Sensitivity Analysis

The study of how uncertainty in model outputs can be attributed to different input parameters, examining local and global sensitivity, illustrated with reactor kinetics problems and sensitivity measures.

• Sensitivity Analysis
• Uncertainty
• Model Outputs
• Reactor Kinetics

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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 1.8 =0.08 increased by 50% dP 1.7 = + 0 ( ) ( ) P t C t dt 1.6 dC 1.5 = ( ) ( ) P t C t P(t) 1.4 dt = = P 1.3 ) 0 ( P 1 P 0 1.2 = 0 ) 0 ( C 1.1 1 0 0.5 1 1.5 2 2.5 3 time (s)

4. Results P(t) normalized to P0 Mean lifetime normalized to baseline value (0.001 s) t=3 s 2 -3 3x 10 1 relative change in P(t) 0 -1 -2 -3 -0.1 -0.05 0 0.05 0.1 0.15 relative change in

5. Results P(t) normalized to P0 Mean lifetime normalized to baseline value (0.001 s) t=0.1 s 0.015 0.02 0.01 relative change in P(t) 0.005 0 -0.005 -0.01 -0.015 -0.1 -0.05 0 0.05 0.1 0.15 relative change in

6. Putting all on one chart t=0.1 s 0.025 0 0.02 0.015 dimensionless variation in P(t) 0.01 0.005 0 -0.005 -0.01 -0.015 -0.02 -0.025 -0.2 -0.15 -0.1 dimensionless variation in input variable -0.05 0 0.05 0.1 0.15

7. Putting all on one chart t=3 s 0.15 0 0.1 dimensionless variation in P(t) 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.2 -0.15 -0.1 dimensionless variation in input variable -0.05 0 0.05 0.1 0.15

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 + + = Y C P C P C P s s t t j j Then, if inputs are independent, C P C Y + = = + + P C P s s t t j j + y C p C p C p s s t t j j = + + 2 y 2 s 2 s 2 t 2 t 2 j 2 j C C C

9. Quantifying Sensitivity Most obvious calculation of sensitivity is Y S = x P x This is the slope of the curves we just looked at We can normalize about some point (y0) = + + 0 0 0 0 y C p C p C p s s t t j j 0 p Y = l x x 0 S y P x

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 Y = x S x P y x

11. Rewriting Y = = s s S C s s P y s y = t S C t t y j = S C j j y = + + 2 s 2 s 2 t 2 t 2 j 2 j C C C y 2 j 2 s 2 t = + + 2 s 2 t 2 j 1 C C C 2 y 2 y 2 y

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 (Px) constant Find output variance V(Y|Px=px) This will vary as we vary px So now do this for a variety of values of pxand find expected value E(V(Y|Px)) Note: V(Y)=E(V(Y|Px))+V(E(Y|Px))

13. Now normalize ( ( | )) V E Y P = x S x V y This is often called the importance measure, sensitivity index, correlation ratio, or first order effect

14. Variance-Based Methods Assume ( ) k ( ) x ( ) = = = + + + + ( ) , ... , ,..., Y f x f f f x x f x x x 0 2 , 1 ,..., 1 2 i i ij i j k k 1 i i j i Choose each term such that it has a mean of 0 Hence, f0is average of f(x) ( ) ( ( , , x Y E x x f i j i ij = ) = f x E Y x f 0 i i i ( ) ) ( ) x ( ) x x f f f 0 j i i j j

15. Variance Methods Since terms are orthogonal, we can square everything and integrate over our domain ( ) x Y E i V i | = = 2 k = + + + + ... V V V V V 2 , 1 ,..., f i ij ijk k 1 i i j i j k ( ) x = 2 V f dx i i i i V = i S i V f k = = + + + + 1 ... S S S S 2 , 1 ,..., i ij ijk k 1 i i j i j k

16. Variance Methods Siis first order (or main) effect of xi Sijis 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=S1+S2+S3+S4+S12+S13+S14+S23+S24+S34+ S123+S124+S134+S234+S1234 Total # of terms is 4+6+4+1=15=24-1