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1 Avalable onlne at wwwscencedrectcom Mathematcs and Computers n Smulaton xxx 29 xxx xxx Dervatve based global senstvty measures and ther lnk wth global senstvty ndces IM Sobol a, S Kucherenko b, a Insttute for Mathematcal Modellng of the Russan Academy of Scences, 4 Musskaya Square, Moscow 12547, Russa b Centre for Process Systems Engneerng, Imperal College London, London SW7 2AZ, UK Receved 11 Aprl 28; accepted 3 January 29 Abstract A model functon fx 1,,x n defned n the unt hypercube wth Lebesque measure dx = dx 1 dx n s consdered If the functon s square ntegrable, global senstvty ndces provde adequate estmates for the nfluence of ndvdual factors x or groups of such factors Alternatve estmators that requre less computer tme can also be used If the functon f s dfferentable, functonals dependng on f/ x have been suggested as estmators for the nfluence of x The Morrs mportance measure modfed by Campolongo, Carbon and Saltell μ* s an approxmaton of the functonal μ = f / x dx In ths paper a smlar functonal s studed 2 f ν = dx H x n Evdently, μ ν, and ν Cμ f f / x C A lnk between ν and the senstvty ndex S tot s establshed: ν π 2 D where D s the total varance of fx 1,,x n Thus small ν mply small S tot, and unessental factors x that s x correspondng to a very small S tot can be detected analyzng computed values ν 1,,ν n However, rankng nfluental factors x usng these values can gve false conclusons Generalzed S tot and ν can be appled n stuatons where the factors x 1,,x n are ndependent random varables If x s a normal random varable wth varance σ 2, then Stot ν σ 2/D 29 IMACS Publshed by Elsever BV All rghts reserved Keywords: Global senstvty ndex; Morrs method; Quas Monte Carlo method; Dervatve based global senstvty measure 1 Introducton Let fx 1,,x n be a model functon defned n the unt hypercube If the functon f s square ntegrable, global senstvty ndces provde an adequate tool for estmatng the effect of ndvdual factors x or groups of such factors on f However, numercal algorthms for computng these ndces nvolve evaluaton of f at a large number of random Correspondng author Tel: ; fax: E-mal addresses: hq@mamodru IM Sobol, skucherenko@cacuk S Kucherenko /$36 29 IMACS Publshed by Elsever BV All rghts reserved do:1116/jmatcom29123 Please cte ths artcle n press as: IM Sobol, S Kucherenko, Dervatve based global senstvty measures and ther lnk wth global senstvty ndces, Math Comput Smul 29, do:1116/jmatcom29123
2 2 IM Sobol, S Kucherenko / Mathematcs and Computers n Smulaton xxx 29 xxx xxx or quas-random ponts, and f one model evaluaton requres more than several mnutes of computer tme, a drect applcaton of these algorthms wthout usng multprocessor schemes s mpractcal Alternatve approaches were developed that provde less expensve estmates of the nfluence of ndvdual factors x on f They can be appled to more complex models These alternatve estmates sometmes dsagree wth the ones obtaned from senstvty ndces If the functon f s dfferentable, the partal dervatve f/ x s used for estmatng the local senstvty of f wth respect to x at the pont x 1,,x n Naturally that attempts were made to construct global senstvty measures as functonals dependng on f/ x [1,3,5] The Morrs mportance measure μ s a measure of ths type: fnte dfference approxmatons to f/ x are computed at a dscrete set of ponts nsde and μ s defned as a weghted mean of ths approxmate values of f/ x [3] Campolongo et al suggested a modfed Morrs measure based on absolute values f/ x called μ* [1] It was notced that for some practcal problems ths measure has smlartes wth global total senstvty ndces S tot n that t gves a rankng of the varables very smlar to that based on the S tot but no formal proof of the lnk between μ* and S tot was gven In ths paper we consder partal dervatve based global senstvty measures and establsh the lnk between them and global total senstvty ndces It s organzed as follows: the next secton gves a bref descrpton of the Morrs method Secton 3 gves an overvew of the theory of global senstvty ndces A lnk between