Variance-based sensitivity analysis using harmonic analysis

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Variance-based sensitivity analysis using harmonic analysis

Jean-Yves Tissot, Clémentine Prieur

To cite this version:

Jean-Yves Tissot, Clémentine Prieur. Variance-based sensitivity analysis using harmonic analysis.

2012. �hal-00680725�

analysis

Jean-YvesTissot

∗

and Clémentine Prieur

Université de Grenoble LJK/INRIA

51, rue des Mathématiques

Campus de Saint Martin d'Hères, BP53

38041 Grenoble edex 09 (Frane)

(jean-yves.tissotimag.fr)

(lementine.prieurimag.fr)

Phone: +33 (0)4 76 635447

Fax: +33 (0)4 76 6312 63

Marh20, 2012

Abstrat

FourierAmplitudeSensitivityTest(FAST)andRandomBalaneDesign(RBD)arepopular

methodsofestimatingvariane-basedsensitivityindies. Werevisittheminlightofthedisrete

Fourier transform (DFT) on nite subgroups of the torus and randomized orthogonal array

sampling. Wethen study theestimation errorofboththese methods. This allowsto improve

FAST and to deriveexpliit rates of onvergene of its estimators byusing theframework of

lattie rules. We also give a natural generalization of the lassi RBD by using randomized

orthogonal arrays having any parameters, and we provide a bias orretion method for its

estimators.

Keywords: global sensitivity analysis, random balane design, Fourier amplitude sensitivity test,

orthogonalarrays,lattie rules

∗

Correspondingauthor.

Variane-based sensitivity analysis onsists in omputing indies the so-alled variane-based

sensitivity indies (SI)or Sobol' indies (see [34 ℄) that areessentially multiple integrals. Many

numerial tehniques have been developed to estimate these quantities. This inludes the rude

MonteCarloestimator(see[34℄,and[18℄for areent work),thepolynomialhaos-basedestimators

(see[37 ℄and[2 ℄)andthe FASTmethod(see [9 ℄and[30 ℄)aswellasits derivedapproah,RBD(see

[38 ℄), and their hybrid approah, RBD-FAST (see [38 ℄ and [24 ℄), and many others (see [29℄ for a

review).

The main purpose of this paper is to revisit FAST and RBD by using the disrete harmoni

analysis framework, in order to arry out a theoretial error analysis. In these methods the SI

estimation amountsto omputinga nite numberof the omplex Fourier oeients of themodel

ofinterestdenedontheunithyperube. Intheorytheseomputationsouldbedonebyperforming

a rude Monte Carlo integration or a ubature on a regular grid. But the rate of onvergene of

the Monte Carlo method is low, and ubatures aregenerally unfeasible inhigh dimension beause

ofthe exponential growthof thenumberofnodes, alsoknownastheurse ofdimensionality.

A rst possible starting point to overome these drawbaks is to note that the disrete om-

plex Fourier oeients omputed by using the ubature approah are exatly the oeients in

the representation of the trigonometri interpolation polynomial of the model of interest on the

regular grid. Consequently this approah onsists of a trigonometri interpolation issue and an

be generalized by using Smolyak algorithm on sparse grids (see [12℄). Suh interpolation shemes

are quiteeient as long as the model of interest is suiently smooth (see [3℄). But thematrix

ofthe interpolationoperator insuha methodsuers froman inreaseof its onditionnumber for

both inreasing renement oftheregulargridand inreasingmodeldimension, andthus makesthe

interpolationshemeunstable(see [19 ℄).

Asa onsequene,it turnsout to be obviousthat, inorder to avoidthestabilityissue, one has

to fous onunitary operators. Thus DFT operators onnite subgroups of thetorus (see e.g. [23℄)

i.e. theunithyperubeviewasagroupwhosematrieshaveaperfetonditionnumberequal

to

1

^arepartiularly well-suited inthepresent framework. Thisleadsto theuseof lattierules(see [33 ℄ for a review)to whih FAST, asshown inSubsetion 4.1, is loselyrelated. In a seondtime,

by viewing nite subgroups of the torus asorthogonal arrays (see [16 ℄ for a review), theprevious

method an be generalized by performing a randomization proess on these arrays. This leads to

to whih RBD,asshowninSubsetion 4.2,is loselyrelated.

Thepaperproeedsasfollows. InSetion2,wesetupthenotation,wegivebakgroundmaterials

relatedtotheANOVAdeompositionandtotheFourierseriesrepresentation,andweintroduethe

lassof estimators ofinterest. InSetion 3, we rst reviewboth FAST and RBD,andthen revisit

them. Setion 4isdevotedtotheerroranalysisbyusingtherevisiteddenitionprovidedinSetion

3. At last, Setion 5 gives numerial illustrations of RBDestimates on an analytial model. Most

ofthe proofsof the propositions aregiveninappendix A.

2 Bakground

2.1 Notation

First,

E[Y ]

^,

E[Y |X]

^andVar

[Y ]

^denotethe unonditional expetation of

Y

^{,theonditional expe-}

tation of

Y

^given

X

^{and the variane of}

Y

^,respetively. By onvention, we dene

E[Y |∅] = E[Y ]

^.

Seondly,onsider a parameter

d

ⁱⁿ

N ^∗

^{the dependeneon whih is omitted for onveniene}

anddene for any

u ∈ {1, . . . , d}

^,

Z u = {k ∈ Z ^d | ∀i ∈ u , k _i ∈ Z

^and

∀i / ∈ u , k _i = 0}

Z ^∗ u = {k ∈ Z ^d | ∀i ∈ u , k _i ∈ Z ^∗

^and

∀i / ∈ u , k _i = 0}

andfor all

i ∈ N ^∗

^,

Z u (i) = Z u ∩

− i 2 , i

2 i d

Z ^∗ u (i) = Z ^∗ u ∩

− i 2 , i

2 i d

.

Lastly,adesign ofexperimentsisommonlydenotedby

D

^and,for

i ∈ N ^∗

^,thenotation

D(i)

^refers

to the regulargridin

[0, 1) ^d

D(i) =

0, 1

i , . . . , i − 1 i

d

.

2.2 Variane-based sensitivity indies

Let

X = (X ₁ , . . . , X _d ) ∈ [0, 1] ^d

^{be a}

d

-dimensional random vetor and let us onsider

Y = f (X)

where

f : [0, 1] ^d → R

^{isa measurable funtion suh that}

E[Y ² ] < +∞

^{. Under theassumption that}

X

^hasindependentomponents,theHoedingdeomposition[17,41℄statesthat

Y

^anbeuniquely

deomposed into summandsof inreasing dimensions

Y − E[Y ] = X d m=1

X

u ⊆{1,...,d}

| u |=m

f u (X _i , i ∈ u )

⁽¹⁾

where the

2 ^d − 1

^{randomvariables ontheright-hand sideof (1)shouldsatisfytheproperty}

∀ v u , E

f u (X _i , i ∈ u )|X i , i ∈ v

= 0 .

