Download - Propagation of Uncertainty Jake Blanchard Spring 2010 Uncertainty Analysis for Engineers1
Uncertainty Analysis for Engineers 2
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
Uncertainty Analysis for Engineers 3
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)]
Uncertainty Analysis for Engineers 4
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
Uncertainty Analysis for Engineers 5
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
Uncertainty Analysis for Engineers 6
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
Uncertainty Analysis for Engineers 7
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
Uncertainty Analysis for Engineers 8
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
Uncertainty Analysis for Engineers 9
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
Uncertainty Analysis for Engineers 10
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
Uncertainty Analysis for Engineers 11
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
Uncertainty Analysis for Engineers 12
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
Uncertainty Analysis for Engineers 13
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
Uncertainty Analysis for Engineers 14
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
Uncertainty Analysis for Engineers 15
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
Uncertainty Analysis for Engineers 16
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
Uncertainty Analysis for Engineers 18
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
Uncertainty Analysis for Engineers 19
Example (continued)
g-1 is multi-valued
2221
222
221
2
01
01
wLwfwff
LwwL
wf
LwwL
wf
w
</2
>/2
Uncertainty Analysis for Engineers 20
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
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
Uncertainty Analysis for Engineers 21
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),(
, ),(
Uncertainty Analysis for Engineers 22
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
Uncertainty Analysis for Engineers 23
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
Uncertainty Analysis for Engineers 25
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
Uncertainty Analysis for Engineers 26
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
Uncertainty Analysis for Engineers 27
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
Uncertainty Analysis for Engineers 28
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
Uncertainty Analysis for Engineers 29
Example (continued)
tvz
tvf
zxzx
v
z
z
z
allx
xzx
exp!
!)!(!
The sum of n independent Poisson processes is Poisson
Uncertainty Analysis for Engineers 30
Continuous Variables
z
yxz
z
yxz
g
yxz
zyxg
yxz
dydzdz
dgygfzF
dzdydz
dgygfzF
gyzgx
dxdyyxfzF
dxdyyxfzF
YXgZ
11
,
11
,
11
,
),(
,
),()(
),()(
),(
),()(
),()(
),(
1
Uncertainty Analysis for Engineers 31
Continuous Variables
dYYa
bYZf
af
adz
dg
a
bYZX
bYaXZ
if
dXdz
dggXf
dYdz
dgYgfzf
yxz
yx
yxz
),(1
1
),(
),()(
,
1
11
,
11
,
Uncertainty Analysis for Engineers 32
Continuous Variables (cont.)
dyYfa
bYZf
af
tindependenyx
dXb
aXZXf
bf
dYYa
bYZf
af
yxz
yxz
yxz
)()(1
,
),(1
),(1
,
,
Uncertainty Analysis for Engineers 33
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
Uncertainty Analysis for Engineers 34
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
Uncertainty Analysis for Engineers 35
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
Uncertainty Analysis for Engineers 36
ExampleW, F, E are lognormal
2222
2
1
2
1
)ln(2
1)ln()ln()ln(
EFWC
EFWC
EFWC
E
WFC
Uncertainty Analysis for Engineers 37
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)
Uncertainty Analysis for Engineers 38
GeneralizationMore than two variables…
nnxxZ
n
dxdxdxz
gxxxgfzf
xxxxgZ
n...,...,,,...)(
),...,,,(
32
1
321
,...,
321
1
Uncertainty Analysis for Engineers 39
MomentsSuppose Z=g(X1, X2, …,Xn)
bXaEdxxfbdxxfxaYE
dxxfbaxdyyfYYE
baXY
imagine
dXdXdXXXXfXXXgZE
dXdXdXXXXfzZE
xx
xy
nnXXXn
nnXXX
n
n
)()()()(
)()()()(
...),...,,(),...,,(...)(
...),...,,(...)(
2121,...,,21
2121,...,,
21
21
Uncertainty Analysis for Engineers 40
Moments
)()(
)()()(
)()()(
)()(
2
22
2
22
XVaraYVar
dxxfxExaYVar
dxxfbxaEbaxYVar
dxxfbaxYEYVar
x
x
xYY
Uncertainty Analysis for Engineers 41
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
Uncertainty Analysis for Engineers 42
Approximation
)()(
)()()()(
)()()()(
)()()(
)()(
)()()(
)(
x
xxxx
xxxx
xxx
xx
x
gYE
dXXfXdx
dgdXXfgYE
dXXfdx
dgXdXXfgYE
dXXfdx
dgXgYE
dx
dgXgXg
dXXfXgYE
XgY
Uncertainty Analysis for Engineers 43
Approximation
2
22
2
2
2
)(
)()(
)()(
)()()(
)()(
)()()(
xx
xx
xx
xyxx
xx
xy
dx
dgXVarYVar
dXXfXdx
dgYVar
dXXfdx
dgXYVar
dXXfdx
dgXgYVar
dx
dgXgXg
dXXfXgYVar
Uncertainty Analysis for Engineers 44
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
Uncertainty Analysis for Engineers 45
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
Uncertainty Analysis for Engineers 46
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
Uncertainty Analysis for Engineers 47
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
Uncertainty Analysis for Engineers 48
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