global senstvty ndces and partal dervatves s establshed n Secton 4 Secton 5 ntroduces dervatves based mportance crtera Secton 6 contans examples A counterexample showng that n some cases dervatve based mportance estmates suggest false conclusons s presented n Secton 7 Secton 8 brefly revews the case when x 1,,x n are ndependent random varables Fnally, conclusons are presented n the last secton In the Appendx A a lmt for μ* s consdered 2 The Morrs method The senstvty measures proposed n the orgnal work of Morrs [3] are based on what s called an elementary effect The general scheme of the Morrs method s defned as follows The range of each nput varable s dvded nto p levels Then the elementary effect of the th nput factor s defned as fnte dfference EE x = f x1,,x 1,x + Δ, x +1,,x n f x, 21 Δ where Δ s a predetermned multple of 1/p 1 and pont x* s such that x + Δ 1 The dstrbuton of elementary effects F s obtaned by randomly samplng N ponts from Two senstvty measures are evaluated for each factor: μ an estmate of the mean of the dstrbuton F, and σ an estmate of the standard devaton of F A hgh value of μ ndcates an nput varable wth an mportant overall nfluence on the output A hgh value of σ ndcates a factor nvolved n nteracton wth other factors or whose effect s nonlnear The total computatonal cost for ths scheme s N F =2Nn Morrs suggested a more economcal algorthm by usng already computed values of functons n calculaton of more than one elementary effects Hs algorthm nvolves a calculaton of the so-called samplng matrx whch s used for generatng trajectores of n steps n the nput varables space These trajectores are such that on each step only one component of a statng pont x =x 1,x 2,,x n taken from grd-levels s ncreased by Δ The computatonal cost of the Morrs method s N F = Nn + 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 n [1] Non-monotonc functons have regons of postve and negatve values of EE x*, hence due to the effect of averagng values of μ can be very small or even zero For ths reason Campolongo et al [1] consdered another senstvty measure called μ*, whch estmates the mean of the dstrbuton of elementary effect absolute values 3 Global senstvty ndces Global senstvty ndces are often classfed as varance based However, they can be defned wthout assumng that the varable x s random Consder a functon fx defned and square ntegrable n the unt hypercube wth the Please cte ths artcle n press as: IM Sobol, S Kucherenko, Dervatve based global senstvty measures and ther lnk wth global senstvty ndces, Math Comput Smul 29, do:1116/jmatcom29123
3 IM Sobol, S Kucherenko / Mathematcs and Computers n Smulaton xxx 29 xxx xxx 3 Lebesque measure dx = dx 1 dx n Accordng to [7] the dentty n f x = f + x1,, x s f 1,, s s=1 1 < < s s called ANOVA-decomposton of fxf f = f x dx and for all p = 1,2,,s 1 f 1,, s x1,, x s dxp = The nteror sum n 31 s extended over all dfferent groups of ndces 1,, s such that 1 1 < 2 < < s n Thus 31 can be rewrtten as f x = f + f x + <j f,j x,x j + +f1,2,,n x 1,x 2,, x n It follows from 32 and 33 that all the terms n 31 are orthogonal Constants D 1,, s = f 2 1,, x1 s,, x s dx are called partal varances and the constant D = f 2 x dx f 2 s called total varance Squarng 31 and ntegratng over, we obtan n D = D 1,, s s=1 1 < < s Global senstvty ndces are defned as ratos S 1,, s = D 1,, s D Obvously n S 1,, s = 1 s=1 1 < < s One-dmensonal ndex S = D /D shows the effect of the sngle factor x on the output fx but t does not account for the hgh dmensonal terms n 31 For estmatng the total nfluence of the factor x, total partal varances are ntroduced: D tot = D 1 s, where the sum s extended over all dfferent groups of ndces satsfyng 34 at 1 s n, where one of the ndces s equal The correspondng total senstvty ndex s = Dtot D In general S 1 The output fx does not depend on the factor x f and only f S tot = If the value x s somehow fxed, the error n fx depends on S tot for more detals see [9] Indces are often used for rankng varables x Please cte ths artcle n press as: IM Sobol, S Kucherenko, Dervatve based global senstvty measures and ther lnk wth global senstvty ndces, Math Comput Smul 29, do:1116/jmatcom29123