⁽²⁾

Notethat inthis ase the randomvariables

f u (X _i , i ∈ u )

^{have meanzero and are mutually unor-}

related. Thereforetakingthevarianeofbothsidesin(1)givesthevariane deomposition [14 ,34℄

of

Y

Var

[Y ] = X d m=1

X

u ⊆{1,...,d}

| u |=m

Var

f u (X _i , i ∈ u ) .

Finally,ifVar

[Y ] 6= 0

^{,we dene theso-alled} variane-basedsensitivityindies or Sobol' indies as

S

u (f, X) =

^Var

f u (X _i , i ∈ u )

Var

[Y ] .

Inpratie,global sensitivityanalysisfousesonomputingtherst-order (

| u | = 1

^{)andtheseond-}

order(

| u | = 2

^{) terms.}

2.3 Fourier series representation

From here on let us assume that the

X _i

^{'s are} independent and uniformly distributed on

[0, 1]

^.

Thereforethejoint probabilitydensity funtionof

X

^on

[0, 1] ^d

^{isequal to}

1

^{and, denoting}

P _n (f, X) =

n 1

X

k 1 =−n 1

· · ·

n d

X

k d =−n d

k (f)

^exp

(2iπk · X)

where

k (f ) = Z

[0,1] ^d

f (X)

^exp

(−2iπk · X)dX ,

the Riesz-Fishertheoremyields

P _n (f, X) −→ ^{L 2} Y .

Inpartiular, wehave

Y = X

k 1 ∈Z

· · · X

k d ∈Z

k (f)

^exp

(2iπk · X)

^{a.s. (3)}

andasthefollowingpropositionshows,thisFourierseriesrepresentationgivesanharmoniapproah

to handlethevariane-basedsensitivity indies.

Proposition1. Let

X ₁ , . . . , X _d

^be independentrandomvariables uniformlydistributed on

[0, 1]

^and

letus onsider

Y = f (X)

^where

f : [0, 1] ^d → R

^{isa measurable funtionsuh that}

E[Y ² ] < +∞

^and

Var

[Y ] 6= 0

^{. Thenfor any non-emptysubset}

u

of

{1, . . . , d}

^{we have}

S

u (f, X) = X

k∈Z ^∗ u

_k (f ) ² X

k∈(Z ^d ) ^∗

_k (f ) ² .

⁽⁴⁾

Proof. Inviewof(3),itiseasytonotiethattheomponentsintheHoedingdeompositionsatisfy

f u (X _i , i ∈ u ) = X

k∈Z ^∗ u

k (f )

^exp

(2iπk · X)

^a.s.

andtheonlusion follows fromParseval'sidentity.

As in (4) the index S

u (f, X)

^{does no more depend on}

X

^{we now simply denote the sensi-}

tivity indies by S

u (f )

^{. In the same way, we now denote V}

u (f )

^{and V}

(f )

^{the parts of variane}

Var

f u (X _i , i ∈ u )

and the total variane Var

[Y ]

^, respetively. Lastly, when

u = {i 1 , . . . , i _s }

^is

expliitelygiven,weuse themore ommon notation V

i 1 ...i s (f )

^andS

_{i 1 ...i s} (f )

^.

2.4 Estimation

We now dene basi estimatorsbased onProposition 1. For anynon-emptysubset

u

of

{1, . . . , d}

^,

let

K u

^{be anite subset of}

Z ^∗ u

^and

D

^{anite subset of}

[0, 1) ^d

^with

|D| = n

^{. Denoting}

b

^k (f, D) = 1

n X

x∈D

f(x)

^exp

(−2iπk · x),

⁽⁵⁾

we dene theestimatorof V

u (f )

^{asthetrunated series}

b

V

u (f, K u , D) = X

k∈K u

| b

k (f, D)| ² ,

⁽⁶⁾

theestimatorof V

(f )

^{asthe empirialvariane}

b

V

(f, D) = 1 n

X

x∈D

f (x) − 1 n

X

y∈D

f (y) 2

(7)

andtheestimator ofS

u (f )

^{naturally as}

b

S

u (f, K u , D) =

^V

b u (f, K u , D) b

V

(f, D) .

⁽⁸⁾

Example 1. If the design of experiments

D

^{is a set of} independent random pointsuniformly dis- tributed on

[0, 1] ^d

^and

K = G

u ⊆{1,...,d}

u 6=∅

K u ,

we have

b

V

u (f, K u , D) =

^V

u ( ˜ f )

where

f ˜ (X) = X

k∈K∪{0}

b

k (f, D)

^e

^2iπk·X

istheapproximationof

f (X)

^usingthequasi-regressionapproah [1℄basedon therandomsample

D

^.

Note that

| b

^k (f, D)| ²

^{isa biased estimatorof}

|

k (f, D)| ²

^{andit isreommended touse the unbiased}

estimator

n

n − 1 | b

k (f, D)| ² − 1 n ²

X

x∈D

f ² (x)

!

(see e.g. [22℄). In the sameway, the empirial variane ^V

b (f, D)

^{should be replaed by the unbiased}

sample variane

n

n−1

^V

b (f, D)

^.

Example2. If thedesignof experiments

D

^{istheregular grid}

D(q)

^with

n = q ^d

^,

q ∈ N ^∗

^and

iffor all non-emptysubsets

u

of

{1, . . . , d}

^,

K u = Z ^∗ u (q)

^and

K = G

u ⊆{1,...,d}

u 6=∅

K u

then by Parseval's identity,it an be easily shown that

b

S

u f, K u , D(q)

=

^S

u ( ˜ f)

where

f ˜ (x) = X

k∈K

b

^k f, D(q)

e

2iπk·x

is the trigonometri interpolation polynomial of

f (x)

^{(see e.g. [11 ℄) at the}

n = q ^d

^{equally spaed}

nodes

x ∈ D(q)

^.

3 New introdution to FAST and RBD

Inthe sequel,sine the

X _i

^'sareindependent and uniformlydistributedon

[0, 1]

^{, we have}

E

f (X)

= Z

[0,1] ^d

f (x)dx

sowe use no more probabilisti notation. Moreover, theintegrability assumption on

f

^{now reads}

f ∈ L ² ([0, 1] ^d )

^.

