propagation of uncertainty jake blanchard spring 2010 uncertainty analysis for engineers1

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 Variables

Example:◦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

...,2,1

00)(

...,2,1,0!

)(

2)ln(

)ln(

2

CCw

Cw

e

we

wp

CCwC

w

w

x

xC

xxhw

xx

exf

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

ixXY

i

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

yey

tp

xex

tp

YXYg

XgXY

ty

y

tx

x


Continuous DistributionsIf fx is continuous, it takes a bit

more work

dy

dggf

dy

dFyf

or

dydy

ydgygfyF

dydy

ydgdx

ygx

dxxfdxxfyF

xy

y

xY

yg

x

ygx

xY

11

11

1

1

)(

)(

)(

)()(

)(

)()()(

1

1


Example

2exp

2

1

2exp

2

1

2

1exp

2

1

)(

2

2

2

1

1

yf

yf

Xf

imagine

dy

dg

YygX

XY

y

y

x

Normal distribution

Mean=0, =1


Example

X is lognormal

2

2

2

1

1

2

1exp

2

1

)exp(2

1exp)exp(2

1

)ln(

2

1exp

2

1

)exp(

)exp()(

)ln(

yf

yy

yf

x

xf

imagine

Ydy

dg

YygX

XY

y

y

x

Normal distributi

on


If g-1(y) is multi-valued…

),(

2

1

2

1

)(

2

1

11

ognormallS

cuc

uf

c

uff

cudu

dS

c

uS

cSU

Example

dy

dggfyf

ssu

k

iixY


Example (continued)

2

2ln

2

2lnln

2

1exp

22

1

2

1ln

2

1exp

2

1

)ln(

2

1exp

2

1

2

2

2

u

u

u

u

s

c

cu

uf

cu

cu

cu

f

s

sf

lognormal


Example

00

00

22

exp1

2

1

2

1

2

1

2

1

0exp1

1400

vaz

vaz

aza

zf

aza

zf

a

zff

azdz

dV

a

Zv

vv

v

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

11

1

sin

,0

sin

wLL

w

Ldw

d

L

ww

Uf

Lhw


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

jiyxz

ji

yxff),(

, ),(


ExampleZ=X+Y

0.5 1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

fx

5 10 15 20 25 30 350

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

y

fy


AnalysisX Y Z P Z-rank

1 10 11 .08 1

1 20 21 .04 4

1 30 31 .08 7

2 10 12 .24 2

2 20 22 .12 5

2 30 32 .24 8

3 10 13 .08 3

3 20 23 .04 6

3 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

0.1

0.2

0.3

0.4

0.5

0.6

0.7

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-rank

1 2 3 .08 1

1 3 4 .04 2

1 4 5 .08 3

2 2 4 .24 2

2 3 5 .12 3

2 4 6 .24 4

3 2 5 .08 3

3 3 6 .04 4

3 4 7 .08 5


Compiled Dataz fz

3 .08

4 .28

5 .28

6 .28

7 .08

2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.50

0.05

0.1

0.15

0.2

0.25

0.3

fz

z


Example

allx

xzxz

z

allx

xzx

xallyxz

y

y

x

x

xzx

vtvtf

tvxzx

tvtxzfxff

ty

tf

vtx

vtf

YXZ

)!(!exp

exp)!(!

)()(

)exp(!

)exp(!

x and y are integers


Example (continued)

tvz

tvf

zxzx

v

z

z

z

allx

xzx

exp!

!)!(!

The sum of n independent Poisson processes is Poisson


Continuous Variables

z

yxz

z

yxz

g

yxz

zyxg

yxz

dydzdz

dgygfzF

dzdydz

dgygfzF

gyzgx

dxdyyxfzF

dxdyyxfzF

YXgZ

11

,

11

,

11

,

),(

,

),()(

),()(

),(

),()(

),()(

),(

1


Continuous Variables

dYYa

bYZf

af

adz

dg

a

bYZX

bYaXZ

if

dXdz

dggXf

dYdz

dgYgfzf

yxz

yx

yxz

),(1

1

),(

),()(

,

1

11

,

11

,


Continuous Variables (cont.)

dyYfa

bYZf

af

tindependenyx

dXb

aXZXf

bf

dYYa

bYZf

af

yxz

yxz

yxz

)()(1

,

),(1

),(1

,

,


Example

m

w

mf

uwu

du

m

w

mf

duuwum

w

mf

dum

uw

uwmm

u

muduuwfuff

m

v

mvf

m

u

muf

VUVUW

w

w

w

w

vuw

v

u

2exp

2

1

)(2exp

2

1

11

2exp

2

1

2

)(exp)(2

1

2exp

2

1)()(

2exp

2

1

2exp

2

1

0,;

0


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

dist.

