Propagation of Uncertainty

Uncertainty Analysis for Engineers 1

Propagation of UncertaintyJake BlanchardSpring 2010


IntroductionWe’ve discussed single-variable

probability distributionsThis lets us represent uncertain

inputsBut what of variables that

depend on these inputs? How do we represent their uncertainty?

Some problems can be done analytically; others can only be done numerically

These slides discuss analytical approaches


Functions of 1 Random VariableSuppose we have Y=g(X) where

X is a random input variableAssume the pdf of X is

represented by fx.If this pdf is discrete, then we can

just map pdf of X onto YIn other words X=g-1(Y)So fy(Y)=fx[g-1(y)]


ExampleConsider Y=X2.Also, assume discrete pdf of X is

as shown belowWhen X=1, Y=1; X=2, Y=4; X=3,

Y=9

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4


Discrete VariablesExample:

◦Manufacturer incurs warranty charges for system breakdowns

◦Charge is C for the first breakdown, C2 for the second failure, and Cx for the xth breakdown (C>1)

◦Time between failures is exponentially distributed (parameter ), so number of failures in period T is Poisson variate with parameter T

◦What is distribution for warranty cost for T=1 year


Formulation

...,,!)ln(

)ln(

0)(

...,,)ln()ln(

00

...,2,100

)(

...,2,1,0!

)(

2)ln(

)ln(

2

CCw

Cw

e

wewp

CCwCw

wx

xCx

xhw

xxexf

Cw

x

x


Plots

C=2=1

1 1.5 2 2.5 3 3.5 4 4.5 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

0 5 10 15 20 25 30 350

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

w


CDF For Discrete DistributionsIf g(x) monotonically increases,

then P(Y<y)=P[X<g-1(y)]If g(x) monotonically decreases,

then P(Y<y)=P[X>g-1(y)]…and, formally,

)(

1

1

)()()(ygx

ixXYi

xpygFyF

x

y

x

y


Another ExampleSuppose Y=X2 and X is Poisson

with parameter

,...9,4,1,0

!

,...3,2,1,0!

)(

)(1

2

yeytp

xextp

YXYg

XgXY

ty

y

tx

x


Continuous DistributionsIf fx is continuous, it takes a bit

more work

dydggf

dydFyf

or

dydy

ydgygfyF

dydy

ydgdx

ygx

dxxfdxxfyF

xy

y

xY

yg

xygxxY

11

11

1

1

)(

)(

)(

)()(

)(

)()()(1

1


Example

2exp

21

2exp

21

21exp

21

)(

2

2

2

1

1

yf

yf

Xf

imaginedydg

YygX

XY

y

y

x

Normal distribution

Mean=0, =1


ExampleX is

lognormal

2

2

2

1

1

21exp

21

)exp(21exp

)exp(21

)ln(21exp

21

)exp(

)exp()(

)ln(

yf

yyy

f

xx

f

imagine

Ydydg

YygX

XY

y

y

x

Normal distributi

on


If g-1(y) is multi-valued…

),(

21

21

)(

2

1

11

ognormallS

cucuf

cuff

cududS

cuS

cSU

Exampledydggfyf

ssu

k

iixY


Example (continued)

