Jake Blanchard 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: 398901
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Slide1
Sensitivity Analysis
Jake Blanchard
Fall
2010Slide2
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 inputsSlide3
Typical Approach
Consider a Point Reactor Kinetics problemSlide4
Results
P(t) normalized to P
0
Mean lifetime normalized to baseline value (0.001 s)
t=3 sSlide5
Results
P(t) normalized to P
0
Mean lifetime normalized to baseline value (0.001 s)
t=0.1 sSlide6
Putting all on one chart – t=0.1 sSlide7
Putting all on one chart – t=3 sSlide8
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, Slide9
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
)Slide10
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 variancesSlide11
Rewriting…Slide12
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
))Slide13
Now normalize
This is often called the
importance measure,
sensitivity index,
correlation ratio, or
first order effectSlide14
Variance-Based Methods
Assume
Choose each term such that it has a mean of 0
Hence, f
0
is average of f(x)Slide15
Variance Methods
Since terms are orthogonal, we can square everything and integrate over our domainSlide16
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 wellSlide17
Suppose k=4
1=S
1
+S
2
+S
3+S4+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