Then Z is also normal with 222yxz

yxz

2

2222

22

22

2

1exp

2

1

2

1

2

1exp

2

1

2

1exp

2

1

2

1exp

2

1

)()(

yx

yx

yx

z

y

y

x

x

yxz

y

y

yx

x

x

z

yxz

zf

dyyyz

f

dyyyz

f

dyyfyzff


Products

n

iXZ

n

iXZ

n

ii

n

ii

i

YXz

i

i

XZ

XZ

andognormallallXimagine

dyyy

zf

YZf

YZ

XY

ZX

XYZ

1

22

1

1

1

,

)ln()ln(

),(1

)(

1


ExampleW, F, E are lognormal

2222

2

1

2

1

)ln(2

1)ln()ln()ln(

EFWC

EFWC

EFWC

E

WFC


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

dxdxdxz

gxxxgfzf

xxxxgZ

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

),...,,,(

32

1

321

,...,

321

1


MomentsSuppose Z=g(X1, X2, …,Xn)

bXaEdxxfbdxxfxaYE

dxxfbaxdyyfYYE

baXY

imagine

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

XEXEXXEXXEXXCov

XXabCovXVarbXVaraYVar

dxdxxxfxxab

dxdxxxfxb

dxdxxxfxaYVar

dxdxxxfbabxaxYVar

dxdxxxfyYVar

XbEXaEYE

bXaXY

imagine

XX

xxxx

xxx

xxx

xxxx

xxy


Approximation

)()(

)()()()(

)()()()(

)()()(

)()(

)()()(

)(

x

xxxx

xxxx

xxx

xx

x

gYE

dXXfXdx

dgdXXfgYE

dXXfdx

dgXdXXfgYE

dXXfdx

dgXgYE

dx

dgXgXg

dXXfXgYE

XgY


Approximation

2

22

2

2

2

)(

)()(

)()(

)()()(

)()(

)()()(

xx

xx

xx

xyxx

xx

xy

dx

dgXVarYVar

dXXfXdx

dgYVar

dXXfdx

dgXYVar

dXXfdx

dgXgYVar

dx

dgXgXg

dXXfXgYVar


Second Order Approximation

)(2

1)()(

)(2

1)()(

)(2

1)()(

2

1)()(

)()()(

)(

2

2

2

2

2

2

22

2

22

xVardx

gdgYE

dXXfXdx

gdgYE

dXXfdx

gdXgYE

dx

gdX

dx

dgXgXg

dXXfXgYE

XgY

xx

x

xxx

xxx

xxx

x


Approximation for Multiple Inputs

n

i iX

i

n

ixXXXX

n

X

gYVar

x

ggYE

XXXXgY

i

in

1

2

2

2

2

1

2

321

)(

2

1,...,,,)(

),...,,,(

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 cov

W 2000 ton/y .2

F $20/ton .15

E 1.6 .125

E

WFC


Solving…

84771exp

620,322

1exp

25563.04

1

36.102

1ln

ln2

1lnlnln

124516.0cov1ln

149166.0cov1ln

19804.0cov1ln

4700.0ln

9957.2ln

6009.7ln

2

2

222

2

2

2

CCC

CCC

EFWC

EFWC

EE

FF

WW

medianE

medianF

medianW

CE

EFWC

E

F

W


Approximation

%16.032620

3262032673

326732

1

4

3

0

%4.032620

3248332620;483,32

2016.0;033.3;915.407

6124.1;223.20;6.2039

,,

2

1

2

1

2

1,,)(

2

22

2/52

2

2

2

2

2

2

22

2

22

2

22

error

E

g

E

g

F

g

W

g

error

g

E

g

F

g

W

ggCE

E

E

FW

E

FW

E

FW

EFW

EFW

E

FWEFW

EFWEFW


Variance

%3.18477

83708477

8370

2)(

)(

2

2/32

2

2

2

2

22

22

22

error

CVar

E

g

F

g

W

gCVar

C

E

FWE

E

WF

E

FW

EFW

propagation of uncertainty jake blanchard spring 2010 uncertainty analysis for engineers1

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