variance vs entropy base sensitivity indices
DESCRIPTION
Variance vs Entropy Base Sensitivity Indices. Julius Harry Sumihar. Outline. Background Variance-based Sensitivity Index Entropy-based Sensitivity Index Estimates from Samples Results Conclusions. Background. - PowerPoint PPT PresentationTRANSCRIPT
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Variance vs Entropy BaseSensitivity Indices
Julius Harry Sumihar
![Page 2: Variance vs Entropy Base Sensitivity Indices](https://reader034.vdocument.in/reader034/viewer/2022042608/568134aa550346895d9bbb6d/html5/thumbnails/2.jpg)
Outline
• Background
• Variance-based Sensitivity Index
• Entropy-based Sensitivity Index
• Estimates from Samples
• Results
• Conclusions
![Page 3: Variance vs Entropy Base Sensitivity Indices](https://reader034.vdocument.in/reader034/viewer/2022042608/568134aa550346895d9bbb6d/html5/thumbnails/3.jpg)
Background
• Applications of computational models to complex real situations are often subject to uncertainty
• The aim of sensitivity analysis is to quantitatively express the degree of impact of the uncertainty from the specific sources on the resulting uncertainty of final model output
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Variance Base Sensitivity Index
• Result from the principle of “expected reduction in variance”
• This principle leads to the expression:
• Interpreted as “the amount of variance of output Y that is expected to be removed if the true value of parameter Xi will become known”
varY – E[var(Y|Xi)]
![Page 5: Variance vs Entropy Base Sensitivity Indices](https://reader034.vdocument.in/reader034/viewer/2022042608/568134aa550346895d9bbb6d/html5/thumbnails/5.jpg)
• Main characteristic: it considers the variance of a probability distribution as an overall scalar measure of the uncertainty represented by this distribution
• Intuitively, over a bounded interval, the highest possible degree of uncertainty is expressed by the uniform distribution
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• A scalar measure of uncertainty should attain its maximum value for uniform distribution
• Inconsistency: This is not the case for variance
p=1/3 p=1/3
p=1/6 p=1/6
0 1/3 2/3 1
p=1/4 p=1/4p=1/4 p=1/4
0 1/3 2/3 1
Var(X) = 19/108 Var(X) = 15/108
H(X) = 1.32966 H(X) = 1.38629
![Page 7: Variance vs Entropy Base Sensitivity Indices](https://reader034.vdocument.in/reader034/viewer/2022042608/568134aa550346895d9bbb6d/html5/thumbnails/7.jpg)
• Entropy: an overall scalar uncertainty measure maximized by the uniform distribution
• ‘a measure of the total uncertainty of Y coming from all parameters’
dyyfyfYH ))(ln()()(
Entropy Base Sensitivity Index*
*Bernard Krzykacz-Hausmann,”Epistemic Sensitivity Analysis Based On The Concept Of Entropy”
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• ‘a measure of uncertainty of Y coming from the other parameters if the value of parameter X is known to be x’:
• ‘expected uncertainty of Y if the true value of parameter X will become known’:
dyxyfxyfxXYH ))|(ln()|()|(
dydxxfxyfxyfXYH )())|(ln()|()|(
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• ‘the amount of entropy of output Y that is expected to be removed if the true value of parameter X will become known’:
• By some manipulations:
)|()( XYHYH
dxdyyfxf
yxfyxfXYHYH
)()(
),(ln),()|()(
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b1 b2 b3 bjmaxbj-1 bj
a1
a2
ai-1
ai
aimax
Y
X
Estimates From Samples
i
aa xi
ai
ani
nxf
ii)(1
1
1
..
.)( ),[ 1
j
bb yj
bj
bnj
nyf
jj)(1
1
1
..
.)( ),[ 1
ji
bbaa yxj
bj
bi
ai
anij
nyxf
jjii
,
),[),[ )(1)(11
1
1
1
..),(
11
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ji ji
ij
ij
nn
nn
nn
n
nXYHYH
,
..
.
..
.
..
..
]))((
ln[)|()(
2
..
.
2
...
