Local and Global Sensitivity Analysis...Local and Global Sensitivity Analysis Omar Knio1,2 1Duke...

Local and Global Sensitivity Analysis

Omar Knio1,2

1Duke UniversityDepartment of Mechanical Engineering & Mate-rials [email protected]

2KAUSTDivision of Computer, Electrical, MathematicalScience & [email protected]

Knio Local and Global Sensitivity Analysis

[email protected]

[email protected]

Motivation

In this section, we will be mainly interested with the following, fairlystraightforward but as it turns out delicate and important, question : Givenf (x , y) does f vary more with x or with y ?

Why is this important ?

PlanningDecision analysis and supportDesigning experiments or field campaigns, etc...

For the discussion that follows, assume we have a minimum of information,specifying :

shape of f (equation, table, spreadsheet, simulation code, etc...)range of x and yoptionally a design point, say x0, y0


L2 functions over unit-hypercubes

Let L2(Ud ) be the space of real-valued squared-integrable functions over thed-dimensional hypercube U :

f : x 2 Ud 7! f (x) 2 R, f 2 L2(Ud ) ,Z

Udf (x)2dx < 1.

L2(Ud ) is equipped with the inner product h·, ·i,

8f , g 2 L2(Ud ), hu, vi :=Z

Udf (x)g(x)dx ,

and norm k · k2,8f 2 L2(Ud ), kfk2 := hf , f i1/2 .


L2 functions over unit-hypercubes

Let L2(Ud ) be the space of real-valued squared-integrable functions over thed-dimensional hypercube U :

f : x 2 Ud 7! f (x) 2 R, f 2 L2(Ud ) ,Z

Udf (x)2dx < 1.

NB : all subsequent developments immediately extend to product-typesituations, where

x 2 A = A1 ⇥ · · ·⇥ Ad ✓ Rd ,

and weighted spaces L2(A, ⇢),

⇢ : x 2 A 7! ⇢(x) � 0, ⇢(x) = ⇢1(x1)⇥ · · ·⇥ ⇢d (xd ).

(e.g. : ⇢ is a pdf of a random vector x with mutually independentcomponents.)


Ensemble notations

Let D = {1, 2, . . . , d}.Given i ✓ D, we denote i⇠ := D \ i its complement set in D, such that

i [ i⇠ = D, i \ i⇠ = ;.For instance

i = {1, 2} and i⇠ = {3, . . . , d},i = D and i⇠ = ;.


Ensemble notations

Let D = {1, 2, . . . , d}.Given i ✓ D, we denote i⇠ := D \ i its complement set in D, such that

i [ i⇠ = D, i \ i⇠ = ;.Given x = (x1, . . . , xd ), we denote x i the vector having for components the

xi2i, that isD ◆ i = {i1, . . . , i|i|} ) x i = (xi1 , . . . , xi|i|),

where |i| := Card(i). For instanceZ

U|i|f (x)dx i =

Z

U|i|f (x1, . . . , xd )

Y

i2i

dxi ,

and Z

Ud�|i|f (x)dx i⇠ =

Z

Ud�|i|f (x1, . . . , xd )

i /2iY

i2D

dxi ,


Sobol-Hoeffding decomposition

Any f 2 L2(Ud ) has a unique hierarchical orthogonal decomposition ofthe form

f (x) = f (x1, . . . , xd ) =f0 +dX

i=1

fi(xi) +dX

i=1

dX

j=i+1

fi,j(xi , xj)+

dX

i=1

dX

j=i+1

dX

k=j+1

fi,j,k (xi , xj , xk ) + · · ·+ f1,...,d (x1, . . . , xd ).

Hierarchical : 1st order functionals (fi ) ! 2nd order functionals (fi,j ) ! 3rdorder functionals (fi,j,l ) ! · · · ! d-th order functional (f1,...,d ).

Decomposition in a sum of 2k functionals

Using ensemble notations :

f (x) =X

i✓D

fi(x i).


Sobol-Hoeffding decomposition

Any f 2 L2(Ud ) has a unique hierarchical orthogonal decomposition ofthe form

f (x) =X

i✓D

fi(x i).

Orthogonal : the functionals if the S-H decomposition verify the followingorthogonality relations :

Z

Ufi(x i)dxj = 0, 8i ✓ D, j 2 i,

Z

Udfi(x i)fj(x j)dx = hfi, fji = 0, 8i, j ✓ D, i 6= j.

It follows the hierarchical construction

f; =

Z

Udf (x)dx = hf i;⇠=D

f{i} =

Z

Ud�1f (x)dx{i}⇠ � f; = hf iD\{i} � f; i 2 D

fi =Z

U|i⇠|f (x)dx i⇠ �

X

j(i

fj = hf ii⇠ �X

j(i

fj i 2 D, |i| � 2.


Example

Consider a 3D example :Y = f (X1,X2,X3)

Then SH yields :

f? = E[Y ]

f1(X1) = E[Y |X1]� f?

f2(X2) = E[Y |X2]� f?

f3(X3) = E[Y |X3]� f?

f12(X1,X2) = E[Y |X1,X2]� f1 � f2 � f?

f13(X1,X3) = E[Y |X1,X3]� f1 � f3 � f?

f23(X2,X3) = E[Y |X2,X3]� f2 � f3 � f?

f123(X1,X2,X3) = E[Y |X1,X2,X3]� f1 � f2 � f3� f12 � f13 � f23 � f?

= f (X1,X2,X3)� f1 � f2 � f3� f12 � f13 � f23 � f?


