Derivative based global sensitivity

Transcription

1 Ttle: Dervatve based global senstvty measures Name: Sergeï Kucherenko 1 and Bertrand Iooss 2,3 Affl./Addr. 1: Imperal College London London, SW7 2AZ, UK E-mal: s.kucherenko@mperal.ac.uk Affl./Addr. 2: EDF R&D 6 qua Water, Chatou, France arxv: v3 [math.st] 22 Jul 2015 E-mal: bertrand.ooss@edf.fr Affl./Addr. 3: Insttut de Mathématques de Toulouse Unversté Paul Sabater 118 route de Narbonne, Toulouse, France Dervatve based global senstvty measures Abstract The method of dervatve based global senstvty measures (DGSM) has recently become popular among practtoners. It has a strong lnk wth the Morrs screenng method and Sobol senstvty ndces and has several advantages over them. DGSM are very easy to mplement and evaluate numercally. The computatonal tme requred for numercal evaluaton of DGSM s generally much lower than that for estmaton of Sobol senstvty ndces. Ths paper presents a survey of recent advances n DGSM concernng lower and upper bounds on the values of Sobol total senstvty ndces S tot. Usng these bounds t s possble n most cases to get a good practcal estmaton of the values of S tot. Several examples are used to llustrate an applcaton of DGSM.

2 Keywords: Senstvty analyss, Sobol ndces, Morrs method, Model dervatves, DGSM, Poncaré nequalty 2 Introducton Global senstvty analyss (SA) offers a comprehensve approach to the model analyss. Unlke local SA, global SA methods evaluate the effect of a factor whle all other factors are vared as well and thus they account for nteractons between varables and do not depend on the choce of a nomnal pont. Revews of dfferent global SA methods can be found n Saltell et al [30] and Sobol and Kucherenko [37]. The method of global senstvty ndces suggested by Sobol [33, 34], and then further developed by Homma and Saltell [11] s one of the most effcent and popular global SA technques. It belongs to the class of varance-based methods. These methods provde nformaton on the mportance of dfferent subsets of nput varables to the output varance. There are two types of Sobol senstvty ndces: the man effect ndces, whch estmate the ndvdual contrbuton of each nput parameter to the output varance, and the total senstvty ndces, whch measure the total contrbuton of a sngle nput factor or a group of nputs. The total senstvty ndces are used to dentfy non-mportant varables whch can then be fxed at ther nomnal values to reduce model complexty. Ths approach s known as factors fxng settng [30]. For hgh-dmensonal models the drect applcaton of varance-based global SA measures can be extremely tmeconsumng and mpractcal. A number of alternatve SA technques have been proposed. One of them s the screenng method by Morrs [21]. It can be regarded as global as the fnal measure s obtaned by averagng local measures (the elementary effects). Ths method s consderably cheaper than the varance based methods n terms of computatonal tme. The Morrs method can be used for dentfyng unmportant varables. However, the Morrs

3 3 method has two man drawbacks. Frstly, t uses random samplng of ponts from the fxed grd (levels) for averagng elementary effects whch are calculated as fnte dfferences wth the ncrement delta comparable wth the range of uncertanty. For ths reason t can not correctly account for the effects wth characterstc dmensons much less than delta. Secondly, t lacks the ablty of the Sobol method to provde nformaton about man effects (contrbuton of ndvdual varables to uncertanty) and t can t dstngush between low and hgh order nteractons. Ths paper presents a survey of dervatve based global senstvty measures (DGSM) and ther lnk wth Sobol senstvty ndces. DGSM are based on averagng local dervatves usng Monte Carlo or Quas Monte Carlo samplng methods. Ths technque s much more accurate than the Morrs method as the elementary effects are evaluated as strct local dervatves wth small ncrements compared to the varable uncertanty ranges. Local dervatves are evaluated at randomly or quas randomly selected ponts n the whole range of uncertanty and not at the ponts from a fxed grd. The so-called alternatve global senstvty estmator defned as a normalzed ntegral of partal dervatves was frstly ntroduced by Sobol and Gershman [36]. Kucherenko et al [17] ntroduced some other DGSM and coned the acronym DGSM. They showed that DGSM can be seen as the generalzaton of the Morrs method [21]. Kucherenko et al [17] also establshed emprcally the lnk between DGSM and Sobol senstvty ndces. They showed that the computatonal cost of numercal evaluaton of DGSM can be much lower than that for estmaton of Sobol senstvty ndces. Sobol and Kucherenko [38] proved theoretcally that, n the cases of unformly and normally dstrbuted nput varables, there s a lnk between DGSM and the Sobol total senstvty ndex S tot for the same nput. They showed that DGSM can be used as an upper bound on total senstvty ndex S tot. Small values of DGSM mply small