11X)]|E[var(Y– varY
j
jj
i j
ijj
i n
nyny
nn
Entropy Base:
Variance Base:
![Page 12: Variance vs Entropy Base Sensitivity Indices](https://reader034.vdocument.in/reader034/viewer/2022042608/568134aa550346895d9bbb6d/html5/thumbnails/12.jpg)
Results• Model:
o Y = U1 + U2
o Y = U1 + 2U2
o Y = N1 + N2
o Y = N1 + 2N2
U1,U2 ~ U[0,1], N1,N2 ~ N(0.5, 0.3)
• Number of samples: 1,000 and 10,000 (@10 times)
• Grid Size: 0.025, 0.05, 0.1, 0.2
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Model: Y = U1 + 2U2
0
0,02
0,04
0,06
0,08
0,1
0,12
0.025 0.05 0.1 0.2
Grid Size
var(
E[Y
|U1]
)
Analytical
1,000 samples
10,000 samples
Model: Y = U1 + 2U2
00,20,40,60,8
11,21,41,61,8
0.025 0.05 0.1 0.2
Grid Size
H(Y
)-H
(Y|U
1)
analytical
1,000 samples
10,000 samples
• 10,000 samples is better than 1,000 samples
• use 10,000 samples from now on
Effect of Sample Number
![Page 14: Variance vs Entropy Base Sensitivity Indices](https://reader034.vdocument.in/reader034/viewer/2022042608/568134aa550346895d9bbb6d/html5/thumbnails/14.jpg)
Model Xi 0.025 0.05 0.1 0.2 Analytical
Y = U1 + U2
U1 0,5629084 0,4847560 0,4381850 0,3740526 0,5
U2 0,5618695 0,4838189 0,4378267 0,3759858 0,5
Y = U1 + 2U2
U1 0,4104407 0,2723420 0,2263539 0,1890477 0.25
U2 0,9948456 0,9062422 0,83651837 0,7288735 0.943147
Y = N1 + N2
N1 0,5493945 0,4147645 0,35574700 0,3236304 0,346573
N2 0,5524757 0,4150046 0,35787368 0,3255338 0,346573
Y = N1 + 2N2
N1 0,5112249 0,2500562 0,15475538 0,1155110 0.111572
N2 0,9992094 0,8611518 0,79986998 0,7280601 0.804719
H(Y)-H(Y|Xi)
Effect of Grid Size
![Page 15: Variance vs Entropy Base Sensitivity Indices](https://reader034.vdocument.in/reader034/viewer/2022042608/568134aa550346895d9bbb6d/html5/thumbnails/15.jpg)
Model Xi 0.025 0.05 0.1 0.2 Analytical
Y = U1 + U2
U10,08324767 0,0829774 0,0822966 0,0796876 0,083333
U20,08371050 0,0833495 0,0825837 0,0802683 0,083333
Y = U1 + 2U2
U10,08395208 0,0832341 0,0823905 0,0794308 0,083333
U20,33438259 0,3335535 0,3309643 0,3212748 0,333333
Y = N1 + N2
N10,08959163 0,0892577 0,0884411 0,0856991 0,09
N20,08977242 0,0894751 0,0887430 0,0860685 0,09
Y = N1 + 2N2
N10,09116694 0,0901768 0,0886759 0,0859255 0,09
N20,35729517 0,3565734 0,3537290 0,3437929 0,36
Var(Y)-E[Var(Y|Xi)]
![Page 16: Variance vs Entropy Base Sensitivity Indices](https://reader034.vdocument.in/reader034/viewer/2022042608/568134aa550346895d9bbb6d/html5/thumbnails/16.jpg)
Model: Y = U1 + U2
0
0,1
0,2
0,3
0,4
0,5
0,6
0,025 0,05 0,1 0,2
Grid Size
H(Y
)-H
(Y|X
i)
Analytical
Numerical U1
Numerical U2
Model: Y = N1 + N2
0
0,1
0,2
0,3
0,4
0,5
0,6
0,025 0,05 0,1 0,2
Grid Size
H(Y
)-H
(Y|X
i)
Analytical
Numerical N1
Numerical N2
Model: Y = U1 + U2
0,01
0,03
0,05
0,07
0,09
0,025 0,05 0,1 0,2