22ln

22lnln

21exp

221

21

ln

21exp

2

1

)ln(21exp

21

2

2

2

u

u

u

u

s

c

cuu

f

cu

cu

cu

f

ss

f

lognormal


Example

00

00

22

exp121

21

21

21

0exp1

1400

vaz

vaz

azazf

azazf

azff

azdzdV

aZv

vvv

vf

imagine

aVd

FVZ

vvvz

v


A second exampleSuppose we are making strips of

sheet metalIf there is a flaw in the sheet, we must

discard some materialWe want an assessment of how much

waste we expectAssume flaws lie in line segments (of

constant length L) making an angle with the sides of the sheet

is uniformly distributed from 0 to


Schematic

L

w


Example (continued)Whenever a flaw is found, we

must cut out a segment of width w

22

2/12

1

111

sin

,0sin

wLLw

Ldwd

Lww

UfLhw


Example (continued)g-1 is multi-

valued

2221

222

221

2

01

01

wLwfwff

LwwL

wf

LwwL

wf

w

</2

>/2


Results

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5

1

1.5

2

2.5

3

3.5

4

4.5

5

wL=1

cdf

pdf

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

w


Functions of Multiple Random VariablesZ=g(X,Y)For discrete variables

If we have the sum of random variables

Z=X+Y iji xall

iiyxzyx

jiyxz xzxfyxff ,),( ,,

zyxg

jiyxzji

yxff),(

, ),(


ExampleZ=X+Y

0.5 1 1.5 2 2.5 3 3.50

0.10.20.30.40.50.60.7

x

fx

5 10 15 20 25 30 350

0.050.1

0.150.2

0.250.3

0.350.4

0.45

y

fy


AnalysisX Y Z P Z-rank1 10 11 .08 11 20 21 .04 41 30 31 .08 72 10 12 .24 22 20 22 .12 52 30 32 .24 83 10 13 .08 33 20 23 .04 63 30 33 .08 9


Result

5 10 15 20 25 30 350

0.05

0.1

0.15

0.2

0.25

0.3

Z

fz


ExampleZ=X+Y

0.5 1 1.5 2 2.5 3 3.50

0.10.20.30.40.50.60.7

x

fx

1.5 2 2.5 3 3.5 4 4.50

0.050.1

0.150.2

0.250.3

0.350.4

0.45

fy

y


AnalysisX Y Z P Z-rank1 2 3 .08 11 3 4 .04 21 4 5 .08 32 2 4 .24 22 3 5 .12 32 4 6 .24 43 2 5 .08 33 3 6 .04 43 4 7 .08 5


Compiled Dataz fz3 .084 .285 .286 .287 .08

2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.50

0.050.1

0.150.2

0.250.3

fz

z


Example

allx

xzxz

z

allx

xzx

xallyxz

y

y

x

x

xzxvtvtf

tvxzxtvtxzfxff

tytf

vtxvtf

YXZ

)!(!exp

exp)!(!

)()(

)exp(!

)exp(!

x and y are integers


Example (continued)

tvz

tvf

zxzxv

z

z

z

allx

xzx

exp!

!)!(!

The sum of n independent Poisson processes is Poisson


Continuous Variables

z

yxz

z

yxz

g

yxz

zyxgyxz

dydzdzdgygfzF

dzdydzdgygfzF

gyzgx

dxdyyxfzF

dxdyyxfzF

YXgZ

11

,

11

,

11

,

),(,

),()(

),()(

),(

),()(

),()(

),(

1


Continuous Variables

dYYabYZf

af

adzdg

abYZX

bYaXZif

dXdzdggXf

dYdzdgYgfzf

yxz

yx

yxz

),(1

1

),(

),()(

,

1

11

,

11

,


Continuous Variables (cont.)

dyYfabYZf

af

tindependenyx

dXbaXZXf

bf

dYYabYZf

af

yxz

yxz

yxz

)()(1

,

),(1

),(1

,

,


Example

mw

mf

uwudu

mw

mf

duuwum

wm

f

dumuw

uwmmu

muduuwfuff

mv

mvf

mu

muf

VUVUW

w

w

w

w

vuw

v

u

2exp

21

)(2exp

21

112

exp21

2)(exp

)(21

2exp

21)()(

2exp

21

2exp

21

0,;

0


In General…If Z=X+Y and X and Y are normal dist.

Then Z is also normal with222yxz

yxz

2

2222

22

22

21exp

2

1

21

21exp

21

21exp

21

21exp

21

)()(

yx

yx

yx

z

y

y

x

x

yxz

y

y

yx

x

xz

yxz

zf

dyyyzf

dyyyzf

dyyfyzff


Products

n

iXZ

n

iXZ

n

ii

n

ii

i

YXz

i

i

XZ

XZ

andognormallallXimagine

dyyyzf

YZf

YZX

YZX

XYZ

1

22

1

1

1

,

)ln()ln(

),(1)(

1


ExampleW, F, E are lognormal

2222

2121

)ln(21)ln()ln()ln(

EFWC

EFWC

EFWC

EWFC


Central Limit TheoremThe sum of a large number of

individual random components, none of which is dominant, tends to the Gaussian distribution (for large n)