4 4 IM Sobol, S Kucherenko / Mathematcs and Computers n Smulaton xxx 29 xxx xxx In [7] Theorem 3 a general formula for Dy tot s gven, where y s an arbtrary subset of the varables x 1,,x n In the case when y =x, ths formula can be rewrtten as where D tot = 1 1 [ x ] 2dxdx f x f 2, 35 x = x 1,, x 1,x,x +1,, x n 4 Senstvty ndces and partal dervatves In ths secton two theorems that establsh lnks between the ndex S tot and the dervatve f/ x are proved In the frst theorem the lmtng values of f/ x and n the second theorem the mean value of f/ x 2 are used Theorem 1 Assume that c f/ x C Then c 2 12D Stot C2 12D The constant factor 12 n 41 cannot be mproved Proof Consder the ncrement of fxn35: x f ˆx f x f = x x, 42 x where ˆx s a pont between x and x Substtutng 42 nto 35 we obtan D tot = 1 1 f ˆx 2 x x 2dxdx 2 x 43 In 43 c 2 f/ x 2 C 2 whle the remanng ntegral s 1 1 x 2dx x dx = 1 6 Thus we obtan nequaltes that are equvalent to 41 Fnally consder the functon f = f + c x 1/2 In ths case C = c, D = 1/12, = 1 and the nequaltes 41 become equaltes Theorem 2 Assume that f/ x L 2 Then 1 f 2 π 2 dx 44 D x Proof Denote by ux the sum of all terms n 31 that depend on x : u x = <>f 1,, s x1,, x s 41 Obvously D tot = u 2 x dx and f x = u x Consder ux as a functon of x only It follows from 33 that the mean 1 u x dx =, therefore an nequalty for one-dmensonal functons from [6] can be appled: 1 u 2 x dx 1 1 u 2 π 2 dx x Please cte ths artcle n press as: IM Sobol, S Kucherenko, Dervatve based global senstvty measures and ther lnk wth global senstvty ndces, Math Comput Smul 29, do:1116/jmatcom29123
5 IM Sobol, S Kucherenko / Mathematcs and Computers n Smulaton xxx 29 xxx xxx 5 Integratng ths nequalty over all other varables we obtan D tot 1 f 2 π 2 dx x Ths s equvalent to 44 To complete the proof of Theorem 2, consder an example: f x = sn π x 1/2 In ths case = 1, D = 1/2 and 1 f/ x 2 dx = π 2 /2, so the rght-hand sde n 44 s also equal to 1 Remark From the relaton 42 we conclude that D tot = 12[ 1 f/ x 2], where [ ]* s a rather sophstcated mean value If [ ]* s replaced by an ordnary mean value we get an approxmate relaton that yelds an approxmate formula 1 f 2 dx 12D x Unfortunately, we have no relable error estmate for ths approxmaton We can only expect ths approxmaton to be correct n stuatons n whch the second dervatve 2 f/ x 2 s neglgble 5 Dervatve based mportance crtera Consder the set of values ν 1,,ν n, where f 2 ν = dx, 1 n x 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 see Appendx A μ = f 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, μ ν Therefore the results of Secton 4 can be regarded as support for both ν and μ The only pont that can be nterpreted as an advantage of ν s the nequalty 44: ν π 2 D that provdes the estmaton of S tot wthout knowng the upper bound C of the partal dervatve It s been shown n Kucherenko et al [2] that the computatonal tme requred for MC evaluaton of dervatve based mportance crtera s much lower than that for estmaton of the Sobol senstvty ndces It s also lower than that for the Morrs method The effcency agan s especally dramatc for the Quas MC ntegraton method based on Sobol sequences It s also been shown that the Morrs method can produce naccurate measures for non-monotonc functons such as g-functon for whch characterstc length of functon varaton s much smaller than Δ 6 Functons wth separated varables Consder f x = Then A = D = 1 =1 ϕ x, where ϕ t L 2, ϕ t L 2 Denote =1 ϕ t dt, D = D + A 2 1 A 2, =1 ϕ 2 t dt A2 Please cte ths artcle n press as: IM Sobol, S Kucherenko, Dervatve based global senstvty measures and ther lnk wth global senstvty ndces, Math Comput Smul 29, do:1116/jmatcom29123