4 S tot, and hence unessental factors x. However, rankng nfluental factors usng DGSM can be smlar to that based on S tot only for the case of lnear and quas-lnear models. For hghly non-lnear models two rankngs can be very dfferent. They also ntroduced modfed DGSM whch can be used for both a sngle nput and groups of nputs [39]. From DGSM, Kucherenko and Song [16] have also derved lower bounds on total senstvty ndex. Lambon et al [19] extended results of Sobol and Kucherenko for models wth nput varables belongng to the general class of contnuous probablty dstrbutons. In the same framework, Roustant et al [28] have defned crossed-dgsm, based on second-order dervatves of model output, n order to bound the total Sobol ndces of an nteracton between two nputs. All these DGSM measures can be appled for problems wth a hgh number of nput varables to reduce the computatonal tme. Indeed, the numercal effcency of the DGSM method can be mproved by usng the automatc dfferentaton algorthm for calculaton DGSM as was shown n Kparssdes et al [15]. However, the number of requred functon evaluatons stll remans to be proportonal to the number of nputs. Ths dependence can be greatly reduced usng an approach based on algorthmc dfferentaton n the adjont or reverse mode [9] ( Varatonal Methods). It allows estmatng all dervatves at a cost at most 4-6 tmes of that for evaluatng the orgnal functon [13]. Ths paper s organsed as follows: the Morrs method and DGSM are frstly descrbed n the followng secton. Sobol global senstvty ndces and useful relatonshps are then ntroduced. Therefore, DGSM-based lower and uppers bounds on total Sobol senstvty ndces for unformly and normally dstrbuted random varables are presented, followed by DGSM for groups of varables and ther lnk wth total Sobol senstvty ndces. Another secton presents the upper bounds results n the general case of varables wth contnuous probablty dstrbutons. Then, computatonal costs 4

5 5 are consdered, followed by some test cases whch llustrate an applcaton of DGSM and ther lnks wth total Sobol senstvty ndces. Fnally, conclusons are presented n the last secton. From Morrs method to DGSM Bascs of the Morrs method The Morrs method s tradtonally used as a screenng method for problems wth a hgh number of varables for whch functon evaluatons can be CPU-tme consumng (see Desgn of Experments for Screenng). It s composed of ndvdually randomzed one-factor-at-a-tme (OAT) experments. Each nput factor may assume a dscrete number of values, called levels, whch are chosen wthn the factor range of varaton. The senstvty measures proposed n the orgnal work of Morrs [21] are based on what s called an elementary effect. It s defned as follows. The range of each nput varable s dvded nto p levels. Then the elementary effect (ncremental rato) of the -th nput factor s defned as [ ( G x EE (x ) = 1,..., x 1, x +, x +1,..., xd) G (x ) ], (1) where s a predetermned multple of 1/(p-1) and pont x = (x 1,..., x d ) Hd s such that x + 1. One can see that the elementary effect are fnte dfference approxmatons of the model dervatve wth respect to x and usng a large perturbaton step. The dstrbuton of elementary effects EE s obtaned by randomly samplng R ponts from H d. Two senstvty measures are evaluated for each factor: µ an estmate of the mean of the dstrbuton EE, and σ an estmate of the standard devaton of EE. A hgh value of µ ndcates an nput varable wth an mportant overall nfluence

6 6 on the output. A hgh value of σ ndcates a factor nvolved n nteracton wth other factors or whose effect s nonlnear. The computatonal cost of the Morrs method s N F = R (d+1 ). The revsed verson of the EE (x ) measure and a more effectve samplng strategy, whch allows a better exploraton of the space of the uncertan nput factors was proposed by Campolongo et al [3]. To avod the cancelng effect whch appears n non-monotonc functons Campolongo et al [3] ntroduced another senstvty measure µ based on the absolute value of EE (x ): EE (x ). It was also notced that µ has smlartes wth the total senstvty ndex S tot n that t can gve a rankng of the varables smlar to that based on the S tot but no formal proof of the lnk between µ and S tot was gven [3]. Fnally, other extensons of the ntal Morrs method have been ntroduced for the second-order effects analyss [2] [4] [6], for the estmaton of Morrs measures wth any-type of desgn [26] [32] and for buldng some 3D Morrs graph [26]. The local senstvty measure Consder a dfferentable functon G (x), where x = (x 1,..., x d ) s a vector of nput varables defned n the unt hypercube H d (0 x 1, = 1,..., d). Local senstvty measures are based on partal dervatves E (x ) = G(x ) x. (2) Ths measure E s the lmt verson of the elementary effect EE defned n (2) when tends to zero. It s ts generalzaton n ths sense. In SA, usng the partal dervatve G / x s well known as a local method (see Varatonal Methods). In ths paper, the goal s to take advantage of ths nformaton n global SA. The local senstvty measure E (x ) depends on a nomnal pont x and t changes wth a change of x. Ths defcency can be overcome by averagng E (x ) over

