slides - complete variance decomposition methods

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)

5

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

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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

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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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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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( ) ( ) ( )[ ]

( )

=

+

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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