Sensitivity AnalysisJake BlanchardFall 2010

IntroductionSensitivity 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

Typical ApproachConsider a Point Reactor Kinetics

problem

0

0

0

)0(

1)0(

)()(

)()(

PC

PP

tCtPdt

dC

tCtPdt

dP

0 0.5 1 1.5 2 2.5 31

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

time (s)

P(t

)

=0.08 increased by 50%

ResultsP(t) normalized to P0

Mean lifetime normalized to baseline value (0.001 s)

t=3 s

-0.1 -0.05 0 0.05 0.1 0.15-3

-2

-1

0

1

2

3x 10

-3

relative change in

rela

tive

chan

ge in

P(t

)

ResultsP(t) normalized to P0

Mean lifetime normalized to baseline value (0.001 s)

t=0.1 s

-0.1 -0.05 0 0.05 0.1 0.15-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

relative change in

rela

tive

chan

ge in

P(t

)

Putting all on one chart – t=0.1 s

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

dimensionless variation in input variable

dim

ensi

onle

ss v

aria

tion

in P

(t)

0

Putting all on one chart – t=3 s

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

dimensionless variation in input variable

dim

ensi

onle

ss v

aria

tion

in P

(t)

0

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 and

Then, if inputs are independent,

jjttss PCPCPCY

2222222jjttssy

jjttss

jjttss

CCC

pCpCpCy

PCPCPCY

Quantifying SensitivityMost obvious calculation of

sensitivity is

This is the slope of the curves we just looked at

We can normalize about some point (y0)

xx P

YS

x

xlx

jjttss

P

Y

y

pS

pCpCpCy

0

0

0000

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 end

Normalize to input variances

xy

xx P

YS

Rewriting…

2

22

2

22

2

22

222222

1y

jj

y

tt

y

ss

jjttssy

y

jjj

y

ttt

y

ss

sy

ss

CCC

CCC

CS

CS

CP

YS

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 px

So 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))

Now normalize

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

y

xx V

PYEVS

))|((

Variance-Based Methods

Assume

Choose each term such that it has a mean of 0

Hence, f0 is average of f(x)

kki ij

jiij

k

iii xxxfxxfxffxfY ,...,,...,)( 21,...,2,1

10

0

0

,, fxfxfxxYExxf

fxYExf

jjiijijiij

iii

Variance MethodsSince terms are orthogonal, we

can square everything and integrate over our domain

ki j k

ijki j

ij

k

ii

f

ii

iiii

ki j k

ijki j

ij

k

iif

xYEi

SSSS

V

VS

dxxfV

VVVVV

Vi

,...,2,11

2

,...,2,11

2|

...1

...

Variance MethodsSi is first order (or main) effect of

xi

Sij 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

Suppose k=41=S1+S2+S3+S4+S12+S13+S14+S23

+S24+S34+S123+S124+S134+S234+S12

34

Total # of terms is 4+6+4+1=15=24-1