3.1 Review of FAST

3.1.1 Numerial integration

FAST is essentially an appliation of the following result due to Weyl [43 ℄ (see also the Weyl's

ergoditheorem [42℄ingerman or [32℄)

Theorem 1. [Weyl℄ Let

g

^{be a bounded Riemann integrable funtion on}

[0, 1] ^d

^{and for all}

i = 1, . . . , d

^,

x _i (t) = {ω i t}

^{where the}

ω _i

^{'s are real numberslinearly} independent over

Q

^and

{·}

^denotes

the frational part,then

Z

[0,1] ^d

g(x)dx = lim

T →∞

1 2T

Z T

−T

g x ₁ (t), . . . , x _d (t)

dt.

⁽⁹⁾

Inpartiular, for any

k ∈ Z ^d

^and

g : x 7→ f (x)

^exp

− 2iπk · x)

^,(9)reads

k (f ) = lim

T →∞

1 2T

Z T

−T

f ◦ x(t)

^exp

− 2iπ(k · ω)t

dt.

⁽¹⁰⁾

ThenFASTonsistsinreplaing

x i (t) = {ω i t}

^withsemiparametri funtions

x i (t) = G i sin(ω i t)

(see [8 ℄) where the

ω _i

^{'s are positive integers and the} transformations

G _i

^{are hosen to preserve}

themarginal distributions of the

X _i

^{'s. If the latterare uniformly} distributed asin the present paper , it an be shown (see [9℄ and [30℄) that

G _i (·) = _π ¹ arcsin(·) + ¹ ₂

^{. Saltelli et al. [30℄ also}

propose to add a random phase-shift

ϕ _i ∈ [0, 2π)

^{, getting the} semiparametri funtions

x ^∗ _i (t) =

1

π arcsin sin(2πω _i t + ϕ _i )

+ ¹ ₂

^{. Hene, replaing}

x

^with

x ^∗

^in(10)gives

k (f ) ≈ lim

T →∞

1 2T

Z _T

−T

f ◦ x ^∗ (t)

^exp

− 2iπ(k · ω)t dt.

Thus,sine the funtions

x ^∗ _i

^are

1

^{-periodi, itomes}

k (f ) ≈ Z 1

0

f ◦ x ^∗ (t)

^exp

− 2iπ(k · ω)t dt

andapplying the retanglerule to theright-hand sideintegral gives

k (f ) ≈ b

k·ω (f ◦ x ^∗ ).

⁽¹¹⁾

where

b

^k·ω (f ◦ x ^∗ ) = 1 n

n−1 X

j=0

f ◦ x ^∗ j n

exp

− 2iπj k · ω n

istheomplex disreteFourieroeient oftheone-dimensionalfuntion

f ◦ x ^∗

^{. Inthesequel,the}

dependene on

n

^,

ω

^and

ϕ

^{is generallyomitted foronveniene.}

The estimators of V

u (f )

^{, V}

(f )

^and onsequently of S

u (f )

^{were introdued by using the approx-}

imation in (11) (see [8℄ and Appendix C in [9℄). On the one hand, for any non-empty subset

u ⊆ {1, . . . , d}

^{andanynitesubset}

K u ⊆ Z ^∗ u

^{,(11)leadsto thedenitionoftheestimator ofV}

u (f ) b

V FAST

u (f, K u , x ^∗ ) = X

k∈K u

b

^k·ω (f ◦ x ^∗ ) ² .

⁽¹²⁾

Ontheotherhand, (11)gives

V

(f ) =

0 (f ² ) −

0 (f) ²

≈ b

0 (f ² ◦ x ^∗ ) − b

0 (f ◦ x ^∗ ) ²

andParseval'sidentity leads to the denitionoftheestimator ofV

(f ) b

V

FAST

(f, x ^∗ ) =

n−1 X

k=1

b

k (f ◦ x ^∗ ) ² .

Thisnaturally leads to theestimatorof thevariane-based sensitivityindies S

u (f ) b

S FAST

u (f, K u , x ^∗ ) = X

k∈K u

b

k·ω (f ◦ x ^∗ ) ²

n−1 X

k=1

b

k (f ◦ x ^∗ ) ² .

Asin Example 2,note that by Parseval's identity ^V

b

^FAST

(f, x ^∗ )

^{is equal to the empirial variane}

b

V

(f, {x ^∗ ( _n ^j )} j=0..n−1 )

^.

3.1.3 Choie of parameters

ω

^and

n

Asdisussed byShaiblyandShuler[31℄andCukier etal. [10 ℄,

ω

^and

n

^{shouldbeorretlyhosen}

soas to minimize theubature error inthe approximationin (11). In orderto avoid interferenes

i.e.

k · ω − k ^′ · ω = 0

^for

k

^,

k ^′ ∈ Z ^d

^,

k 6= k ^′

andaliasing i.e.

k · ω − k ^′ · ω = jn

^for

k

^,

k ^′ ∈ Z ^d

^,

k 6= k ^′

^and

j ∈ Z ^∗

that both lead to

b

^k·ω (f ◦ x ^∗ ) = b

^{k ′ ·ω} (f ◦ x ^∗ )

^{Shaiblyand Shuler[31℄ proposeto hoose}

ω 1

^,

...,

ω _d

^{free of}interferenes up toorder

N ∈ N ^∗

^:

(k − k ^′ ) · ω 6= 0

^{for all}

k, k ^′ ∈ Z ^d , k 6= k ^′

^,s.t.

X d i=1

|k i − k _i ^′ | ≤ N + 1

⁽¹³⁾

and

n

^{suiently large}

n ≈ N max(ω ₁ , . . . , ω _d ).

⁽¹⁴⁾

More reently, referring to the lassial information theory, Saltelli et al. [30 ℄ suggest to replae

(14)withNyquist-Shannon sampling theorem(see e.g. [24 ℄)

n > 2N max(ω ₁ , . . . , ω _d ).

⁽¹⁵⁾

Inour opinion,the riterion stated in(13) shouldbe written

(k − k ^′ ) · ω 6= 0

^{for all}

k, k ^′ ∈ Z ^d , k 6= k ^′

^,s.t.

X d i=1

|k i | ≤ N ^′

^and

X d i=1

|k ^′ _i | ≤ N ^′

⁽¹⁶⁾

sinethemainobjetiveistoavoidinterfereneswithinanitesubsetof

Z ^d

^{outofwhihtheFourier}

oeients of

f

^{are a priori negligible in (16), this subset is the losed}

l ¹

^{-norm ball of radius}

N ^′

^{. Thus we may} reformulate the wholeriterion stated in (13) and (15) with respet to the set

K = ⊔ u K u

^{where the}

K u

^{'s are thetrunation sets in the FAST estimator of V}

u (f )

^{given in(12).}

We proposeto hoose

ω ₁

^,...,

ω _d

^freeof interferenes within

K

^i.e.

