1. Sensitivity Analysis Jake Blanchard Fall 2010

2. Introduction Sensitivity Analysis = the study of how uncertainty in the output of a model can be apportioned to different input parameters Local sensitivity = focus on sensitivity at a particular set of input parameters, usually using gradients or partial derivatives Global or domain-wide sensitivity = consider entire range of inputs

3. Typical Approach Consider a Point Reactor Kinetics problem

4. Results P(t) normalized to P 0 Mean lifetime normalized to baseline value (0.001 s) t=3 s

5. Results P(t) normalized to P 0 Mean lifetime normalized to baseline value (0.001 s) t=0.1 s

6. Putting all on one chart – t=0.1 s

7. Putting all on one chart – t=3 s

8. Quantifying Sensitivity To first order, our measure of sensitivity is the gradient of an output with respect to some particular input variable. Suppose all variables are uncertain and Then, if inputs are independent,

9. Quantifying Sensitivity Most obvious calculation of sensitivity is This is the slope of the curves we just looked at We can normalize about some point (y 0 )

10. Quantifying Sensitivity This normalized sensitivity says nothing about the expected variation in the inputs. If we are highly sensitive to a variable which varies little, it may not matter in the end Normalize to input variances

11. Rewriting…

12. A Different Approach Question: If we could eliminate the variation in a single input variable, how much would we reduce output variation? Hold one input (P x ) constant Find output variance – V(Y|P x =p x ) This will vary as we vary p x So now do this for a variety of values of p x and find expected value E(V(Y|P x )) Note: V(Y)=E(V(Y|P x ))+V(E(Y|P x ))

13. Now normalize This is often called the ◦ importance measure, ◦ sensitivity index, ◦ correlation ratio, or ◦ first order effect

14. Variance-Based Methods Assume Choose each term such that it has a mean of 0 Hence, f 0 is average of f(x)

15. Variance Methods Since terms are orthogonal, we can square everything and integrate over our domain

16. Variance Methods S i is first order (or main) effect of x i S ij is second order index. It measures effect of pure interaction between any pair of output variables Other values of S are higher order indices “Typical” sensitivity analysis just addresses first order effects An “exhaustive” sensitivity analysis would address other indices as well

17. Suppose k=4 1=S 1 +S 2 +S 3 +S 4 +S 12 +S 13 +S 14 +S 23 +S 24 +S 34 + S 123 +S 124 +S 134 +S 234 +S 1234 Total # of terms is 4+6+4+1=15=2 4 -1