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8/12/2019 Slides - Complete Variance Decomposition Methods
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Complete Variance
Decomposition MethodsCdric J. Sallaberry
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)(xy f=
[ ]nXxxx ,,, 21 =x
[ ]nYyyy ,,,21
=y
f
Sensit ivity Analysis
Question: What part of the uncertainty in y can be explained by the uncertainty
in each element of x ?
is a vector of uncertain inputs
is a vector of results
is a complex function (succession of different codes, systems of pde, ode )
Traditional Sampling-Based Sensitivity Method
Capture linear relationship between one input and one output ( CC, PCC, SRC)
Capture monotonic relationship between one input and one output (RCC, PRCC, SRRC)
2
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Limit on traditional methods: Non-monotonic influence
21 )5.0( = xy
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2121 .xxxxy += 2121 .xxxxy ++=
Such relation will not be captured with traditional sampling-based sensitivity analysis
Limit on traditional methods: Conjoint Influence
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High dimensional Model representation (1/3)
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We would like to find a method that :
capture any kind of relationship between input and output
capture conjoint influence
( ) ( ) ( )nXnXi ij
jiij
nX
i
ii xxxfxxfxfffy ,,,,)( 21,...,2,11
0 ++++== >=
x
Main Idea: Decompose the function into functions depending on any possible combinations of inputs
However, this decomposition is NOT unique
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If all the parameters are orthogonal and if
then the decomposition is unique
[ ]yf E0=
[ ] ( ) ( ) ( )nXnXi ij
jiij
nX
i
ii xxxfxxfxfyy ,,,,E 21,...,2,11
+++= >=
( ) nXi ij
ji
nX
i
i VVVy ,...,2,1,1
V +++= >=
( )
d...d),...,(
with
112,...,2,1,...,2,1
2
=
=
nXnXnXnX
iiii
xxxxfV
dxxfV
Decomposition of the variance ofy
6
High dimensional Model representation (2/3)
Since all terms are orthogonal, the cross
products are all equal to zero
One important consequence is that we have to consider independence between input parameters
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( ) nXi ij
ji
nX
i
i VVVy ,...,2,1,1
V +++= >=
nX
i ij
ji
nX
i
i SSS ,...,2,1,1
1 +++= >=
ofvariancethe to
parametersallofninteractiotheofoncontributi
ofvariancethe to
andofninteractiotheofoncontributiofvariancethetoofoncontributi
with
,...,2,1
y
S
y
xxS
yxS
nX
jiij
ii
Dividing
by )(V y
Sensitivity indices
7
High dimensional Model representation (3/3)
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Difference between the mean
If we knowxi and the mean
If we dont know it.
)(xfy=
)(E)|(E)( yxyxf iii =
)( ii xf
1x
ix
ijjx ,
We calculate the
average of the functionf
for a given value ofxi
Monte Carlo Approach
Two samples of size nSare
created
Same set of value forxi
Different set of values forall otherxj ,j i
Same operation done for
eachxi, i=1,,nX
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Sobol Variance Decomposit ion (1/4)
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Sobol variance decomposition (2/3)
)( ii xf
ix
)(2 ii xf
ix
( )
= iiii dxxfV
2
Vi integration of the square offi on the whole range ofxi.
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Sobol variance decomposition (3/3)
Higher Order
By fixing the value ofxi andxj (j i) the conjoint influence ofxi andxj can be
calculated.
Si,j, representing the influence of the sole interaction ofxi andxj, is defined by
integrating
Higher order, up to V1,2,,nXcan be defined the same way
)(E)|(E)|(E),|(E),(, yxyxyxxyxxf jijijiji +=
calculated in the previous step
Total Order
By fixing the value of all variables butxi one can calculate the influence of all inputs
with their interactions, except withxi (S-i). The difference STi = 1 S-i represents the influence ofxi solely and all its interaction
with the others inputs.
This index is called total sensitivity index ofxi.