(k − k ^′ ) · ω 6= 0

^{for all}

k, k ^′ ∈ K, k 6= k ^′

^and

n > max

k,k ^′ ∈K (k − k ^′ ) · ω

.

⁽¹⁷⁾

Inthe sequel,we refer tothelatter asthe"lassi"riterion of FAST.

3.2 Review of RBD

RBD makes use of the previous framework setting

ϕ = 0

^,

ω ₁ = · · · = ω _d = ω ∈ N ^∗

^usually

set to

1

^{and applying random} permutations on the oordinates of the resulting points

x ^∗ ( _n ^j )

^.

Morepreisely, let

σ ₁

^{, ...,}

σ _d

^{be random} permutations on

{0, . . . , n − 1}

^and

S

denote the set of all possible

σ = (σ 1 , . . . , σ _d )

^{. Given}

σ ∈ S

, onsider the funtion

x ^× = (x ^× ₁ , . . . , x ^× _d )

^{dened on}

{0, _n ¹ , . . . , ⁿ⁻¹ _n }

^{suh thatfor all}

i ∈ {1, . . . , d}

^and

j ∈ {0, . . . , n − 1}

^,

x ^× _i j n

= 1

π arcsin sin

2πω σ _i (j) n

+ 1 2 .

Thus denoting

σ _i ⁻¹

^{the inverse} permutation of

σ _i

^,dene

x ^×,i j

n

= x ^× σ ⁻¹ _i (j) n

.

Finallythrougha heuristi argument Tarantola etal. [38℄ introdue theRBDestimators of V

u (f )

^,

V

(f )

^{and S}

u (f )

^{for rst-order terms i.e.}

u = {i}

^,

i ∈ {1, . . . , d}

^{. For any nite subset}

K _{i} ⊆ Z ^∗ _{i}

^{,we have}

b

V RBD

i (f, K _{i} , x ^× ) = X

k∈K _{i}

b

k i ω (f ◦ x ^×,i ) ² ,

b

V

RBD

(f, x ^× ) =

n−1 X

k=1

b

k (f ◦ x ^× ) ² .

and

b

S RBD

i (f, K _{i} , x ^× ) = X

k∈K {i}

b

^k i ω (f ◦ x ^×,i ) ²

n−1 X

k=1

b

k (f ◦ x ^× ) ² .

Asin FAST note that by Parseval's identity, the estimator ^V

b

^RBD

(f, x ^× )

^{is equal to the empirial}

variane ^V

b (f, {x ^× ( _n ^j )} j=0..n−1 )

^{. In thesequel,the dependene on}

ω

^and

σ

^{isgenerally omittedfor}

onveniene.

3.3 FAST and RBD revisited

3.3.1 Main result

Firstweintrodue morenotation. For any

p ∈ N ^∗

^,let

r _p : [0, 1] −→ [0, 1]

x 7−→

2{px}

^if

0 ≤ {px} < ¹ ₂ 2 − 2{px}

^if

¹ ₂ ≤ {px} ≤ 1

andfor any

ϕ ∈ [0, 2π)

t _ϕ : [0, 1] −→ [0, 1]

x 7−→ {x + ˜ ϕ}

^with

ϕ ˜ = ¹ ₄ + _2π ^ϕ .

Then we dene the linear operators

R _p

^and

T ϕ

^{(see Figure 1) on}

L ² ([0, 1] ^d )

^{suh that for all}

x ∈ [0, 1] ^d

^,

R p f (x) = f r _p (x ₁ ) . . . , r _p (x _d )

et

T ϕ f (x) = f t _{ϕ 1} (x ₁ ), . . . , t _{ϕ d} (x _d ) .

andnote that

R p = R 1 ◦ · · · ◦ R 1

| {z }

p

^times

. We also introdue two lassialdesigns ofexperiments. For any

ω ∈ (N ^∗ ) ^d

^{,we denote}

G(ω) = n j n ω ₁ o

, . . . , n j n ω _d o

, j ∈ {0, . . . , n − 1}

.

theylisubgroupoforder

n/gcd(ω ₁ , . . . , ω _d , n

ofthetorus

T ^d = (R/Z) ^d ≃ [0, 1) ^d

^generated

by

({ ^ω _n ¹ }, . . . , { ^ω _n ^d })

^{(see e.g. [15 ℄). For any}

σ ∈ S

we alsodenote

A(σ) = σ 1 (j)

n , . . . , σ _d (j) n

, j ∈ {0, . . . , n − 1}

0 0.5 1 0

2 4 6

(a)Plotof

f : x 7→ x + sin(x)

0 0.5 1

0 2 4 6

(b)Plotof

R ₁ f

0 0.5 1

0 2 4 6

()Plotof

T ^π

10 f

0 0.5 1

0 2 4 6

(d)Plotof

( T ^π

30 ◦ R ₁ )f

Figure1: Examples of operators

R p

^and

T ϕ

^indimension

1

^.

theorthogonal array of strength

1

^{and index unity with elements taken from}

{0, ¹ _n , . . . , ⁿ⁻¹ _n }

^and

basedon thepermutation

σ

^{(see e.g. [16℄). FAST andRBD methods are now introdued ina new}

way byusing the basi estimatorin(8).

Proposition 2. Let

f : [0, 1] ^d → R

^{be a} square-integrable funtion. For any non-empty subset

u ⊆ {1, . . . , d}

^{, any nite subset}

K u ⊆ Z ^∗ u

^,

ϕ ∈ [0, 2π) ^d

^and

ω ∈ (N ^∗ ) ^d

^{, we have}

b

S FAST

u (f, K u , x ^∗ ) = b

^S

u (T ϕ ◦ R ₁ )f, K u , G(ω)

.

⁽¹⁸⁾

For any

i ∈ {1, . . . , d}

^{, any nite subset}

K _{i} ⊆ Z ^∗ _{i}

^,

σ ∈ S

and

ω ∈ N ^∗

^{,we have}

b

S RBD

i (f, K _{i} , x ^× ) = b

^S

_i (T ω ˜ ◦ R ω )f, ωK _{i} , A(σ)

.

⁽¹⁹⁾

where

ω ˜ = _(1−ω)π

2ω , · · · , ^(1−ω)π _2ω

and

ωK _{i} = {(ωk 1 , . . . , ωk _d ), k ∈ K _{i} }

^.

Proof. Itessentiallyonsistsinshowing thatfor all

j ∈ {0, . . . n − 1}

f ◦ x ^∗ j n

= (T ϕ ◦ R 1 )f n j n ω ₁ o

, . . . , n j

n ω _d o

and

f ◦ x ^× j n

= (T ω ˜ ◦ R ω )f σ ₁ (j)

n , . . . , σ _d (j) n

.