Grid Size
var(
E[Y
|Xi]
)
Analytical
Numerical U1
Numerical U2
Model: Y = N1 + N2
0,01
0,03
0,05
0,07
0,09
0,025 0,05 0,1 0,2
Grid Sizeva
r(E
[Y|X
i])
Analytical
Numerical N1
Numerical N2
• Entropy base is very sensitive to grid size
• No rule exist for choosing grid size
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Model XiH(Y)-H(Y|Xi) Analytical var(E[Y|Xi]) Analytical
Y = U1 + U2
U1 0.484756 0.5 0.083248 0.083333
U2 0.483819 0.5 0.083350 0.083333
Y = U1 + 2U2
U1 0.272342 0.25 0.083234 0.083333
U2 0.906242 0.943147 0.333553 0.333333
Y = N1 + N2
N1 0.355747 0.346573 0.089591 0.09
N2 0.357874 0.346573 0.089772 0.09
Y = N1 + 2N2
N1 0.115511 0.111572 0.090177 0.09
N2 0.799870 0.804719 0.357295 0.36
Best Estimates
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Model: Y = U1 + U2
0,04
0,05
0,06
0,07
0,08
0,09
0,1
1 2 3 4 5 6 7 8 9 10
Sample
var(
E[Y
|Xi]
)
U1
Average U1
U2
Average U2
Analytical
Model Y = U1 + U2
0,45
0,46
0,47
0,48
0,49
0,5
0,51
1 2 3 4 5 6 7 8 9 10
Sample
H(Y
)-H
(Y|X
i)
U1
U2
Average U1
Average U2
Analytical
Model: Y = U1 + 2U2
0,23
0,24
0,25
0,26
0,27
0,28
0,29
1 2 3 4 5 6 7 8 9 10
Sample
H(Y
)-H
(Y|U
1)
U1
Average
Analytical
Model: Y = U1 + 2U2
0,04
0,05
0,06
0,07
0,08
0,09
0,1
1 2 3 4 5 6 7 8 9 10
Sample
var(
E[Y
|U1]
)
U1
Average
Analytical
Model: Y = U1 + 2U2
0,88
0,89
0,9
0,91
0,92
0,93
0,94
0,95
1 2 3 4 5 6 7 8 9 10
Sample
H(Y
)-H
(Y|U
2)
U2
Average
Analytical
Model: Y = U1 + 2U2
0,31
0,32
0,33
0,34
0,35
0,36
0,37
1 2 3 4 5 6 7 8 9 10
Sample
var(
E[Y
|U2]
)
U2
Average
Analytical
H(Y)-H(Y|Xi) Var(Y)-E[Var(Y|Xi)]
![Page 19: Variance vs Entropy Base Sensitivity Indices](https://reader034.vdocument.in/reader034/viewer/2022042608/568134aa550346895d9bbb6d/html5/thumbnails/19.jpg)
Model: Y = N1 + N2
0,3
0,32
0,34
0,36
0,38
1 2 3 4 5 6 7 8 9 10
Sample
H(Y
)-H
(Y|X
i)
U1
U2
Average U1
Average U2
Analytical
Model: Y = N1 + N2
0,05
0,06
0,07
0,08
0,09
0,1
1 2 3 4 5 6 7 8 9 10
Sample
var(
E[Y
|Xi]
)
N1
Average N1
N2
Average N2
Analytical
Model: Y = N1 + 2N2
0,08
0,09
0,1
0,11
0,12
0,13
0,14
1 2 3 4 5 6 7 8 9 10
Sample
H(Y
)-H
(Y|N
1)
N1
Average
Analytical
Model: Y = N1 + 2N2
0,06
0,07
0,08
0,09
0,1
0,11
0,12
1 2 3 4 5 6 7 8 9 10
Sample
var(
E[Y
|N1]
)
N1
Average
Analytical
Model: Y = N1 + 2N2
0,75
0,77
0,79
0,81
0,83
0,85
1 2 3 4 5 6 7 8 9 10
Sample
H(Y
)-H
(Y|N
2)
N2
Average
Analytical
Model: Y = N1 + 2N2
0,3
0,32
0,34
0,36
0,38
0,4
1 2 3 4 5 6 7 8 9 10
Sample
var(
E[Y
|N2]
)
N2
Average
Analytical
Var(Y)-E[Var(Y|Xi)]H(Y)-H(Y|Xi)
![Page 20: Variance vs Entropy Base Sensitivity Indices](https://reader034.vdocument.in/reader034/viewer/2022042608/568134aa550346895d9bbb6d/html5/thumbnails/20.jpg)
Conclusions
• Entropy-based sensitivity index is difficult to estimate