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0
)()()()|(
)()()()|(
)()()(
00
0
1
321133221
111
03233221
111111
1 3 2
1 3 2
1
=
=
=
=
=
ff
fdxdxdxxpxpxpxxf
dxxpfdxdxxpxpxxf
dxxpxffE
S S S
S S S
S
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Properties of Sobol Variance Decomposit ion (1/2)
Examples in dimension 3: Expected value of f1 equal to zero
Indeed, we have
and f0 is constant
relatively to x1
Since the function f is
integrated on , we can
replace with
=1
111 1)(S
dxxp
)|( 1xxf
1S)(
xf
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Properties of Sobol Variance Decomposit ion (2/2)
0
)().()()(
)()(
,
2 3 1
232233
111110
110
10
=
=
=
S S S
S
dxdxxpxpdxxpxff
dVxpxff
ff
0
)()()()()(
)()()(
,
3 21
333
22222
11111
2211
21
=
=
=
S SS
S
dxxpdxxpxfdxxpxf
dVxpxfxf
ff
Examples in dimension 3: Two proofs of orthogonality
== S
dVxpxhxghg
0)()()(,
Definition: Two functions g and h defined on are said to be
orthogonal if their inner (or dot) product is equal to 0 :321 SSSS =
=0 as shown in
previous slide
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Fourier Amplitude Sensitivity Test (1/3)
( ) ( ) ( )[ ]
=
sGsGsGf
xxxpxxxfpf
nXnX
r
nXnX
rr
sin,,sin,sin
2
1
d...dd)(),,,(d)()(
2211
2121
xxxx
Basic Idea
In the moments calculation, converting the output from a function of nXvariables (i.e.,
the elementsxi of x) to a function of one variable (i.e., s) lead to convert the multi-
dimensional integral to a mono-dimensional integral.
Each inputxi is associated with a unique frequency iThe functions Gi are used to provide a better coverage of the domain
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Fourier Amplitude Sensitivity Test (2/3)
Small frequencies
Good approximation of
search curve with a small
number of points
Good coverage of the
domain
Large frequencies
But
Bad coverage of the
domain
But
Need large number of
points for approximating
the search function 14
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Fourier Amplitude Sensitivity Test (3/3)
( ) ( ) ( )[ ]
( )
=
+
1
22
2211
2
sin,,sin,sin2
1)(V
k
kk
nXnX
r
BA
sGsGsGfy
Fourier Series Representation
where
( ) ( ) ( )[ ]
( ) ( ) ( )[ ]
skssGsGsGfB
skssGsGsGfA
nXnXk
nXnXk
d)sin(sin,,sin,sin1
d)cos(sin,,sin,sin1
2211
2211
( )
=
+1
22i 2)(V
k
kk iiBAy
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Example Function for illustration
[ ])25.1(3cos)(),( 22 ++++= UVgVUVUVUf
with
+
+
= 10,10,
43
331
43
221
43
111
maxmin)(
VVV
Vg
Highly nonlinear and non-monotonic function
Involving complex interaction between Uand V
singularities for V=11/43, V=22/43 and V=33/43
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Example Traditional Sensitivi ty Results
CC RCC PCC PRCC SRC SRRC
U 0.1035 0.1103 0.1162 0.1217 0.1051 0.1110
V 0.4267 0.4100 0.4294 0.4127 0.4271 0.4102
R2 equal to ~ 0.19 (19% of the variance explained)
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Example Sobol variance decomposit ion
Parameter Sj STj
U(~j = 1) 8.53x10-4 0.686
V(~j = 2) 0.295 0.979
Almost 98% of the variance is explained
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Example FAST
Parameter Sj STj
U(~j = 1) 1.18 x 10-2 0.700
V(~j = 2) 0.225 0.973
92% to 98% of the variance is explained
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Conclusion
Strong Points of Variance Decomposition Methods
Capture nonlinear and nonmonotonic relationship between input and output
Allows calculation of conjoint influence of two or more inputs
Weak Points of Variance Decomposition Methods
Non negligible cost in number of simulations required
Suppose input parameters are independent to each other
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References
This work has been performed at Sandia National Laboratories (SNL), which is a multiprogram
laboratory operated by Sandia Corporation, a Lockheed Martin company, for the United States
Departement of Energys National Nuclear Security Administration under contract DE-AC04-
94AL-85000. Review provided at SNL by Rob Rechard and Kathryn Knowles.
FAST Cukier, R.I., H.B. Levine, and K.E. Shuler,Nonlinear Sensitivity Analysis of Multiparameter Model
Systems. Journal of Computational Physics, 1978. 26(1): p. 1-42
Saltelli, A., S. Tarantola, and K.P.-S. Chan,A Quantitative Model-Independent Method for Global
Sensitivity Analysis of Model Output. Technometrics, 1999. 41(1): p. 39-56.
SOBOL
Sobol', I.M., Sensitivity Estimates for Nonlinear Mathematical Models. Mathematical Modeling &Computational Experiment, 1993. 1(4): p. 407-414.
Calculation done with the software Simlab, available at
http://webfarm.jrc.cec.eu.int/uasa/
see the related paper for many more
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