GeneralizationMore than two variables…

nnxxZ

n

dxdxdxzgxxxgfzf

xxxxgZ

n...,...,,,...)(

),...,,,(

32

1

321

,...,

321

1


MomentsSuppose Z=g(X1, X2, …,Xn)

bXaEdxxfbdxxfxaYE

dxxfbaxdyyfYYE

baXYimagine

dXdXdXXXXfXXXgZE

dXdXdXXXXfzZE

xx

xy

nnXXXn

nnXXX

n

n

)()()()(

)()()()(

...),...,,(),...,,(...)(

...),...,,(...)(

2121,...,,21

2121,...,,

21

21


Moments

)()(

)()()(

)()()(

)()(

2

22

2

22

XVaraYVar

dxxfxExaYVar

dxxfbxaEbaxYVar

dxxfbaxYEYVar

x

x

xYY


Moments

)()()(),(),(2)()()(

),(2

),(

),()(

),()(

),()(

)()()(

21212121

2122

12

2121,21

2121,2

22

2121,2

12

2121,2

21

2121,2

21

21

21

2121

212

211

2121

21

XEXEXXEXXEXXCovXXabCovXVarbXVaraYVar

dxdxxxfxxab

dxdxxxfxb

dxdxxxfxaYVar

dxdxxxfbabxaxYVar

dxdxxxfyYVar

XbEXaEYEbXaXY

imagine

XX

xxxx

xxx

xxx

xxxx

xxy


Approximation

)()(

)()()()(

)()()()(

)()()(

)()(

)()()(

)(

x

xxxx

xxxx

xxx

xx

x

gYE

dXXfXdxdgdXXfgYE

dXXfdxdgXdXXfgYE

dXXfdxdgXgYE

dxdgXgXg

dXXfXgYE

XgY


Approximation

2

22

2

2

2

)(

)()(

)()(

)()()(

)()(

)()()(

xx

xx

xx

xyxx

xx

xy

dxdgXVarYVar

dXXfXdxdgYVar

dXXfdxdgXYVar

dXXfdxdgXgYVar

dxdgXgXg

dXXfXgYVar


Second Order Approximation

)(21)()(

)(21)()(

)(21)()(

21)()(

)()()(

)(

2

2

22

2

2

22

2

22

xVardxgdgYE

dXXfXdxgdgYE

dXXfdxgdXgYE

dxgdX

dxdgXgXg

dXXfXgYE

XgY

xxx

xxx

xxx

xxx

x


Approximation for Multiple Inputs

n

i iX

i

n

ixXXXX

n

XgYVar

xggYE

XXXXgY

i

in

1

22

2

2

1

2

321

)(

21,...,,,)(

),...,,,(

321


ExampleExample 4.13Do exact and then use

approximation and compareWaste Treatment Plant – C=cost,

W=weight of waste, F=unit cost factor, E=efficiency coefficient

median covW 2000 ton/y .2F $20/ton .15E 1.6 .125

EWFC


Solving…

84771exp

620,3221exp

25563.041

36.1021ln

ln21lnlnln

124516.0cov1ln

149166.0cov1ln

19804.0cov1ln

4700.0ln9957.2ln6009.7ln

2

2

222

2

2

2

CCC

CCC

EFWC

EFWC

EE

FF

WW

medianE

medianF

medianW

CE

EFWC

EFW


Approximation

%16.032620

3262032673

3267321

43

0

%4.032620

3248332620;483,32

2016.0;033.3;915.4076124.1;223.20;6.2039

,,

21

21

21,,)(

2

22

2/52

2

2

2

2

2

2

22

2

22

2

22

error

Eg

Eg

Fg

Wg

error

g

Eg

Fg

WggCE

EE

FW

E

FW

E

FW

EFW

EFW

E

FWEFW

EFWEFW


Variance

%3.1847783708477

8370

2)(

)(

2

2/32

2

2

2

2

22

22

22

error

CVar

Eg

Fg

WgCVar

C

E

FWE

E

WF

E

FW

EFW

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Propagation of Uncertainty