6 6 IM Sobol, S Kucherenko / Mathematcs and Computers n Smulaton xxx 29 xxx xxx D tot = k/= D k + A 2 k D, Thus f 2 dx = x f x 2dx D tot = k/= D k + A 2 k 1 [ ϕ t ] 2 dt D 1 [ ϕ t ] 2 dt 61 Example 1 Consder the so-called g-functon that s often used for numercal experments n senstvty analyss f = 4x 2 + a /1 + a Surprsngly, for the g-functon the rato 61 s constant and does not depend on =1 the parameter a Thus for all mportant or non-mportant varables x the rght-hand sde n 44 s proportonal to the left-hand sde However, the value of ths constant 48 s consderably larger than 2 or 12 1 [ Example 2 If ϕ s strongly nonlnear, the rato ϕ t ] 2 dt /D can be very large Assume that t=t m Then A = 1/m + 1, D = m 2 /2m + 1m + 1 2, 1 [ ϕ t ] 2 dt = m 2 1 [ /2m 1 The rato ϕ t ] 2 dt /D = m m + 1/2m 1 At m = 1 the rato s 12, but for large m t wll be m Counterexample Example 3 f = =1 Consder a functon f whch has the followng ANOVA decomposton: 4 c x 1 + c 12 x 1 1 x 2 1 5, where c =1,1 4, c 12 = 5 For ths functon all S = 237, 1 4, S 12 = 523 and S1 tot = S2 tot = 289, 4 = 237, so varables 1, 2 and varables 3, 4 have the same mportance However, for dervatve based mportance crtera varables 1 and 2 have dfferent mportance ν 1 = 122, ν 2 = 326, whle varables 3 and 4 stll have equal mportance ν 3 = ν 4 = 1 Moreover, ν 2 > ν 1 + ν 3 + ν 4 Comparng left and rght-hand sde of nequalty 44 Table 1, one can see that ν / 2 D s much hgher than only for varable 2 It s caused by the strong nonlnearty of the term f 1,2 x 1,x 2 wth respect to x 2 compare wth test functon of Example 2 3 = Ths example shows that rankng of nfluental varables based on ν may result n false conclusons: n our example x 2 seems more mportant than all the other varables together Table 1 S tot and ν /π 2 D for the Counterexample ν /π 2 D Please cte ths artcle n press as: IM Sobol, S Kucherenko, Dervatve based global senstvty measures and ther lnk wth global senstvty ndces, Math Comput Smul 29, do:1116/jmatcom29123
7 8 Random varables IM Sobol, S Kucherenko / Mathematcs and Computers n Smulaton xxx 29 xxx xxx 7 Consder a model functon fx 1,,x n, where x 1,,x n are ndependent random varables wth dstrbuton functons F 1 x 1,,F n x n Thus the pont x =x 1,,x n s defned n the Eucldan space R n and ts measure s df 1 x 1 df n x n The theory of global senstvty ndces can be easly generalzed and appled n ths case see eg [8] The followng asserton s a generalzaton of Theorem 1 Theorem 3 Assume that c f/ x C and that the varance of x s fnte σ 2 = varx < Then σ 2c2 /D σ 2C2 /D The constant factor σ 2 cannot be mproved Proof One can repeat the proof of Theorem 1 wth three changes: 1 The startng pont s the relaton D tot = 2 1 [ R n f x f x ] 2 df k x k df x 2 The remanng ntegral n ths case s x 2dF x x df x = 2σ2 3 Fnally consder the functon fx=f + cx Ex, where Ex s a mean value of x In ths case C = c, D = σ 2, Stot = 1 and the nequaltes become equaltes The followng results are smlar to Theorem 2 but t s not a generalzaton of Theorem 2 Theorem 4 Then Assume that x s a normal random varable wth parameters a ;σ and the ntegral n 81 s fnte σ2 f 2 df k x k D R n x The constant factor σ 2 cannot be reduced Proof The logc of the proof s the same as n Theorem 2 However, the nequalty from [6] that was used n Theorem 2 must be replaced by a new one: Inequalty Denote pt = 1 σ 2π exp[ t a2 /2σ 2 ], < t < If both ut and u t are square ntegrable wth weght pt, and Then utptdt = u 2 tptdt σ 2 [ u t ] 2 ptdt 83 The smple example fx=x a shows that n 81 equalty s possble: = 1, f/ x = 1, D = σ 2 Proof of the nequalty Let Φ = Φ[u] be a functonal dependng on ut: [ Φ = σ 2 u 2 u 2] ptdt Consder a typcal problem n calculus of varatons: mnmze Φ[u] whle ut satsfes 82 The extremal functon u*=t a satsfes the Euler Lagrange equaton and condton 82 The mnmum value of the functonal 84 s mn Φ[u]=Φ[t a] = Thus Φ[u] and ths s equvalent to 83 Example 4 We consder the quadratc polynomal Oakley and O Hagan functon defned as follows [4]: f x = a T 1 x + at 2 cos x + at 3 sn x + xt Mx Please cte ths artcle n press as: IM Sobol, S Kucherenko, Dervatve based global senstvty measures and ther lnk wth global senstvty ndces, Math Comput Smul 29, do:1116/jmatcom29123