• Variance-based sensitivity index is better than the Entropy-based one
![Page 21: Variance vs Entropy Base Sensitivity Indices](https://reader034.vdocument.in/reader034/viewer/2022042608/568134aa550346895d9bbb6d/html5/thumbnails/21.jpg)
dxxyfxfyf
UXXXXY
xxy )()()(
]1,0[~,
21
2121
otherwise
xyxxyf
otherwise
xxf
x
x
,0
1,1)(
,0
10,1)(
2
1
otherwise
yydxxyfxf
yydxxyfxf
yfy
xx
y
xx
y
,0
21,2)()(
10,)()(
)(
1
1
0
21
21
Model: Y = U1 + U2
![Page 22: Variance vs Entropy Base Sensitivity Indices](https://reader034.vdocument.in/reader034/viewer/2022042608/568134aa550346895d9bbb6d/html5/thumbnails/22.jpg)
otherwise
xyxyf
xxUxXY
UXXXXY
xy ,0
1,1)(
]1,[~|
]1,0[~,
|
1
2121
01)1ln(1)|(
1
0
1
1 x
x
dydxXYH
2
1)2ln()2()ln()(
1
0
2
1 dyyydyyyYH
2
1)|()( 1 XYHYH
Model: Y = U1 + U2
![Page 23: Variance vs Entropy Base Sensitivity Indices](https://reader034.vdocument.in/reader034/viewer/2022042608/568134aa550346895d9bbb6d/html5/thumbnails/23.jpg)
)())|((
))|(()]|([)(22 YEXYEE
XYEVarXYVarEYVar
1
0
2
1
2 1)2()()( dyyydyydyyyfYE
xdyydyyfyxXYE
x
x
xy
2
11)()|(
1
|
12
1)]|([)( XYVarEYVar
12
131)
2
1()()|())|((
1
0
222 dxxdxxfxXYEXYEE x
Model: Y = U1 + U2
![Page 24: Variance vs Entropy Base Sensitivity Indices](https://reader034.vdocument.in/reader034/viewer/2022042608/568134aa550346895d9bbb6d/html5/thumbnails/24.jpg)
]2,0[~]1,0[~,2 3213121 UXUXXXXXXY
dxxyfxfyf xxy )()()(31
otherwise
yxyxyf
otherwise
xxf
x
x
,0
2,2
1)(
,0
10,1)(
3
1
Model: Y = U1 + 2U2
![Page 25: Variance vs Entropy Base Sensitivity Indices](https://reader034.vdocument.in/reader034/viewer/2022042608/568134aa550346895d9bbb6d/html5/thumbnails/25.jpg)
otherwise
yydxxyfxf
ydxxyfxf
yydxxyfxf
yf
y
xx
xx
y
xx
y
,0
32,2
1
2
3)()(
21,2
1)()(
10,2
1)()(
)(1
2
1
0
0
31
31
31
Model: Y = U1 + 2U2
![Page 26: Variance vs Entropy Base Sensitivity Indices](https://reader034.vdocument.in/reader034/viewer/2022042608/568134aa550346895d9bbb6d/html5/thumbnails/26.jpg)
otherwise
xyxyf
xxUxXY
UXXXXY
xy
,0
2,2
1)(
]2,[~|
]1,0[~,2
1|
1
2121
)2ln(1)2
1ln(
2
1)|(
1
0
2
1 x
x
dydxXYH
9431471807.0
)22
3ln()
22
3()
2
1ln(
2
1)
2ln(
2)(
1
0
2
1
3
2
dyyy
dydyyy
YH
25.0)|()( 1 XYHYH
Model: Y = U1 + 2U2
![Page 27: Variance vs Entropy Base Sensitivity Indices](https://reader034.vdocument.in/reader034/viewer/2022042608/568134aa550346895d9bbb6d/html5/thumbnails/27.jpg)
1
0
2
1
3
2
22
2
3)
2
1
2
3(
2
1
2
1)()( dyyyydydyydyyyfYE
xdyydyyfyxXYE
x
x
xy
12
1)()|(
2
|1 1
12
1
2
3
3
7)]|([)(
2
1
XYVarEYVar
3
71)1()()|())|((
1
0
21
21
2
1 dxxdxxfxXYEXYEE x
Model: Y = U1 + 2U2
![Page 28: Variance vs Entropy Base Sensitivity Indices](https://reader034.vdocument.in/reader034/viewer/2022042608/568134aa550346895d9bbb6d/html5/thumbnails/28.jpg)
otherwise
xyxyf
xxUxXY
