Probabilistic Engineering Mechanics. Do Rosenblatt and Nataf isoprobabilistic transformations really differ?

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1 Probabilistic Egieerig Mechaics 4 (009) Cotets lists available at ScieceDirect Probabilistic Egieerig Mechaics joural homepage: wwwelseviercom/locate/probegmech Do Roseblatt ad Nataf isoprobabilistic trasformatios really differ? Régis Lebru a,, Ae Dutfoy b a EADS Iovatio Wors, System Egieerig - Applied Mathematics, Frace b EDF R&D, Idustrial Ris Maagemet, Frace a r t i c l e i f o a b s t r a c t Article history: Received 9 September 008 Received i revised form 9 February 009 Accepted 3 April 009 Available olie April 009 Keywords: Roseblatt trasformatio Elliptical copula Nataf trasformatio This article is the third i a series dedicated to the mathematical study of isoprobabilistic trasformatios ad their relatioship with stochastic depedece modellig, see [R Lebru, A Dutfoy, A iovatig aalysis of the Nataf trasformatio from the viewpoit of copula, Probabilistic Egieerig Mechaics (008) doi: 006/jprobegmech ] for a iterpretatio of the Nataf trasformatio i term of ormal copula ad [R Lebru, A Dutfoy, A geeralizatio of the Nataf trasformatio to distributios with elliptical copula, Probabilistic Egieerig Mechaics (4) (009), 7 78 doi:006/jprobegmech ] for a geeralisatio of the Nataf trasformatio to ay elliptical copula I this article, we explore the relatioship betwee two isoprobabilistic trasformatios widely used i the commuity of reliability aalysts, amely the Geeralised Nataf trasformatio ad Roseblatt trasformatio First, we recall the elemetary results of the copula theory that are eeded i the remaiig of the article, as a prelimiary sectio to the presetatio of both the Geeralized Nataf trasformatio ad the Roseblatt trasformatio i the light of the copula theory The, we show that the Roseblatt trasformatio usig the caoical order of coditioig is idetical to the Geeralised Nataf trasformatio i the ormal copula case, which is the most usual case i reliability aalysis sice it correspods to the classical Nataf trasformatio At this step, we also show that it is ot possible to exted the Roseblatt trasformatio to distributios with geeral elliptical copula the way the Nataf trasformatio has bee geeralised Furthermore, we explore the effect of the coditioig order of the Roseblatt trasformatio o the usual reliability idicators obtaied from a FORM or SORM method We show that i the ormal copula case, all these reliability idicators, excepted the importace factors, are uchaged whatever the coditioig order oe chooses I the last sectio, we coclude the article with two umerical applicatios that illustrate the previous results: the equivalece betwee both trasformatios i the ormal copula case, ad the effect of the coditioig order i the ormal ad o-ormal copula case 009 Published by Elsevier Ltd Itroductio I reliability aalysis, the ucertaity related to the iput variables is modelled with a multivariate probability distributio that defies a radom vector X of dimesio We suppose i this article that this distributio is absolutely cotiuous ad thus admit a joit desity fuctio These distributios are propagated through a physical model f to compute the distributio of the output variable Y Staeholders aim at taig a decisio o the Correspodig address: EADS Iovatio Wors, System Egieerig - Applied Mathematics, rue Pasteur, BP76, 95 Sureses, Frace Tel: ; fax: addresses: regislebru@eadset, regislebru@m4xorg (R Lebru), aedutfoy@edffr (A Dutfoy) basis of the realisatio of criteria, which may be the probability that the output variable overpasses a give threshold The evaluatio of this probability might require a high umber of simulatios, which might forbid the use of simulatio algorithms, particularly if the physical model