7 the parameter space H d. Ths s done just below, allowng to defne new senstvty measures, called DGSM for Dervatve-based Global Senstvty Measures. 7 DGSM for unformly dstrbuted varables Assume that G/ x L 2. Three dfferent DGSM measures are defned: where m > 0 s a constant, and ν = w (m) = H d ( ) 2 G(x) dx, (3) H d x m x G(x) x dx, (4) ς = 1 ( ) 2 G(x) x (1 x ) dx. (5) 2 H x d DGSM for randomly dstrbuted varables Consder a functon G (X 1,..., X d ), where X 1,..., X d are ndependent random varables, defned n the Eucldan space R d, wth cumulatve densty functons (cdfs) F 1 (x 1 ),..., F d (x d ). The followng DGSM was ntroduced n Sobol and Kucherenko [38]: ( ) [ 2 ( G(x) ) ] 2 G(x) ν = df (x) = E, (6) x x R d wth F the jont cdf. A new measure s also ntroduced: ( ) G(x) G(x) w = df (x) = E. (7) R x d x In (3) and (6), ν s n fact the mean value of ( G/ x ) 2. In the followng and n practce, t wll be the most useful DGSM. Sobol global senstvty ndces Defntons The method of global senstvty ndces developed by Sobol (see Varance-based Senstvty Analyss: Theory and Estmaton Algorthms) s based on ANOVA decom-

8 poston [10]. Consder a square ntegrable functon G(x) defned n the unt hypercube H d. It can be expanded n the followng form 8 G(x) = g 0 + g (x ) + <j g j (x, x j ) g 12...d (x 1, x 2,..., x d ). (8) Ths decomposton s unque f condtons satsfed. Here 1 1 < < s d. 1 0 g 1... s dx k = 0 for 1 k s, are The varances of the terms n the ANOVA decomposton add up to the total varance of the functon d V = where V 1... s = s=1 1 0 d g s (x 1,..., x s )dx 1,..., x s 1 < < s V 1... s, are called partal varances. Sobol defned the global senstvty ndces as the ratos S 1... s = V 1... s /V. All S 1... s are non negatve and add up to one: d S + =1 S j + j S jk... + S 1,2,...,d = 1. j k Sobol also defned senstvty ndces for subsets of varables. Consder two complementary subsets of varables y and z: x = (y, z). Let y = (x 1,..., x m ), 1 1 <... < m d, K = ( 1,..., m ). The varance correspondng to the set y s defned as V y = m V 1... s. s=1 ( 1 < < s) K V y ncludes all partal varances V 1, V 2,..., V 1... s such that ther subsets of ndces ( 1,..., s ) K.

9 The total senstvty ndces were ntroduced by Homma and Saltell [11]. The 9 total varance V tot y s defned as V tot y = V V z. V tot y conssts of all V 1... s such that at least one ndex p K whle the remanng ndces can belong to the complmentary to K set K. The correspondng global senstvty ndces are defned as S y = V y /V, S tot y = Vy tot /V. The mportant ndces n practce are S and S tot, = 1,..., d: S = V /V, S tot = V tot /V. (9) (10) Ther values n most cases provde suffcent nformaton to determne the senstvty of the analyzed functon to ndvdual nput varables. Varance-based methods generally requre a large number of functon evaluatons (see Varance-based Methods: Theory and Algorthms) to acheve reasonable convergence and can become mpractcal for large engneerng problems. Useful relatonshps To present further results on lower and upper bounds of S tot, new notatons and useful relatonshps have to be frstly presented. Denote u (x) the sum of all terms n the ANOVA decomposton (8) that depend on x : u (x) = g (x ) + d j=1,j From the defnton of ANOVA decomposton t follows that g j (x, x j ) + + g 12 d (x 1,, x d ). (11) H d u (x)dx = 0. (12)