UXXXXY
xy ,0
1,1)(
]1,[~|
]1,0[~,2
2|
2
2121
02
1)1ln(1)|(
2
0
1
2 x
x
dydxXYH
9431471807.0)|()( 2 XYHYH
9431471807.0
)22
3ln()
22
3()
2
1ln(
2
1)
2ln(
2)(
1
0
2
1
3
2
dyyy
dydyyy
YH
Model: Y = U1 + 2U2
![Page 29: Variance vs Entropy Base Sensitivity Indices](https://reader034.vdocument.in/reader034/viewer/2022042608/568134aa550346895d9bbb6d/html5/thumbnails/29.jpg)
1
0
2
1
3
2
22
2
3)
2
1
2
3(
2
1
2
1)()( dyyyydydyydyyyfYE
xdyydyyfyxXYE
x
x
xy
2
11)()|(
1
|2 2
3333.02
3
12
31)]|([)(
2
2
XYVarEYVar
12
31
2
1)
2
1()()|())|((
1
0
22
22
2
2 dxxdxxfxXYEXYEE x
Model: Y = U1 + 2U2
![Page 30: Variance vs Entropy Base Sensitivity Indices](https://reader034.vdocument.in/reader034/viewer/2022042608/568134aa550346895d9bbb6d/html5/thumbnails/30.jpg)
Model: Y = a1N1+a2N2+a3N3+…
),(~,...,,... 212211 NXXXXaXaXaY nnn
Bernard Krzykacz-Hausmann:
22)]|([)( iii aXYVarEYVar
22
22
1ln2
1)|()(
kk
iii
a
aXYHYH
![Page 31: Variance vs Entropy Base Sensitivity Indices](https://reader034.vdocument.in/reader034/viewer/2022042608/568134aa550346895d9bbb6d/html5/thumbnails/31.jpg)
dyyfyfYH ))(ln()()(
)(
),()|(| xf
yxfxyf xy
dxyxfyf ),()(
dxdyyfyxf ))(ln(),(
dxdyxfxyfxyf
dxxfxYHXYH
)())|(ln()|(
)()|()|(
dxdyxf
yxfyxf )
)(
),(ln(),(
dxdyyfxf
yxfyxfXYHYH )
)()(
),(ln(),()|()(
;
;
Derivation of H(Y)-H(Y|X)
![Page 32: Variance vs Entropy Base Sensitivity Indices](https://reader034.vdocument.in/reader034/viewer/2022042608/568134aa550346895d9bbb6d/html5/thumbnails/32.jpg)
dxdyyfxf
yxfyxfXYHYH )
)()(
),(ln(),()|()(
1
1
1
1
1
1
..
.
1
1
..
.
1
1
1
1
..ln
1
1
1
1
..
ia
ia
jb
jb
jb
jbn
jn
ia
ian
in
jb
jb
ia
ian
ijn
jb
jb
ia
ian
ijn
j i
i
aa xi
ai
ani
nxf
ii)(1
1
1
..
.)( ),[ 1
j
bb yj
bj
bnj
nyf
jj)(1
1
1
..
.)( ),[ 1
ji
bbaa yxj
bj
bi
ai
anij
nyxf
jjii
,
),[),[ )(1)(11
1
1
1
..),(
11
j i
nj
n
ni
n
nij
n
nij
n
..
.
..
.
..ln..
Estimate of H(Y)-H(Y|X)
![Page 33: Variance vs Entropy Base Sensitivity Indices](https://reader034.vdocument.in/reader034/viewer/2022042608/568134aa550346895d9bbb6d/html5/thumbnails/33.jpg)
))|(())|(()( XYEVarXYVarEYVar
))|(())|(( 22 XYEEXYEE
)())|(( 22 YEXYEE
j j
ijjjj
jj
ijjy n
nybb
bbn
nydyyfyYE
..
1
1..
1)()(
dyxf
yxfydyxyfyxXYE xy )(
),()|()|( |
j
iiaa
ii
i
iiii
ijj
bbx
aan
nbbaan
ny
ii 1),[
1..
.
11.. )(11
11
1
j
aa
i
ijj x
n
ny
ii)(1 ),[
.1
Estimate of Var(Y)-E(Var(Y|X))
![Page 34: Variance vs Entropy Base Sensitivity Indices](https://reader034.vdocument.in/reader034/viewer/2022042608/568134aa550346895d9bbb6d/html5/thumbnails/34.jpg)
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Estimate of Var(Y)-E(Var(Y|X))
![Page 35: Variance vs Entropy Base Sensitivity Indices](https://reader034.vdocument.in/reader034/viewer/2022042608/568134aa550346895d9bbb6d/html5/thumbnails/35.jpg)