Effects Method

  30. Base runs Clones Let do some more counting. We have 4 base runs, 16 runs in total, six factors and four effects for factor. Efficiency= 24/16=3/2

  31. For 6 base runs, we have 15 factors, 36 runs in total, again four effects for factor. Efficiency= 60/36 ~ 2 for increasing number of factors … It would be nice to stop here! … but let us go back to the 6 factors example

  32. There are many more effects hidden in the scheme: e.g. three more effects for run 16. Most of these effects are of the X~i type The number of extra terms is between 2k and 4 k

  33. The number of extra terms grows with k Some of these need only one more point to close a square Most of these need two extra points to close a square

  34. Let us forget about the additional terms for the moment and let us try screening …

  35. Numerical Experiment: g-function where

  36. Results: g-function (180 runs)

  37. Number of runs: EE(2007)= 25; EE =22 K=10, a=(0.01,0.02,0.015,99,78,57,89,97,96,87)

  38. Test function Book (2007) The last two Z’s and the last two omegas are the most important factors

  39. Number of runs: new method= 64; old method =58

  40. g function 25 replicas of EE1(2007)

  41. g function 25 replicas of EE2(2007)

  42. g function 25 replicas of EE

  43. book function 25 replicas of EE2(2007)

  44. book function 25 replicas of EE1(2007)

  45. book function 25 replicas of EE

  46. What next? Good for Si, STi ?

  47. Si couple STi couple STi couple Si couple

  48. Try to exploit this design for the improvement of the Saltelli 2002 method for the STi Si couple STi couple STi couple Si couple

  49. The number of extra terms grows with k Some of these need only one more point to close a square Most of these need two extra points to close a square (closed squares give 4 effects, 2 Si & 2 STi)

  50. Conclusions The new scheme (aka il matricione) has promises for EE and STi Work on the algorithms is needed to make a sizeable difference with best available practices …

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