Local Probabilistic Sensitivity Measure

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

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

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 Xi.

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. Using Monte Carlo • Algorithm: • Save a large number of samples • Compute E(X|Z=z0± ) 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. Alternative: Isaco’s method An alternative to calculating is proposed by Isaco Meilijson. The idea is to expand E(X|Z) around z0 using the Taylor expansion:

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

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. 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. 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. Entropy optimization (EO)

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

13. Newton’s method There are a number of reasons not to use Newton’s method for solving the EO problem: • 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.

14. MOSEK A much easier way of solving the EO problem is by using MOSEK created by Erling Andersen: • 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.

15. Some results

16. Even worse results…

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 z0. • A few different approaches to this problem have been tried, but these seem to give similar results.

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!