See detailsinAppendix A.1.

Remark1.IntheRBDmethod,theparameter

ω

^{isusuallysetto}

1

^{butitsroleisnotwellunderstood}

upto now. In our opinion there isno reason toset

ω 6= 1

^{sine if}

gcd(ω, n) = 1

^{then it leads to the}

ase

ω = 1

^{, and otherwise the estimatorin (19) is} potentially less eient than in the ase

ω = 1

(see details in Appendix A.2.).

3.3.2 What FAST and RBDare

It is lear from Proposition 2 that FAST and RBD only onsists in applying the basi estimator

introdued in(8)to a partiular transform

(T ϕ ◦ R p )f

^{ofthe funtion}

f

^{anda partiular designof}

experiments

G(ω)

^or

A(σ)

^{. Now it is also lear that the basi estimator generates an error term}

due to trunations in(6) and an other one due to numerial integrations in(5) and (7).

Moreover, theuse of

(T ϕ ◦ R p )f

^{instead of}

f

^{ould also have an impat on the} sensitivity indies estimationerror. Wenowinvestigatethis latterissuebyintroduingthenotionof invariane ofthe

variane deomposition.

Denition 1. Let

L

^{be a linear operator on}

L ² ([0, 1] ^d )

^{. The variane} deomposition is said to be

L

^{-invariant on}

L ² ([0, 1] ^d )

^{if for any non-empty set}

u ⊆ {1, . . . , d}

^{and any funtion}

f ∈ L ² ([0, 1] ^d )

we have

V

u (Lf ) =

^V

u (f ).

Thisleads to thefollowing result

Lemma1. Forany

p ∈ N ^∗

^andany

ϕ ∈ [0, 2π) ^d

^,thevarianedeompositionis

R p

^and

T ϕ

^-invariant

on

L ² ([0, 1] ^d )

^.

Proof. See Appendix A.3.

Asa onsequene,for anynon-emptysubset

u ⊆ {1, . . . , d}

^{,we have}

S

u (T ϕ ◦ R _p )f

=

^S

u (f )

andthisassertsthevalidityofFASTandRBDmethods. Notethatthelinearoperator

R p

"regular- ize"thefuntion

f

^{inthesense thatif}

x 7→ f (x)

^{isontinuouson}

[0, 1] ^d

^and

x → f ({x 1 }, . . . , {x d })

is disontinuous on

R ^d

^then

x → R p f ({x 1 }, . . . , {x d })

^{is ontinuous on}

R ^d

^{. This is an impor-}

tant property sine by Riemann-Lebesgue lemma

|

k (f )|

^{onverges to}

0

^as

||k||

^{tends to}

∞

^{, and}

the smoother the funtion

f

^{, the faster the onvergene (see e.g. [45℄). The other operator}

T ϕ

essentially allows to dene randomized estimatorsin FAST.

3.3.3 Potential generalizations

To endwith, we listthree natural generalizationsthatarefurther disussed inthenextsetion:

- theestimator

b

^S

u (T ϕ ◦ R 1 )f, K u , G(ω)

an alsobe dened fora group

G

^{of any rank}

r ≤ d

- the estimator

b

^S

_i (T ω ˜ ◦ R ω )f, ωK _{i} , A(σ)

an also be dened for a sensitivity index of any

order:

b

^S

u (T ω ˜ ◦ R ω )f, ωK u , A(σ)

,notethatithasbeen alreadyapplied in[44℄

- thelatterestimator

b

^S

u (T ω ˜ ◦ R ω )f, ωK u , A(σ)

analsobedened foranorthogonalarray

A

having anyparameters.

4 Error analysis

For onveniene, operators

T ϕ

^and

R p

^{are now omitted. Moreover, we assumethat thefuntion}

f

hasanabsolutely onvergent Fourier representation,i.e.

X

k∈Z ^d

|

k (f )| < +∞ .

4.1 Cubature error in FAST

4.1.1 Two points of view

Inthis setionwe mainly fous onthe error term

e _k (f, G) = b

k (f, G) −

k (f )

⁽²⁰⁾

where

G

^{is a subgroup of}

T ^d

^{of order}

n

^and

k ∈ Z ^d

^{. By its denition, the term}

b

k (f, G)

^onsists

of an equal weight ubature rule at the

n

^{nodes of the group}

G

^{, also known as a lattie rule (see}

[33 ℄forasurvey). MoreoverbythegeneralizedPoissonsummationformula(seee.g. [23℄),theerror

term in(20)ispreisely

e k (f, G) = X

h∈G ^⊥ \{0}

k+h (f )

⁽²¹⁾

where

G ^⊥ = {h ∈ Z ^d | ∀x ∈ G, h · x ≡ 0 (

^mod

1)}

^{is the subgroup of}

Z ^d

^{orthogonal to}

G

^{, also}

knownastheduallattie of

G

^.

In the lattie rules eld,

e ₀ (f, G)

^{is the only term of interest, and there exist two main points of}

viewtoontrolit. Oneonsistsinlookingfor"good"groups

G

^{suhthattheubatureruleisexat}

fora set oftrigonometri polynomials,i.e. for a nitesubset

K

^of

Z ^d

^,

e 0 (f, G) = 0

^{for all}

f

^{suh that}

∀k ∈ / K,

k (f ) = 0 .

Theother point ofview aimsto nd"good"groups

G

^{suh thattheubature rulehasan absolute}

error

|e 0 (f, G)|

^{dominated byan expliitbound forall}

f

^{inapartiular spae ofsmooth funtions.}

Notethatthese approahes areompatible to eahother (see e.g. [7℄ andthereferenes therein).

NowonerningthestudyoferrorinFAST,therstpointofview,whihessentiallyorresponds

to the lassi FAST, onsists of a trigonometri interpolation issue and leads to a metamodel ap-

proah of the estimation of thesensitivityindies. The seond one, whih ismore original, allows

to derive error bounds for ^V

b u (f, K u , G)

^{and V}

b (f, G)

^{in spaes of smooth funtions. Both these}

methods are disussedbelow.

4.1.2 Metamodel approah

Let

K

^{be a nitesubset of}

Z ^d

^{. Thenan immediate onsequene of(21) isthat agroup}

G

^satises

theproperty

e k (f, G) = 0

^{for all}

k ∈ K

^{andfor all}

f

^{suh that}

∀k ∈ / K,

k (f ) = 0

ifandonly if

∀k, k ^′ ∈ K, k 6= k ^′ , ∃x ∈ G, (k − k ^′ ) · x 6≡ 0 (

^mod

1) .