10 It s obvous that 10 G x = u x. (13) Denote z = (x 1,..., x 1, x +1,..., x d ) the vector of all varables but x, then x (x, z) and G(x) G(x, z). The ANOVA decomposton of G(x) (8) can be presented n the followng form G(x) = u (x, z) + v(z), where v(z) s the sum of terms ndependent of x. Because of (12) t s easy to show that v(z) = 1 0 G(x)dx. Hence u (x, z) = G(x) 1 0 G(x)dx. (14) Ths equaton can be found n Lambon [18]. The total partal varance V tot can be computed as V tot = u 2 (x)dx = H d u 2 (x, z)dx dz. H d Then the total senstvty ndex S tot (10) s equal to S tot = 1 V H d u 2 (x)dx. (15) A frst drect lnk between total Sobol senstvty ndces and partal dervatves Consder contnuously dfferentable functon G(x) defned n the unt hypercube H d =[0, 1] d. Ths secton presents a theorem that establshes lnks between the ndex S tot and the lmtng values of G/ x. In the case when y = (x ), Sobol -Jansen formula [14][35][31] for D tot can be rewrtten as D tot = 1 2 H d 0 where x o = (x 1,..., x 1, x, x +1,..., x n ). Theorem 1. Assume that c G x 1 [ ( x )] 2 G (x) G dxdx, (16) C, then

11 Proof: Consder the ncrement of G (x) n (16): 11 c 2 12V Stot C2 12V. (17) ( x ) G (x) G = G (ˆx) x (x x ), (18) where ˆx s a pont between x and x. Substtutng (18) nto (16) leads to V tot = 1 2 H d 1 0 ( G (ˆx) x In (19) c 2 ( G/ x ) 2 C 2 whle the remanng ntegral s ) 2 (x x ) 2 dxdx. (19) (x x ) 2 dx dx = 1 6. Thus obtaned nequaltes are equvalent to (17). Consder the functon G = g 0 +c(x 1/2). In ths case C = c, V = 1/12 and S tot = 1 and the nequaltes n (17) become equaltes. DGSM-based bounds for unformly and normally dstrbuted varables In ths secton, several theorems are lsted n order to defne useful lower and upper bounds of the total Sobol ndces. The proofs of these theorems come from prevous works and papers and are not recalled here. Two cases are consdered: varables x followng unform dstrbutons and varables x followng Gaussan dstrbutons. The general case wll be seen n a subsequent secton. Unformly dstrbuted varables Lower bounds on S tot Theorem 2. There exsts the followng lower bound between DGSM (3) and the Sobol total senstvty ndex:

12 12 (H d [G (1, z) G (0, z)] [G (1, z) + G (0, z) 2G (x)] dx ) 2 4ν V < S tot (20) Proof: The proof of ths Theorem s gven n Kucherenko and Song [16] and s based on equaton (15) and a Cauchy-Schwartz nequalty appled on u (x) u (x) dx. H x d The lower bound number number one (LB1) s defned as [G (1, z) G (0, z)] [G (1, z) + G (0, z) 2G (x)] dx (H ) 2 d. 4ν V Theorem 3. There exsts the followng lower bound, denoted γ(m), between DGSM (4) and the Sobol total senstvty ndex: γ(m) = (2m + 1) [ ] 2 (G(1, z) G(x)) dx w (m+1) H d (m + 1) 2 V < S tot. (21) Proof: The proof of ths Theorem n gven n Kucherenko and Song [16] and s based on equaton (15) and a Cauchy-Schwartz nequalty appled on x m u (x)dx. H d In fact, Theorem 3 gves a set of lower bounds dependng on parameter m. The value of m at whch γ(m) attans ts maxmum s of partcular nterest. Further, star ( ) s used to denote such a value m: m = arg max(γ(m)). γ(m ) s called the lower bound number two (LB2): γ(m ) = (2m + 1) [ ] 2 (G(1, z) G(x)) dx w (m +1) H d (m + 1) 2 V (22) The maxmum lower bound LB* s defned as LB* = max(lb1,lb2). (23) Both lower and upper bounds can be estmated by a set of dervatve based measures: Υ = {ν, w (m), ζ }, m > 0. (24)

13 13 Upper bounds on S tot Theorem 4. There exsts the followng upper bound between DGSM (3) and the Sobol total senstvty ndex: S tot ν π 2 V. (25) Proof: The proof of ths Theorem n gven n Sobol and Kucherenko [38]. It s based on nequalty: 1 and relatonshps (13) and (15). 0 u 2 (x) dx 1 π ( ) 2 u dx x Consder the set of values ν 1,..., ν d, 1 d. One can expect that smaller ν correspond to less nfluental varables x. Ths mportance crteron s smlar to the modfed Morrs mportance measure µ, whose lmtng values are µ = H d G(x) x dx. From a practcal pont of vew the crtera µ and ν are equvalent: they are evaluated by the same numercal algorthm and are lnked by relatons ν Cµ and µ ν. The rght term n (25) s further called the upper bound number one (UB1). Theorem 5. There exsts the followng upper bound between DGSM (5) and the Sobol total senstvty ndex: S tot Proof: The followng nequalty [10] s used: ς V. (26) ( 1 u 2 dx 0 ) 2 udx x(1 x)u 2 dx. (27) The nequalty s reduced to an equalty only f u s constant. Assume that u s gven by (11), then 1 0 udx = 0. From (27), equaton (26) s obtaned.