8 8 IM Sobol, S Kucherenko / Mathematcs and Computers n Smulaton xxx 29 xxx xxx Here x s a vector of ffteen normally dstrbuted varables N, 1 For ths functon the senstvty ndces S tot ncrease monotoncally from S1 tot = 59 up to S15 tot = 154 The estmates on the rght-hand sde of 81 were computed and dvded by In full agreement wth Theorem 4, all these ratos exceed 1: 11; 1; 1; 14; 19; 118; 116; 16; 126; 18; 119; 12; 12; 115; Conclusons The man results of the present paper are 1 A lnk between senstvty ndces and measures based on partal dervatves s establshed 2 It s proved that small values of dervatve based measures mply small values of one-dmensonal total senstvty ndces Ths result supports the recommendaton of [1,3,5] that dervatve based measures can successfully be used for detectng unessental varables 3 The mportance crteron μ* can be mproved by usng squared partal dervatve rather than ts absolute value 4 It s shown that for hghly nonlnear functons the rankng of mportant factors usng dervatve based mportance measures may suggest false conclusons Acknowledgements The authors would lke to thank A Sobol for hs help n preparaton of ths manuscrpt and N Shah for hs support and nterest n ths work One of the authors SK gratefully acknowledges the fnancal support by the EPSRC grant EP/D56743/1 Appendx A A lmt for Morrs mportance measure Let x l,,x k, be a quas-random sequence of ponts nsde so that for an arbtrary Remann ntegrable functon gx 1 N lm gx k = gxdx N N We consder one of the varables, say x, and let h be ts ncrement, < h <1 x Denote x = x 1,, x 1,x + h, x +1,, x n and Δf x = f x f x The followng algorthm s a verson of the modfed Morrs measure μ* Choose N ponts x k,1 k N, and N correspondng ncrements h k Compute 2N values fx k and f x k,1 k N Then μ = 1 N N f x k h k If fx does not depend on x, then μ*= Theorem A Assume that f/ x s Remann ntegrable and 2 f/ x 2 s bounded Then f N and max h k, then μ* μ, where μ = f x dx Proof Gven an arbtrary >, we choose N so large that 1 N f x k N x μ < ε 2 A1 Please cte ths artcle n press as: IM Sobol, S Kucherenko, Dervatve based global senstvty measures and ther lnk wth global senstvty ndces, Math Comput Smul 29, do:1116/jmatcom29123
9 IM Sobol, S Kucherenko / Mathematcs and Computers n Smulaton xxx 29 xxx xxx 9 We choose the ncrements h l,,h N so small that max h k 2 f/ x 2 ε The functon fx can be regarded as one-dmensonal functon of x Its ncrement has a form f x h = f x x + 1 f ˆx 2 h 2 x 2, where ˆx s a pont between x and xifx = x k and h = h k, the last term n ths expresson does not exceed ε/2, therefore we can easly prove that f x k f x k h k = x + r k, where the remander r k ε/2 Averagng the last relaton over 1 k N, we obtan μ 1 N f x k N x < ε 2 A2 From A1 and A2 t follows that μ μ <ε References [1] F Campolongo, J Carbon, A Saltell, An effectve screenng desgn for senstvty analyss of large models, Envronmental Modellng & Software [2] S Kucherenko, M Rodrguez-Fernandez, C Panteldes, N Shah, Monte Carlo evaluaton of dervatve based global senstvty measures, Relablty Engneerng & System Safety 29, n press [3] MD Morrs, Factoral samplng plans for prelmnary computatonal experments, Technometrcs [4] J Oakley, A O Hagan, Probablstc senstvty analyss of complex models: a Bayesan approach, Journal of Royal Statstcal Socety, Seres B [5] A Saltell, S Tarantola, F Campolongo, M Ratto, Senstvty Analyss n Practce, Wley, 24 [6] IM Sobol, The use of 2 -dstrbuton for error estmaton n the calculaton of ntegrals by the Monte Carlo method, Zh Vychsl Mat Mat Fz Englsh translaton: USSR Computatonal Mathematcs and Mathematcal Physcs [7] IM Sobol, Global senstvty ndces for nonlnear mathematcal models and ther Monte Carlo estmates, Maths and Computers n Smulaton [8] IM Sobol, S Kucherenko, Global Senstvty Indces for Nonlnear Mathematcal Models Revew, vol 1, Wlmott, 25, [9] IM Sobol, S Tarantola, D Gatell, S Kucherenko, W Mauntz, Estmatng the approxmaton error when fxng unessental factors n global senstvty analyss, Relablty Engneerng & System Safety Please cte ths artcle n press as: IM Sobol, S Kucherenko, Dervatve based global senstvty measures and ther lnk wth global senstvty ndces, Math Comput Smul 29, do:1116/jmatcom29123
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