⁽²²⁾

Morefundamentally,for any

E ⊆ Z ^d

^,onsiderthe trigonometri polynomial

f ˜ E (x) = X

k∈E

b

^k (f, G)

^exp

(2iπk · x) ,

⁽²³⁾

thentheequivalene above leads tothefollowing result

Proposition 3. Let

G

^{be a subgroup of the torus}

T ^d

^{of order}

|G| = n

^and

K = ∪ u 6=∅ K u

^satisfying

the riterion (22) where for all non-emptysubsets

u

of

{1, . . . , d}

^,

K u ⊆ Z ^∗ u

i) if

|K| = n

^,then

f ˜ _K

^isatrigonometriinterpolationpolynomialof

f

^atthe

n

^nodes

x ∈ G

^and

we have

b

S

u (f, K u , G) =

^S

u ( ˜ f _K ).

ii) if

|K| < n

^{, let}

H

^{be any subset of}

Z ^d

^{suh that}

K ⊆ H

^,

H

^{satises the riterion (22) and}

|H| = n

^{. Then}

f ˜ _H

^isa trigonometriinterpolationpolynomial of

f

^atthe

n

^nodes

x ∈ G

^and

we have

b

V

u (f, K u , G) =

^V

u ( ˜ f _K )

^{and V}

b (f, G) =

^V

( ˜ f _H ).

Proof. Theonlydiultyistoprove thatthetrigonometri polynomials

f ˜ K

^{intheassertion i)and}

f ˜ _H

^{in the assertion ii) are in}terpolation polynomials at the points

x ∈ G

^{. We} demonstrate it for

f ˜ _K

^,theprooffor

f ˜ _H

^{isexatlythesame.}

Sinethefuntion

f

^{hasabsolutely onvergent Fourier} representation, we an write

f (x) = X

k∈Z ^d

k (f )

^exp

(2iπk · x) = X

k∈K

X

h∈G ^⊥

k+h (f )

^exp

2iπ(k + h) · x

(24)

(seedetails inAppendixA.4) and bydenition of

G ^⊥

^{,we have thatfor any}

x ∈ G

^,

f (x) = X

k∈K

X

h∈G ^⊥

k+h (f )

^exp

(2iπk · x).

Theonlusion followsfrom thedenitionin(23)sine (20) and(21) give

X

h∈G ^⊥

k+h (f ) = b

k (f, G).

From this pointof view,FAST returnsanalytial valuesfrom trigonometri metamodels of the

funtion

(T ϕ ◦ R 1 )f

^{and theerroranalysis shouldbeperformedon themetamodelitself.}

In pratie, a set of a priori non-negligible frequenies

K = ∪ u 6=∅ K u

^{is given and a group}

G

satisfyingtheriterion (22)andwiththesmallest order

|G| = n

^{hastobefound. Searhingforthis}

group

G

^isomputationaly expensiveandmayrapidlybeomeunfeasible. Oneoftheheapestway isto look for yli groups

G = G(ω)

^{, oming bakto thelassi FAST. In this ase, theriterion}

(22)simplyreads

∀k, k ^′ ∈ K, k 6= k ^′ , (k − k ^′ ) · ω 6≡ 0 (

^mod

n) .

⁽²⁵⁾

Note that this new riterion plays the same role as the lassi riterion of FAST given in (17).

Themain dierene between these two approahes is thatoptimization on

n

^{is performed in(25),}

onsequently this new riterion allows to nd group

G

^{with smaller order}

n

^{. We illustrate the}

eieny of both riterions by using basi exhaustive algorithms with omputational omplexity

O(n ^d )

^{. TheresultsaregatheredinTable1andshowthatthenewriterionleadstoa}non-negligible improvement.

d = 2 d = 3 d = 4 d = 5 N1 N ₂ |K| n ^old n ^new |K| n ^old n ^new |K| n ^old n ^new |K| n ^old n ^new

4 2 20 41 29 36 65 50 56 105 63 80 177 111

5 3 32 61 48 66 141 102 112 241 173 170 471 302

6 4 48 85 65 108 241 155 192 541 323 300 997 613

7 5 68 113 89 162 421 284 296 1177 586 470 1891 1279

8 6 92 145 120 228 625 429 424 1985 1033 680 3457 2222

9 7 120 181 149 306 937 645 576 3007 1706 930

10 8 152 221 185 396 1281 933 752 4501 2529 1220

11 9 188 265 228 498 1805 1284 952 7261 3684 1550

Table 1: Comparison in dimension

d = 2

^,

3

^,

4

^and

5

^{between the minimum sample size}

n

^given

by thelassi riterion of FAST(denoted

n _old

^{) and thenew one proposed in(25) (denoted}

n _new

^).

Here, the

K _{i}

^{'sare equalto}

Z ^∗ _{i} ∩ {|k i | ≤ N 1 }

^,the

K _{i,j}

^{'sareequal to}

Z ^∗ _{i,j} ∩ {|k i | + |k j | ≤ N 2 }

andfor all

u

suhthat

| u | > 2

^,

K u = ∅

^{. Suhsets}

K

^arepartiularlywell-suitedtoanalysefuntions whose eetive dimensionis lessthan

2

^{seeDenition 4 inSetion 4.2.2.}

Remark 2. Even if yli groups seem to be suitable in the previous issue, the omputational ost

of the researh of a generator

ω

^{an beome} prohibitive in high-dimensionalproblems. In this ase,

alternativealgorithmsan beused insteadofa systematiresearhtehnique (for a reent referene,

see e.g. [20℄).

4.1.3 Error bounds

Searhing for a nite subgroup

G

^{of the torus}

T ^d

^{suh that}

e 0 (f, G)

^{has an expliit bound in a}

partiular funtionspaeisa problemknownastheonstrution ofgoodlattie rules (forasurvey

see [33℄ or more reently [25℄). Most of the results inthis eld are established in Korobov spaes

whih are suitable to handle lattie methods; so we derive error bounds for sensitivity indies in

thesespaes. For

α > 1

^and

γ = (γ u ) u ⊆{1,...,d}

^withnon-negative

γ u

^{'s, dene theweightedKorobov}

spae

H α,γ

^{to bethe Hilbertspae withreproduing kernel}

RK _α,γ (x, y) = 1 + X

k∈(Z ^d ) ^∗

r(k, α, γ) ⁻¹

^exp

2iπk · (x − y)

where for any

k 6= 0

^,

r(k, α, γ) = γ u ⁻¹ _k

Q

i∈ u _k |k i | ^α

^,where

u _k

is suh that

k ∈ Z ^∗ u _k

^{. For}

k

^{suh that}

γ u _k = 0

^{,we set byonvention}

r(k, α, γ) = ∞

^{. Thus thekernel an be rewritten}

RK _α,γ (x, y) = 1 + X

k∈(Z ^d ) ^∗ γ u

k 6=0

r(k, α, γ) ⁻¹

^exp

2iπk · (x − y)

andwe deduethat thenormof

f ∈ H α,γ

^satises

||f|| ² _{H α,γ} =

₀ (f ) ² + X

k∈(Z ^d ) ^∗ γ ^u _k 6=0

r(k, α, γ)|

k (f )| ² < +∞

andonsequently

∀k ∈ (Z ^d ) ^∗ such that γ u _k 6= 0, |

k (f )| ² ≤ γ u _k ||f || ² _{H α,γ} Y

i∈ u _k

|k i | ^α .