14 Further ς /D s called the upper bound number two (UB2). Note that 1 2 x (1 x ) for 0 x 1 s bounded: x (1 x ) 1/8. Therefore, 0 ς ν /8. 14 Normally dstrbuted varables Lower bound on S tot Theorem 6. If X s normally dstrbuted wth a mean µ and a fnte varance σ 2, there exsts the followng lower bound between DGSM (7) and the Sobol total senstvty ndex: σ 4 (µ 2 + σ2 )V w2 S tot. (28) Proof: Usng the equaton (15) and Cauchy-Schwartz nequalty appled on x u (x)df (x) R d (wth F the jont cdf), Kucherenko and Song [16] gve the proof of ths nequalty when µ = 0 (omttng to menton ths condton). The general proof, obtaned by Pett [25], s gven below. Consder a unvarate functon g(x), wth X a normally dstrbuted varable wth mean µ, fnte varance σ 2 and cdf F. Wth adequate condtons on g, the followng equalty s obtaned by ntegratng by parts: DGSM wrtes E[g (X)] = = 1 [ σ g(x) exp 2π [ g (x)df (x) = 1 σ 2π (x µ)2 2σ 2 ]] σ 2π = 1 xg(x)df (x) µ σ 2 g (x) exp [ g(x) x µ σ 2 g(x)df (x). (x µ)2 2σ 2 exp [ ] dx ] (x µ)2 dx 2σ 2 In ths equaton, replacng g(x) by u (x) wth x normally dstrbuted, the w G(x) u (x) w = df (x) = df (x) = 1 R x d R x d σ 2 R d x u (x)df (x), because R d u (x)df (x) = 0 (due to the ANOVA decomposton condton). Moreover, the Cauchy-Schwartz nequalty appled on R d x u (x)df (x) gves

15 [ ] 2 x u (x)df (x) x 2 df (x) [u (x)] 2 df (x). R d R d R d 15 Combnng the two latter equatons leads to the expresson w 2 1 σ 4 (µ 2 + σ 2 )V S tot, whch s equvalent to Eq. (28). Upper bounds on S tot The followng Theorem 7 s a generalzaton of Theorem 1. Theorem 7. If X has a fnte varance σ 2 and c G x C, then σ 2 c 2 V Stot σ2 C 2 V. (29) The constant factor σ 2 cannot be mproved. Theorem 8. If X s normally dstrbuted wth a fnte varance σ 2, there exsts the followng upper bound between DGSM (6) and the Sobol total senstvty ndex: S tot σ2 V ν. (30) The constant factor σ 2 cannot be reduced. Proof: The proofs of these Theorems are presented n Sobol and Kucherenko [38]. DGSM-based bounds for groups of varables Let x = (x 1,..., x d ) be a pont n the d dmensonal unt hypercube wth Lebesgue measure dx = dx 1 dx d. Consder an arbtrary subset of the varables y = (x 1,..., x s ), s d, and the set of remanng complementary varables z, so that x = (y, z), dx = dy dz. Further all the ntegrals are wrtten wthout ntegraton lmts, by assumng that each ntegraton varable vares ndependently from 0 to 1.

16 16 Consder the followng DGSM τ y : τ y = s ( ) 2 G (x) 1 3x p + 3x 2 p dx. (31) 6 p=1 x p Theorem 9. If G (x) s lnear wth respect to x 1,..., x s, then V tot y = τ y, or n other words S tot y = τ y V. Theorem 10. The followng general nequalty holds: V tot y words S tot y 24 π 2 V τ y. ( 24 / π 2) τ y, or n other Proof: The proofs of these Theorems are gven n Sobol and Kucherenko [39]. The second theorem shows that small values of τ y mply small values of S tot y and ths allows dentfcaton of a set of unessental factors y (usually defned by a condton of the type S tot y < ɛ, where ɛ s small). Importance crteron τ Consder the one dmensonal case when the subset y conssts of only one varable y = (x ), then measure τ y = τ has the form ( ) 2 G (x) 1 3x + 3x 2 τ = dx. (32) x 6 It s easy to show that ν /24 τ ν /6. From UB1 t follows that S tot 24 π 2 V τ. (33) Thus small values of τ mply small values of S tot, that are characterstc for non mportant varables x. At the same tme, the followng corollary s obtaned from Theorem 9: f G (x) depends lnearly on x, then S tot than ν. = τ /V. Thus τ s closer to V tot Note that the constant factor 1/π 2 n (25) s the best possble. But n the general nequalty for τ (33) the best possble constant factor s unknown. There s a general lnk between mportance measures τ, ς and ν :

