1. Local Probabilistic Sensitivity Measure By M.J.Kallen March 16 th, 2001

2. 2 Presentation outline Definition of the LPSM Problems with calculating the LPSM Possible solution: Isaco’s method Results Conclusions

3. 3 LPSM Definition The following local sensitivity measure was proposed by R.M. Cooke and J. van Noortwijk: For a linear model this measure agrees with the FORM method. Therefore this measure can be used to capture the local sensitivity of a non-linear model to the variables X i.

4. 4 Problem with calculating the LPSM The derivative of the conditional expectation can only be analytically determined for a few simple models. Using a Monte Carlo simulation introduces many problems resulting in a significant error.

5. 5 Using Monte Carlo Algorithm: 1.Save a large number of samples 2.Compute E(X|Z=z 0 ±  ) and divide by 2  For good results  needs to be small, but then the number of samples used in step 2 is small and a large error is introduced after dividing by 2 .

6. 6 Alternative: Isaco’s method An alternative to calculating is proposed by Isaco Meilijson. The idea is to expand E(X|Z) around z 0 using the Taylor expansion:

7. 7 Isaco’s method (cont.) We can then calculate the covariance:

8. 8 Isaco’s method (cont.) The main idea in this algorithm is to now take a ‘local distribution’ Z* such that the term is equal to zero. By doing this we get

9. 9 Choosing Z* We want to take Z* such that Z* should be as close as possible to Z, therefore we want to minimize the relative information. This results in a entropy optimization problem.

10. 10 Relative information Definition: the relative information of Q with respect to P is given by: “The distribution with minimum information with respect to a given distribution under given constraints is the smoothest distribution with has a density similar to the given distribution.”

11. 11 Entropy optimization (EO)

12. 12 Solving the EO problem 1.Newton’s method 2.the MOSEK toolbox for MATLAB There are a number of ways to implement this entropy optimization problem. We have tried the following:

13. 13 Newton’s method The implementation of Newton’s method requires a lot of work. Since you have to solve a system, a matrix has to be inverted and this introduces large errors in many cases. There are a number of reasons not to use Newton’s method for solving the EO problem:

14. 14 MOSEK The MOSEK toolbox has a special function for entropy optimization problems, therefore the variables and constraints are easily set up. No long calculations needed, constraints can be changed in a few seconds. A much easier way of solving the EO problem is by using MOSEK created by Erling Andersen:

15. 15 Some results Modelz0z0 Correct answer Isaco’s method (10000 samples) X,Y ~ N(0,1) Z = X+Y 10.50.5079 20.50.4886 X,Y ~ N(0,1) Z = 2X+Y 00.40.3998 20.40.4064 X,Y ~ U(0,1) Z = 2X+Y 0.50.250.2513 10.250.3723 1.50.40.4021

16. 16 Even worse results… Modelz0z0 Correct answer Isaco’s method (10000 samples) X,Y ~ U(0,1) Z = -ln(X)/Y 0.5-0.34712-0.43205 0.75-0.28221-0.35226 1-0.22862-0.29663 2-0.10417-0.16034 3-0.05432-0.09732 4-0.03179-0.06767

17. 17 Attempts to fix Isaco We’ve tried many things to get better results. These attempts mostly consisted of adding and/or changing constraints. Using only the samples from a small interval around z 0. A few different approaches to this problem have been tried, but these seem to give similar results.

18. 18 Conclusions Until now the results cannot be trusted, therefore I recommend not to use this method. We need to gain insight into what is going wrong and why it’s behaving in this way. Maybe Isaco Meilijson has an idea!