Note that for any

k ∈ (Z ^d ) ^∗

^{suh that}

γ u _k = 0

^,

f ∈ H α,γ

^implies

k (f ) = 0

^{. We also make a}

restritiononthesetsoffrequenies

K u

^{'s. Hereweassumethatforanynon-emptyset}

u ⊆ {1, . . . , d}

^,

K u

^{isof Zaremba ross-type (seeFigure 2)}

K u = Z u ,β ^u = (

k ∈ Z ^∗ u , Y

i∈ u

|k i | ≤ β u

)

where

β u ≥ 1

^{. Thiskindofsparsegridsis}partiularlywell-suitedfortheanalysisofhigh-dimensional smooth funtions. We now give theresulton error boundsfor ^V

b u (f, K u , G)

^andV

b (f, G)

ⁱⁿ

H α

^.

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

(a)Plotof

Z _{1,2},4

−10 −5 0 5 10

−10

−5 0 5 10

(b)Plotof

Z _{1,3},9

Figure2: Illustrationof rosses

Z u ,β u

^.

Proposition 4. Let

f ∈ H _α,γ

^with

α > 2

^and

γ = (γ u ) u ⊆{1,...,d}

^with non-negative omponents.

Let

G

^{be a subgroup of}

T ^d

^{of order}

n

^{suh that the ubature error related to}

G

^{is dominated by the}

expliit bound

B (α, n, d, γ)

^{on the unit ball of}

H α,γ

^{i.e. for all}

f

ⁱⁿ

H α,γ , | b

⁰ (f, G) −

0 (f )| ≤ B(α, n, d, γ)||f || H α,γ

^{. Then}

i) if there exists

α ^′ > 2

^and

γ ^′ = (γ u ^′ ) u ⊆{1,...,d}

^withnon-negative omponents suhthat

f ² ∈ H _α ^′ _,γ ^′

^,

b

^V

(f, G) −

^V

(f ) ≤ ||f || ² _{H α} B(α, n, d, γ) 2 + B(α, n, d, γ)

+ ||f ² || H _α ′ B (α ^′ , n, d, γ ^′ )

ii) for any non-empty set

u ⊆ {1, . . . , d}

^and

K u = Z u ,β ^u

^{, we have}

b

^V

u (f, K u , G) −

^V

u (f ) ≤ ||f || ² _{H α,γ} h

C(α, γ, β u , | u |) + B (α, n, d, γ) ² S ₁ (α, γ, β u , u ) + B(α, n, d, γ)S 2 (α, γ, β u , u ) i

where

S ₁ (α, γ, β u , u ) = γ _{f rac} X

k∈K u

Y

i∈ u

|k _i | + 1 α

, γ _{f rac} = max

u , v ⊂{1,...,d}

γ ^v 6=0

γ u /γ v

S ₂ (α, γ, β u , u ) = γ _{f rac} γ u ^1/2 2 ^{α| u |/2} |K u |

andfor

| u | ≤ 2

^{, the trunation error term}

C(α, β u , | u |)

^are

C(α, γ, β u , 1) = 2γ _max ζ (α)

β u ^α−1

, γ _max = max

u ⊂{1,...,d} γ u

⁽²⁶⁾

C(α, γ, β u , 2) = 4γ _max

ζ(α) ² + ζ(α)

^log

(β u ) + 2 β u ^α−1

.

⁽²⁷⁾

Proof. See Appendix A.5.

It is also possible to derive expliit formulas of the trunation error term for

| u | > 2

^{, but this}

is more ompliated and of seond interest. Seondly, it has to be noted that, inthe seond item

ofProposition4,the funtions

S ₁

^and

S ₂

^{areinreasing withrespettotheparameter}

β u

^whilethe

funtion

C

^{isdereasing. Asaonsequene,eient boundsonsistofatrade-obetween}

β u

^and

n

suhthat

B(α, n, d, γ) ² S ₁ (α, γ, β u , u )

^,

B(α, n, d, γ)S ₂ (α, γ, β u , u )

^and

C(α, γ, β u , | u |)

^{have thesame}

order. For example,

i) if

| u | = 1

^and

α > 2

^,notethat

|K ^u | = 2β u

^anddedue

S ₁ (α, γ, β u , u ) ≤ 2 ^{α| u |+1} β u ^1+α

^,andreall

that

C(α, γ, β u , 1) = O(β u ^1−α )

^{. Thus thetrade-ogives}

b

^V

u (f, K u , G) −

^V

u (f ) = O

B (α, n, d, γ) ^{1− α 1} .

ii) if

| u | = 2

^and

α > 2

^{, note that}

|K u | ≤ 4β u (

^log

(β u ) + 1)

^{see argument for (A.21)}

in Appendix A.5 and dedue

S ₁ (α, γ, β u , u ) ≤ 2 ^{α| u |+2} β u ^1+α (

^log

(β u ) + 1)

^{and reall that}

C(α, γ, β u , 1) = O β u ^1−α

^log

(β u )

. Thus thetrade-o gives

b

^V

u (f, K u , G) −

^V

u (f ) = O

log

B (α, n, d, γ) ^−1/α

B(α, n, d, γ) ^{1− α 1}

.

Remark 3. In unweighted Korobov spaes i.e.

γ = 1

^{, it is knownthat the optimal rate of onver-}

gene of a rank-1 lattie rule is

B(α, n, d, γ) = O

(log n) ^dα/2 n ^α/2

(see e.g. [33 ℄). For unweighted Korobov spaes, there exist better rates of onvergene for produt

weights i.e.

γ u = Q

i∈ u γ i

^{(see [21 ℄) or for nite-orderweights i.e}

∀ u

with

| u | > d ^∗ (d ^∗ ≤ d)

^,

γ u = 0

(see [13℄). The latter are essentially related toan assumptionon theeetivedimension of

f

^{in the}

trunation sense and in the superposition sense, respetively (see [5℄ for the denition of eetive

dimension).

4.2 Bias in RBD

We now give some results onthewell-known issuerelatedto thebias oftheestimates inRBD.