17 17 τ = ς ν, then ς = 1 6 ν τ. Normally dstrbuted random varables Consder ndependent normal random varables X 1,..., X d wth parameters (µ, σ ) =1...d. Defne τ as [ ( G τ = 1 ) ] 2 (x) 2 E (x x ) 2. x The expectaton over x can be computed analytcally. Then [ ( G τ = 1 ) ] 2 (x) 2 E (x µ ) 2 + σ 2. x 2 Theorem 11. If X 1,..., X d are ndependent normal random varables, then for an arbtrary subset y of these varables, the followng nequalty s obtaned: S tot y 2 V τ y. Proof: The proof s gven n Sobol and Kucherenko [39]. DGSM-based upper bounds n the general case As prevously, consder the functon G (X 1,..., X d ), where X 1,..., X d are ndependent random varables, defned n the Eucldan space R d, wth cdfs F 1 (x 1 ),..., F d (x d ). Assume further that each X admts a probablty densty functon (pdf), denoted by f (x ). In the followng, all the ntegrals are wrtten wthout ntegraton lmts. The developments n ths secton are based on the classcal L 2 -Poncaré nequalty: G(x) 2 df (x) C(F ) G(x) 2 df (x) (34)

18 where F s the jont cdf of (X 1,..., X d ). (34) s vald for all functons G n L 2 (F ) such that G(x)dF (x) = 0 and f L 2 (F ). The constant C(F ) n Eq. (34) s called a Poncaré constant of F. In some cases, t exsts and optmal Poncaré constant C opt (F ) whch s the best possble constant. In measure theory, the Poncaré constants are expressed as a functon of so-called Cheeger constants [1] whch are used for SA n Lambon et al [19] (see Roustant et al [28] for more detals). A connecton between total ndces and DGSM has been establshed by Lambon et al [19] for varables wth contnuous dstrbutons (called Boltzmann probablty measures n ther paper). Theorem 12. Let F and f be respectvely the cdf and the pdf of X, the followng nequalty s obtaned: S tot wth ν the DGSM defned n Eq. (6) and [ C(F ) = 4 18 C(F ) ν, (35) V sup x R ] 2 mn (F (x), 1 F (x)). (36) f (x) Proof: Ths result comes from the drect applcaton of the L 2 -Poncaré nequalty (34) on u (x) (see Eq. (11)). In Lambon et al [19] and Roustant et al [28], the partcular case of log-concave probablty dstrbuton has been developed. It ncludes classcal dstrbutons as for nstance the normal, exponental, Beta, Gamma and Gumbel dstrbutons. In ths case, the constant wrtes C(F ) = 1 f ( m ) 2 (37) wth m the medan of the dstrbuton F. Ths allows to obtan analytcal expressons for C(F ) n several cases [19]. In the case of a log-concave truncated dstrbuton on [a, b], the constant wrtes [28] ( ( )) 2 (F (b) F (a)) 2 F (a) + F (b) /f q (38) 2

19 wth q ( ) the quantle functon of X. Table 1 gves some examples of Poncaré constants for several well-known and often used probablty dstrbutons n practce. 19 Dstrbuton Poncaré constant Optmal constant Unform U[a b] (b a) 2 /π 2 yes Normal N (µ, σ 2 ) σ 2 yes 4 Exponental E(λ), λ > 0 λ 2 yes ( ) 2 2β Gumbel G(µ, β), scale β > 0 no log 2 [ ] 2λ(log 2) (1 k)/k 2 Webull W(k, λ), shape k 1, scale λ > 0 no k Table 1. Poncaré constants for a few probablty dstrbutons. For studyng second-order nteractons, Roustant et al [28] have derved a smlar to (35) nequalty based on the squared crossed dervatves of the functon. Assumng that second-order dervatves of G are n L 2 (F ), t uses the so-called crossed-dgsm ( ) 2 2 G(x) ν j = df (x), (39) x x j ntroduced by Fredman and Popescu [7]. An nequalty lnk s made wth an extenson of the total Sobol senstvty ndces to general sets of varables (called superset mportance or total nteracton ndex) proposed by Lu and Owen [20]. In the case of a par of varables {X, X j }, the superset mportance s defned as V super j = I {,j} V I. (40) The estmaton methods of ths total nteracton ndex have also been studed by Fruth et al [8]. Theorem 13. For all pars {, j} (1 < j d), V j V super j C(F )C(F j )ν j. (41) These nequaltes wth the correspondng Sobol ndces wrte