Parametric sensitivity analysis

Consider x as a set of d independent random parameters uniformlydistributed on Ud , and f (x) a model-output depending on these randomparameters. It is assumes that f is a 2nd order random variable : f 2 L2(Ud ).Thus, f has a unique S-H decomposition

f (x) =X

i✓D

fi(x i).

Further, the integrals of f with respect to i⇠ are in this context conditionalexpectations,

E[f |x i] =

Z

U|i⇠|f (x)dx i⇠ = g(x i) 8i ✓ D,

so the S-H decomposition follows the hierarchical structure

f; = E[f ]f{i} = E

⇥f |x{i}

⇤� E[f ] i 2 D

fi = E[f |x i]�X

j(i

fj i ✓ D, |i| � 2.


Variance decomposition

Because of the orthogonality of the S-H decomposition the variance V [f ] ofthe model-output can be decomposed as

V [f ] =i 6=;X

i✓D

V [fi], V [fi] = hfi, fii .

V [fi] is interpreted as the contribution to the total variance V [f ] of theinteraction between parameters xi2i.

The S-H decomposition thus provide a rich mean of analyzing the respectivecontributions of individual or sets of parameters to model-output variability.However, as there are 2d � 1 contributions, so one needs more "abstract"characterizations.


Sensitivity indices

To facilitate the hierarchization of the respective influence of each parameterxi , the partial variances V [fi] are normalized by V [f ] to obtain the sensitivityindices :

Si(f ) =V [fi]V [f ]

1,i 6=;X

i✓D

Si(f ) = 1.

The order of the sensitivity indices Si is equal to |i| = Card(i).

1st order sensitivity indices. The d first order indices S{i}2D characterize thefraction of the variance due the parameter xi only, i.e. without any interactionwith others. Therefore,

1 �dX

i=1

S{i}(f ) � 0,

measures globally the effect on the variability of all interactions betweenparameters. If

Pdi=1 S{i} = 1, the model is said additive, because its S-H

decomposition is

f (xi , . . . , xd ) = f0 +dX

i=1

fi(xi),

and the impact of the parameters can be studied separately.


Sensitivity indices

To facilitate the hierarchization of the respective influence of each parameterxi , the partial variances V [fi] are normalized by V [f ] to obtain the sensitivityindices :

Si(f ) =V [fi]V [f ]

1,i 6=;X

i✓D

Si(f ) = 1.

The order of the sensitivity indices Si is equal to |i| = Card(i).

Total sensitivity indices. The first order SI S{i} measures the variability due toparameter xi alone. The total SI T{i} measures the variability due to theparameter xi , including all its interactions with other parameters :

T{i} :=X

i3i

Si � S{i}.

Important point : for xi to be deemed non-important or non-influent on themodel-output, S{i} and T{i} have to be negligible.Observe that

Pi2D T{i} � 1, the excess from 1 characterizes the presence of

interactions in the model-output.


Sensitivity indices

In many uncertainty problem, the set of uncertain parameters can benaturally grouped into subsets depending on the process each parameteraccounts for. For instance, boundary conditions BC, material property ',external forcing F , and D is the union of these distinct subsets :

D = DBC [D' [DF .

The notion of first order and total sensitivity indices can be extended tocharacterize the influence of the subsets of parameters. For instance,

SD' =X

i✓D'

Si,

measures the fraction of variance induced by the material uncertainty alone,while

TDF =X

i\DF 6=;

Si.

measures the fraction of variance due to the external forcing uncertainty andall its interactions.


Recap

Sensitivity index Su , total sensitivity index Tu

it is easy to verify that

Su + Tu⇠ = Su⇠ + Tu = 1

it immediately follows thatP

i S{i} 1 ) Pi T{i} � 1

Su high means that Xu has substantial contribution to V (Y )

Tu low means that Xu has weak contribution to V (Y ) (and so it is acandidate to be dropped)


S-H decomposition from PC expansions.Consider the model-output f : ⇠ 2 ⌅ ⇢ Rd 7! R, where ⇠ = (⇠1, . . . , ⇠d ) areindependent real-valued r.v. with joint-probability density function

p⇠(x1, . . . , xd ) =dY

i=1

pi(xi).

Let { ↵} be the set of d-variate orthogonal polynomials,

↵(⇠) =dY

i=1

(i)↵i (⇠i),

with (i)l�0 2 ⇡l the uni-variate polynomials mutually orthogonal with respect to

the density pi .

If f 2 L2(⌅, p⇠), it has a convergent PC expansion

f (⇠) = limNo!1

X

|↵|No

↵(⇠)f↵, |↵| =dX

i=1

|↵i |.


S-H decomposition from PC expansions.Given the truncated PC expansion of f ,

f̂ (⇠) =X

|↵|A

↵(⇠)f↵,

one can readily obtain the PC approximation of the S-H functionals through

f̂i(⇠i) =X

|↵|A(i)

↵(⇠i)f↵,

where the multi-index set A(i) is given by

A(i) = {↵ 2 A;↵i > 0 for i 2 i,↵i = 0 for i /2 i} ( A.

For the sensitivity indices it comes

Si(f̂ ) =P

↵2A(i) f 2↵ h ↵, ↵i

P↵2A f 2

↵ h ↵, ↵i , T{i}(f̂ ) =P

↵2T (i) f 2↵ h ↵, ↵i

P↵2A f 2

↵ h ↵, ↵i ,

whereT (i) = {↵ 2 A;↵i > 0 for i 2 i}


Local and Global Sensitivity Analysis...Local and Global Sensitivity Analysis Omar Knio1,2 1Duke...

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