Fall 2010 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 ID: 1014172
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1. Sensitivity AnalysisJake BlanchardFall 2010
2. IntroductionSensitivity Analysis = the study of how uncertainty in the output of a model can be apportioned to different input parametersLocal sensitivity = focus on sensitivity at a particular set of input parameters, usually using gradients or partial derivativesGlobal or domain-wide sensitivity = consider entire range of inputs
3. Typical ApproachConsider a Point Reactor Kinetics problem
4. ResultsP(t) normalized to P0Mean lifetime normalized to baseline value (0.001 s)t=3 s
5. ResultsP(t) normalized to P0Mean 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 SensitivityTo first order, our measure of sensitivity is the gradient of an output with respect to some particular input variable.Suppose all variables are uncertain andThen, if inputs are independent,
9. Quantifying SensitivityMost obvious calculation of sensitivity isThis is the slope of the curves we just looked atWe can normalize about some point (y0)
10. Quantifying SensitivityThis 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 endNormalize to input variances
11. Rewriting…
12. A Different ApproachQuestion: If we could eliminate the variation in a single input variable, how much would we reduce output variation?Hold one input (Px) constantFind output variance – V(Y|Px=px)This will vary as we vary pxSo now do this for a variety of values of px and find expected value E(V(Y|Px))Note: V(Y)=E(V(Y|Px))+V(E(Y|Px))
13. Now normalizeThis is often called theimportance measure, sensitivity index, correlation ratio, or first order effect
14. Variance-Based MethodsAssumeChoose each term such that it has a mean of 0Hence, f0 is average of f(x)
15. Variance MethodsSince terms are orthogonal, we can square everything and integrate over our domain
16. Variance MethodsSi is first order (or main) effect of xiSij is second order index. It measures effect of pure interaction between any pair of output variablesOther values of S are higher order indices“Typical” sensitivity analysis just addresses first order effectsAn “exhaustive” sensitivity analysis would address other indices as well
17. Suppose k=41=S1+S2+S3+S4+S12+S13+S14+S23+S24+S34+S123+S124+S134+S234+S1234Total # of terms is 4+6+4+1=15=24-1