20 S j 20 S super j C(F )C(F j ) ν j. (42) V Roustant et al [28] have shown on several examples how to apply ths result n order to detect pars of nputs that do not nteract together (see also Muehlenstaedt et al [22] and Fruth et al [8] whch use Sobol ndces). Computatonal costs All DGSM can be computed usng the same set of partal dervatves G(x), = x G(x) 1,..., d. Evaluaton of can be done analytcally for explctly gven easly- x dfferentable functons or numercally: G(x ) x = [ G ( x 1,..., x 1, x + δ, x +1,..., x n) G (x ) ] δ. (43) Ths s called a fnte-dfference scheme (see Varatonal Methods) wth δ whch s a small ncrement. There s a smlarty wth the elementary effect formula (2) of the Morrs method whch s however computed wth large. In the case of straghtforward numercal estmatons of all partal dervatves (43) and computaton of ntegrals usng MC or QMC methods, the number of requred functon evaluatons for a set of all nput varables s equal to N(d + 1), where N s a number of sampled ponts. Computng LB1 also requres values of G (0, z), G (1, z), whle computng LB2 requres only values of G (1, z). In total, numercal computaton of LB* for all nput varables would requre N LB* G = N(d + 1) + 2Nd = N(3d + 1) functon evaluatons. Computaton of all upper bounds requre N UB G = N(d+1) functon evaluatons. Ths s the same number that the number of functon evaluatons requred for computaton of S tot whch s NG S = N(d + 1) [31]. However, the number of sampled ponts N needed to acheve numercal convergence can be dfferent for DGSM and S tot. It s generally lower for the case of

21 21 DGSM. Moreover, the numercal effcency of the DGSM method can be sgnfcantly ncreased by usng algorthmc dfferentaton n the adjont (reverse) mode [9] (see also Varatonal Methods). Ths approach allows estmatng all dervatves at a cost ndependent of d, at most 4-6 tmes of that for evaluatng the orgnal functon G(x) [13]. Test cases In ths secton, three test cases are consdered, n order to llustrate applcaton of DGSM and ther lnks wth S tot. Example 1. Consder a lnear wth respect to x functon: G(x) = a(z)x + b(z). For ths functon S = S tot, V tot = 1 a 2 (z)dz, ν = a 2 (z)dz, LB1 = 12 ) H d 1 H (H d 1 (a 2 (z) 2a 2 2 (z)x d ) dzdx 4V = 0 and γ(m) = (2m + ( 1)m2 a(z)dz ) 2 H d 1. A maxmum value of γ(m) s attaned at m =3.745, whle γ (m ) = ( 2 a H 2 (z)dz 4(m + 2) 2 (m + 1) 2 V d 1 a(z)dz). The V lower and upper bounds are LB* 0.48S tot, UB1 1.22S tot. UB2 = 1 1 a(z) 2 dz = 12V S tot. For ths test functon UB2 < UB1. Example 2. Consder the so-called g-functon whch s often used n global SA for llustraton purposes: G(x) = d v, where v = 4x 2 + a 1 + a, a ( = 1,..., d) are constants. It s easy to see that for ths functon g (x ) = (v 1), u (x) = (v 1) d j=1,j v j and as a result LB1=0. The total d ( varance s V = /3 ). The analytcal values of S (1 + a j ) 2, S tot and LB2 are gven n Table 2. j=1 =1 0

22 22 Table 2. The analytcal expressons for S, S tot and LB2 for g-functon. S 1/3 (1 + a ) 2 V S tot 1/3 (1+a ) 2 d j=1,j V ( ) 1 + 1/3 (1+a j) 2 γ(m) [ (2m + 1) 1 4(1 (1/2)m+1 ) m+2 (1 + a ) 2 (m + 1) 2 V ] 2 By solvng equaton dγ(m) dm = 0, m =9.64 and γ(m ) = It s n- (1 + a ) 2 V terestng to note that m does not depend on a, = 1, 2,..., d and d. In the extreme cases: f a for all, γ(m ) S tot (4/3) d 1, S S tot UB2 are gven n Table 3. γ(m ) S tot 1 (4/3) 0.257, S S tot 1, whle f a 0 for all,. The analytcal expresson for Stot d 1, UB1 and Table 3. The analytcal expressons for S tot, UB1 and UB2 for g-functon. 1/3 (1+a ) 2 d j=1,j S tot UB1 UB2 ( ) 1 + 1/3 (1+a j) 16 d 2 j=1,j 4 d j=1,j V ( ) 1 + 1/3 (1+a j) 2 (1 + a ) 2 π 2 V ( ) 1 + 1/3 (1+a j) 2 3(1 + a ) 2 V For ths test functon Stot UB1 = π2 48, S tot UB2 = 1 4, hence UB2 UB1 = π2 12 < 1. Values of S, S tot, UB1, UB2 and LB2 for the case of a=[0,1,4.5,9,99,99,99,99], d=8 are gven n Table 4 and shown n Fg. 1. One can see that the knowledge of LB2 and UB1 allows to rank correctly all the varables n the order of ther mportance. Table 4. Values of LB*, S, S tot, UB1 and UB1. Example 2, a=[0,1,4.5,9,99,99,99,99], d =8. x 1 x 2 x 3 x 4 x 5...x 8 LB* S S tot UB UB Example 3. Consder the reduced Morrs test functon wth four nputs [3]:

23 23 Fg. 1. Values of S, S tot, LB2 and UB1 for all nput varables. Example 2 wth a = [0, 1, 4.5, 9, 99, 99, 99, 99], d = 8. f(x) = b x + b j x x j + b jk x x j x k (44) =1 j j k wth b =, b j =, b j4 = The ndces b jk k 4 are null. The four nput varables x ( = 1,..., 4) follow unform dstrbuton on [0, 1]. Sobol ndces are computed va the Monte-carlo scheme of Saltell [29] (usng two ntal matrces of sze 10 5 ), whle DGSM are computed wth Monte-Carlo samplng of sze n (usng dervatves computng by fnte dfferences (43) wth δ = 10 5 ), wth n rangng from 20 to 500, Fgure 2 shows that DGSM bounds UB1 are greater than the total Sobol ndces S T (for = 1, 2, 3, 4) as expected, except for n < 30 whch s a too small sample sze. For small S T, UB1 s close to the S T value. It confrms that DGSM bounds are frst useful for screenng exercses. Other numercal tests nvolvng non-unform and non-normal dstrbutons for the nputs can be found n Lambon et al [19] and Fruth et al [8].

24 24 Fg. 2. For the 4 nput varables of the reduced Morrs test functon: Convergence of the DGSM bound estmates (sold lnes) n functon of the sample sze and comparson to theoretcal values of total Sobol ndces S T (dashed lnes). Conclusons Ths paper has shown that usng lower and upper bounds based on DGSM s possble n most cases to get a good practcal estmaton of the values of S tot at a fracton of the CPU cost for estmatng S tot. Upper and lower bounds can be estmated usng MC/QMC ntegraton methods usng the same set of partal dervatve values. Most of the applcatons show that DGSM can be used for fxng unmportant varables and subsequent model reducton because small values of DGSM mply small values of S tot. In a general case varable rankng can be dfferent for DGSM and varance based methods but for lnear functon and product functon, DGSM can gve the same varable rankng as S tot.

25 25 Engneerng applcatons of DGSM can be found for nstance n Kparssdes et al [15] and Rodrguez-Fernandez et al [27] for bologcal systems modelng, Patell et al [24] for structural mechancs, Iooss et al [12] for an aquatc prey-predator model, Pett [25] for a rver flood model and Touzany and Busby [41] for an hydrogeologcal smulator of the ol ndustry. One of the man prospect n practcal stuatons s to use algorthmc dfferentaton n the reverse (adjont) mode on the numercal model, allowng to estmate effcency all partal dervatves of ths model (see Varatonal Methods). In ths case, the cost of DGSM estmatons would be ndependent of the number of nput varables. Obtanng global senstvty nformaton n a reasonable cpu tme cost s therefore possble even for large-dmensonal model (several tens and spatally dstrbuted nputs n the recent and poneerng attempt of Pett [25]). When the adjont model s not avalable, the DGSM estmaton remans a problem n hgh dmenson and novel deas have to be explored [23] [24]. Couplng DGSM wth nonparametrc regresson technques or metamodel-based technque (see Metamodel-based senstvty analyss: Polynomal chaos expansons and Gaussan processes) s another research prospect as frst shown by Sudret and Ma [40] and De Lozzo and Marrel [5]. The authors would lke to thank Prof. I. Sobol, Dr. S. Song, S. Pett, Dr. M. Lambon, Dr. O. Roustant and Prof. F. Gamboa for ther contrbutons to ths work. One of the authors (SK) gratefully acknowledges the fnancal support by the EPSRC grant EP/H03126X/1. References 1. Bobkov SG (1999) Isopermetrc and analytc nequaltes for log-concave probablty measures. The Annals of Probablty 27(4): Campolongo F, Braddock R (1999) The use of graph theory n the senstvty analyss of model output: a second order screenng method. Relablty Engneerng and System Safety 64:1 12

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