4.2.1 Preliminaries

We begin withthe denitionsof an orthogonalarrayand the"oinidene defet"

Denition 2. An orthogonal array in dimension

d

^{, with}

q

^{levels, strength}

t ≤ d

^{and index}

λ

^isa

matrix with

n = λq ^t

^rowsand

d

^{olumns suh that in every}

n

^-by-

t

^{submatrix eah of the}

q ^t

^possible

rows i.e. the distint

t

^-uples

(l ₁ , . . . , l _t )

^{where the}

l _i

^{'s take their values in the set of the}

q

^levels

ours exatly the samenumber

λ

^{of times.}

Denition 3. Let

A

^{be an orthogonal array in dimension}

d

^{, with}

q

^{levels, strength}

t

^{and index}

λ

^{. We say that}

A

^{has the oinidene defet when there exist two rows of}

A

^{that do agree in}

t + 1

olumns; otherwise we say that

A

^is defet-free.

Let

Π(q)

^{be the set of} permutations on

{0, ¹ _q , . . . , ^q−1 _q }

^,

Π = Π(q, d)

^{the artesian produt}

(Π(q)) ^d

^and

µ = µ(q, d)

^{thenormalized ountingmeasureon}

Π(q, d)

^{. Let}

A

^{bean orthogonalarray}

indimension

d

^,with

q

^levels

{0, ¹ _q . . . , ^q−1 _q }

^,strength

t

^andindex

λ

^,anddenote

n = λq ^t

^itsnumber

ofrows. Foranypermutation

π = (π ₁ , . . . π _d ) ∈ Π

^,denote

A(π)

^{theorthogonalarrayobtainedfrom}

A

^{after applyingeah} permutation

π _j

^{on thelevelsof the}orresponding

j

^{-thfator i.e.}

for all

1 ≤ i ≤ n

^and

1 ≤ j ≤ d, A(π)

ij = π _j (A _ij ) .

Notethatthe

A(π)

^'sand

A

^{areorthogonalarrayswiththesameparameters (see[16 ℄).} Conversely, itisalsoeasy toshowthatif

A

^{hasstrengthandindex equalto}

1

^{i.e. asinthelassiRBDwith}

anodd integer

1

n

^{;anyother orthogonalarray}

A ^′

^{withthesame paramaters as}

A

^{isof theform}

A(π)

^{for a}permutation

π ∈ Π

^{. We arenowinterested inthequantities}

E µ

h

^V

b (f, A(π)) i

and

E µ

h

^V

b u (f, K u , A(π)) i ,

where

K u

^{isa nite subsetof}

Z ^∗ u

^.

4.2.2 Bias of the estimator in RBD

Let

b

^k (f ) = b

^k (f, D(q))

^{denote the}

k

^{-th omplex disrete Fourier oeient; we begin with the}

following important lemma

Theorem 2. [Owen℄ Followingthe previous notation, we have

Var

µ

h b

⁰ f, A(π) i

= 1 n ²

X

| u |>t

X ^{| u |}

r=0

B( u , r)(1 − q) ^{r−| u |} X

k∈Z ^∗ u (q)

| b

k (f )| ²

where

B ( u , r) = X n i=1

X n j=1

1 _|{l∈ u , A il =A jl }|=r

onsists of the number of pairs of rows

(A _i , A _j )

^{that mathon exatly}

r

^{of the axes in}

u

.

Proof. This is exatly Theorem 1 given by Owen in [26℄. Just note that, the embedded ANOVA

termsona

q ^d

^{regulargrid denoted}

β u

^{byOwen are}

β u (x) = X

k∈Z ^∗ u (q)

b

k (f )

^exp

(2iπk · x).

Indeed,for all

x

^{inthe regulargrid}

{0, ¹ _q , . . . , ^q−1 _q } ^d

^,

f (x) = X

u ⊆{1,...,d}

β u (x)

by a trigonometri interpolation argument, and it is also easy to show that the random variables

β u (X _i , i ∈ u )

^{satisfy theproperty (2)for} independent random variables

X _i

^uniformly distributed on

{0, ¹ _q . . . , ^q−1 _q }

^.

Thenwehavethe followingpropositioninwhihthebiasofthevarianeestimateisinvestigated

inunweightedKorobovspaes

H α = H α,1

^{(see Setion 4.1.3.)}

1

If

n

^{iseven,thedesignof} experimentsinRBDonsistsof anorthogonalarraywith

n/2

^{levels,strength1and}

index2,andmaybefaedwiththeoinidenedefet.

Proposition5. Let

A

^{be a defet-free orthogonal array in dimension}

d

^{withparameters}

q

^,

t

^and

λ

in

N ^∗

^with

t < d

^{. If there exists}

α > 2t + 1

^{suh that}

f

^and

f ²

^{are in}

H α

^{, we have}

E µ

h

^V

b (f, A(π)) i

=

^V

(f ) − 1 n

X

1≤| u |>t

V

u (f ) + O n ^{−(1+ 1 t )} .

Proof. See Appendix A.6.

As a onsequene, onsidering thelassi denition of eetive dimension inthe superposition

sense(see e.g. [5 ℄)

Denition 4. The eetive dimension of

f

^{, in the} superposition sense, is the smallest

d S (f )

^suh

that

X

1≤| u |≤d _S (f)

V

u (f ) ≥ l S (f )

^V

(f )

where

l _S (f )

^{isan arbitrary onstant generally set at}

0.99

^.

we have theorollary

Corollary1. UndertheassumptionsofProposition5,let

d _S (f )

^and

l _S (f )

^{bedenedasinDenition}

4. If

t ≥ d S

^{, we have}

E µ

h

^V

b (f, A(π)) i

=

1 − ε n

V

(f ) + O n ^{−(1+ 1 t )} ,

where

0 ≤ ε ≤ 1 − l _S (f)

^.

Proof. Straightforward fromProposition 5.

Ina seond time,sine

E µ

h

^V

b u (f, K u , A(π)) i

= X

k∈K u

E µ

h b

k f, A(π) ² i

the analysisofthe biasof the partsof varianeestimates restson thefollowing result

Proposition 6. Let

A

^{be a defet-free orthogonal array in dimension}

d

^{withparameters}

q

^,

t

^and

λ

in

N ^∗

^with

t < d

^{. Let}

u

be a non-empty subset of

{1, . . . , d}

^and

k ∈ Z ^∗ u

^{. If there exists}

α > 2t + 1

suh that

f

^and

f ²

^{are in}

H α

^{, we have}

E µ

h b

k (f, A(π)) ² i

= n − 1

n |

k (f )| ² + 1

n

^V

(f ) +

₀ (f) ² − R ₁ − R ₂

+ O n ^{−(1+ 1 t )}