has a high computatioal cost That is why a alterative to the simulatio methods has bee developed, based o a isoprobabilistic trasformatio that maps the physical space ito a ew space called the stadard space I that space, the evaluatio of the probability ca be doe usig cheap approximatios such as the FORM or SORM approximatios Two isoprobabilistic trasformatios are preseted i the literature: the Nataf trasformatio [,] which has bee exteded ito a Geeralised Nataf trasformatio [] ad the Roseblatt trasformatio [3] /$ see frot matter 009 Published by Elsevier Ltd doi:006/jprobegmech

2 578 R Lebru, A Dutfoy / Probabilistic Egieerig Mechaics 4 (009) The mai objective of this article is to compare the Geeralized Nataf trasformatio with the Roseblatt oe ad to prove that they are idetical i the ormal copula case, which is the most commo case i actual reliability studies as it correspods to the use of the classical Nataf trasformatio We also study the possibility to modify the Roseblatt trasformatio i a similar way that the Nataf trasformatio has bee exteded to lead to a o-ormal stadard space The secod objective of this article is to study the impact of the coditioig order of the Roseblatt trasformatio o the usual reliability idicators obtaied after a aalytical FORM/SORM method, with a focus o the ormal copula case We ote F X the -th uivariate margial cumulative distributio fuctio of the radom vector X ad C X its copula We ote F X, the cumulative distributio fuctio of the sub-vector (X,, X ) of X ad C X, its copula Whe o cofusio is possible, we remove the superscript i order to ease the readig: F X ad C X become F ad C Furthermore, we ote C N R is a ormal copula whose correlatio matrix is R We suppose that R is a symmetric positive defiite matrix We ote M, (R) the set of a real square matrix of dimesio, O (R) the subset of orthogoal matrices of M, (R), ie: Q O (R), Q Q t = I () ad GL (R) the subset of ivertible matrices of M, (R) If R = (r ij ) i,j M, (R), the R is its -leadig sub-bloc: R = (r ij ) i,j () ad R is the ( + )-th partial colum vector: R = (r,+,, r,+ ) t (3) We call stadard space the image space of a isoprobabilistic trasformatio Elemetary results o copula I this sectio, we recall some basic results of the theory of copulas We restrict ourselves to the strict miimum to follow further developmets The iterested reader is ivited to cosult eg [4] for a thorough itroductio to the theory of copulas, icludig a demostratio of the results preseted here The followig theorem lis the joit cumulative distributio fuctio of a multivariate distributio to a copula: Theorem (Slar, 959) Let F be a cumulative distributio fuctio of dimesio whose uivariate margial cumulative distributio fuctios are F i It exists a copula C of dimesio such that for x R, we have: F(x,, x ) = C(F (x ),, F (x )) (4) If the margial distributios F i are cotiuous, the copula C is uique; otherwise, it is uiquely determied o Rage(F ) Rage(F ) I the case of cotiuous margial distributios, for all u [0, ], we have: C(u) = F ( F (u ),, F (u ) ) (5) The copula is uchaged uder almost strictly icreasig margial trasformatio: Propositio If X has as a copula C ad if (α,, α ) are almost strictly icreasig fuctios defied respectively o the supports of the X i, the C is also the copula of (α (X ),, α (X )) If we are iterested i the distributio of the sub-vector (X,, X ) of the radom vector X, we have the followig property: Propositio 3 (-dimesioal Margial Distributios) Let X be a cotiuous radom vector with a distributio defied by its copula C ad its margial cumulative distributio fuctios F i The cumulative distributio fuctio F, of the -dimesioal radom vector (X,, X ) is defied by its margial distributios F i ad the copula C, through the relatio: F, (x,, x ) = C, (F (x ),, F (x )) (6) with C, (u,, u ) = C(u,, u,,, ) (7) Defiitio 4 (-th Coditioal Margial Distributio) Let X = (X,, X ) be a cotiuous radom vector The cumulative distributio fuctio of the coditioal variable X X,, X is defied by: F,, (x x,, x ) = F, (x,, x ) x x / F, (x,, x ) x x (8) Propositio 5 (-th Coditioal Margial Copula) Let X be a cotiuous radom vector with a distributio defied by its copula C ad its margial cumulative distributio fuctios F i The cumulative distributio fuctio of the coditioal variable X X,, X is defied by its margial distributios F i ad the copula C,, through the relatio: F,, (x x,, x ) = C,, (F (x ) F (x ),, F (x )) (9) with C,, (u u,, u ) = C, (u,, u ) u u / C, (u,, u ) u u (0) As a matter of fact, relatio (0) is the direct applicatio of Defiitio 4 to the cumulative distributio C Furthermore, Defiitio 4 ad relatio (4) lead to: F,, (x x,, x ) [ ] i= / C, (F (x ),, F (x )) = p i (x i ) u i= u [ ] i= C, (F (x ),, F (x )) p i (x i ) () u u i= where p i is the probability desity fuctio of X i Relatios () ad (0) lead to relatio (9) 3 The Geeralised Nataf ad Roseblatt trasformatios The Nataf trasformatio [] has bee itroduced as a procedure to trasform the uivariate margial distributios of a multivariate margial distributio Its usage i reliability aalysis has bee popularised by several authors, see eg [5,6] By the way, the Nataf trasformatio has bee preseted as

3 R Lebru, A Dutfoy / Probabilistic Egieerig Mechaics 4 (009) a way to perform reliability computatios usig icomplete depedece iformatio, ad it is oly recetly that its role as a probabilistic modellig tool has bee emphasised, lied to the copula theory [7]: the traditioal usage of the Nataf trasformatio supposes that the radom vector X has a ormal copula A geeralisatio to radom vectors with ay elliptical copulas has bee proposed i [], ad this is the geeralisatio which is recalled here: Defiitio 6 (Geeralised Nataf Trasformatio) Let X i R be a cotiuous radom vector defied by its uivariate margial cumulative distributio fuctios F X i ad its elliptical copula C X, characterized by its geerator ψ The Geeralised Nataf trasformatio T GN is defied by: U = T GN (X) = T GN T GN (X) () where both trasformatios T GN ad T GN are give by: T GN : R R X W = E F X (X ) E F X (X ) T GN : R R W U = Γ W (3) where E is the cumulative distributio fuctio of uivariate stadard elliptical distributio with characteristic geerator ψ ad Γ is the iverse of the Cholesy factor of R Aother widely used isoprobabilistic trasformatio is the Roseblatt trasformatio [3], defied by: Defiitio 7 (Roseblatt Trasformatio) Let X i R be a cotiuous radom vector defied by its uivariate margial cumulative distributio fuctios F X i ad its copula C X The Roseblatt trasformatio T R of X is defied by: U = T R (X) = T R T R (X) (4) where both trasformatios T R, ad T R are give by: T R : R R F X (X ) X Y = F X,, (X X,, X ) F X,, (X X,, X ) T R : R R Φ (Y ) Y U = Φ (Y ) (5) where F X,, is the cumulative distributio fuctio of the coditioal radom variable X X,, X Let us ote that T R maps X ito a uiformly distributed radom vector over [0, ] with idepedet copula For the demostratio, see [5, chap 7, p 3] Furthermore, T R maps Y ito a ormally distributed radom vector with zero mea ad uit covariace matrix: as a mater of fact, Propositio implies that U has the same copula as Y Besides, by costructio, the uivariate margial distributios of U are stadard ormal I order to ease the further compariso betwee the geeralised Nataf trasformatio ad the Roseblatt oe, it is useful to rewrite the Roseblatt trasformatio as follows: Defiitio 8 (Roseblatt Trasformatio, New Formulatio) Let X i R be a cotiuous radom vector defied by its uivariate margial cumulative distributio fuctios F X i ad its copula C X The ew formulatio of the Roseblatt trasformatio T NR is defied by: U = T NR (X) = T R T 0 (X) (6) where T 0 is give by: T 0 : R R X W = Φ F X (X ) (7) Φ F X (X ) where Φ is the cumulative distributio fuctio of the uivariate stadard ormal distributio, T R the Roseblatt trasformatio of Defiitio 7 Let us ote that U NR = T NR (X) = T R T R T 0(X) If W = T 0 (X), the, thas to (5), the th compoet of U NR writes: U NR = Φ F W,, (W W,, W ) (8) Thas to Propositio 5, the cumulative distributio fuctio of the coditioal variable W W,, W writes: F W (w,, w,, w ) ( ) = C W,, F W (w ) F W (w ),, F W (w ) (9) From Propositio, it follows that X ad W have the same copula C X Furthermore, by costructio of W, we have F W = Φ ad Φ(W ) = F X (X ) The, relatio (9) rewrites: F W,, (W W,, W ) ( ) = C X,, F X (X ) F X (X ),, F X (X ) which fially leads to the relatio: (0) F W,, (W W,, W ) = F X,, (X X,, X ) () ad the to: U NR = Φ F X,, (X X,, X ) () which is precisely the expressio of the Roseblatt trasformatio of Defiitio 7: U NR = T R (X) 4 Do geeralised Nataf ad Roseblatt trasformatios really differ? I this sectio, we first cosider the case where the copula of X is ormal, which is the most usual case i reliability aalysis sice it correspods to the case where the classical Nataf trasformatio applies The, we mae the compariso i all the other cases: o-ormal elliptical copulas ad o-elliptical copulas 4 The ormal copula case The ew formulatio (6) of the Roseblatt trasformatio maes it easier to show that whe X has a ormal copula, both trasformatios are idetical:

4 580 R Lebru, A Dutfoy / Probabilistic Egieerig Mechaics 4 (009) Propositio 9 (Isoprobabilistic Trasformatios, Normal Copula) Let X i R be a cotiuous radom vector defied by its uivariate margial cumulative distributio fuctios F X i ad its ormal copula The, the Roseblatt trasformatio ad the geeralised Nataf oe C N R are idetical: T R (X) = T GN (X) (3) We recall without demostratio some well-ow results about ormal vectors ad a easy-to demostrate result o orthogoal matrices that will be used i the demostratio of Propositio 9 Propositio 0 (Coditioal Normal Vector) Let U = (X, X ) t i R + be a ormal radom vector The the coditioal radom vector V = X X is a ormal vector which mea vector ad covariace matrix are defied by: { E[V] = E[X ] + Cov (X, X )[Cov (X, X )] (X E[X ]) Cov (V) = Cov (X, X ) Cov (X, X )[Cov (X, X )] Cov (X, X ) (4) Propositio (Affie Trasformatio) Let X i R be a ormal vector, with mea vector is µ, ad covariace matrix Σ, A a determiistic matrix i M,p (R) ad b i R p a determiistic vector The Y = A X + b is a ormal vector whose mea vector ad covariace matrix are defied by: { E[Y] = A µ + b Cov (Y) = A Σ A t (5) Propositio (Orthogoal Triagular Matrix with Positive Diagoal) Let T + (R) be the set of lower triagular matrix of M, (R) with positive diagoal elemets The T + (R) is a multiplicative subgroup of GL (R) Furthermore, T + (R) O (R) = {I } We ca ow start to demostrate Propositio 9, usig the ew formulatio of the Roseblatt trasformatio of Defiitio 8, whose differet steps are the followig oes: T NR : X T 0 W T R Y T R U (6) Let us ote S = (W,, W ) t ad V = W S As X has a ormal copula, W is a -dimesioal ormal vector whose uivariate margial distributios are stadard ormal ad which correlatio matrix is R Propositio 0 gives that for all, V follows a uivariate ormal distributio ad relatio (4) leads to: E[V ] = E[W ] + Cov (W, S )[Cov (S, S )] (S E[S ]) = Cov (W, S )[Cov (S, S )] S (7) Besides, we have Cov (W) = Cor (W) = R, ad give the otatios () ad (3), we have: E[V ] = ( R ) t [R ] S (8) Furthermore, relatio (4) also leads to: Var[V ] = Var(W ) Cov (W, S )[Cov (S, S )] Cov (S, W ) = ( R ) t [R ] R (9) Give relatios (8) ad (9), the th compoet of Y is defied by: Y = F W,, (W W,, W ) = Φ W ( R )t [R ] S ( (30) R )t [R ] R Fially, we obtai: U = Φ (Y ) = W ( R )t [R ] S ( R )t [R ] R = A W (3) where for all [, ], A = (a,,, a,, 0,, 0) M (R) with: a, = [ ( ] R )t [R ] R i= (3) a,j = a, r i r ji for j [, ] i= As A is a row matrix, U oly depeds o S Let Γ be the lower triagular matrix which lie is A The relatio (3) implies that: U = Γ W (33) which is very close to relatio (3) It remais to show that Γ = Γ Propositio implies that Cov (U) = Γ R Γ t ad Cov (U) = I by costructio of U If L is the Cholesy factor of R, the R = L L t, ad ( Γ L )( Γ L ) t = I, which leads to Γ L O (R) Furthermore, by costructio, Γ T + (R) As L T + (R), Propositio implies that Γ L T + (R) ad Γ L = I, which rewrites Γ = L = Γ I coclusio, we showed that i the case where X has a ormal copula, we have the relatio T R T R T 0(X) = T N T N (X) which leads to: T R (X) = T N (X) (34) Thus, the equivalece of the Roseblatt trasformatio ad the Geeralised Nataf trasformatio i the ormal copula case is show 4 The other cases I the case where the copula of X is elliptical but o-ormal, both isoprobabilistic trasformatios differ as their associated stadard spaces are differet As a matter of fact, the stadard spaces of the Geeralised Nataf is associated with the stadard spherical represetative of the elliptical family that defies the elliptical copula, whereas the stadard space of the Roseblatt trasformatio is associated to the ormal distributio At this step, it is iterestig to chec whether it is possible to modify the Roseblatt trasformatio i order to mae its stadard space be the same as the oe associated with the geeralised Nataf trasformatio I [7] we recall that the essetial characteristic of the stadard space is the spherical symmetry of its associated distributio, which gives a sese to the FORM ad SORM approximatios of the evet probability Let us ote that by costructio, because of the coditioig step T R, the Roseblatt trasformatio leads to a fial vector U with a idepedet copula

5 R Lebru, A Dutfoy / Probabilistic Egieerig Mechaics 4 (009) orthogoal trasformatio i the stadard space of the Roseblatt trasformatio More precisely, if we ote P the permutatio matrix associated to the arbitrary order, T R the Roseblatt trasformatio associated to this orderig, U = T R(X) ad U = T R (X), the we have: Q O (R) / U = Q U (35) Fig Roseblatt trasformatios whe the coditioig of the compoets W follow the caoical order or a arbitrary order Propositio 3 (Spherical Distributio with Idepedet Copula) The oly spherical distributios with idepedet compoets are the ormal distributios with zero mea ad scalar covariace matrix λi with λ > 0 See [8] for a demostratio Thus, the oly way to map a radom vector with a idepedet copula ito a radom vector followig a spherical distributio, is to map it ito a ormal vector such as described i this propositio: thus, the stadard space of the Roseblatt trasformatio is ecessarily the ormal oe Therefore, the stadard space of the Roseblatt trasformatio ad the stadard space of the Geeralised Nataf trasformatio oly coicide i the ormal copula case At last, for all the other cases where the copula of X is ot elliptical, the Geeralised Nataf trasformatio is ot defied ad the compariso with the Roseblatt trasformatio ot possible 5 Impact of the coditioig order i the Roseblatt trasformatio i the ormal copula case I the literature [5], the presetatio of the Roseblatt trasformatio is give with the warig that the coditioig order i step T R has a impact o the results obtaied from a FORM/SORM method Let us call caoical order the order preseted i the relatio (5) I that sectio, we study the impact of a chage i the coditioig order of the Roseblatt trasformatio o the quatities evaluated i the cotext of the use of the FORM or SORM methods: the desig poit, which is used through its orm (reliability idex) ad its ormalised squared coordiates (importace factors), ad the curvatures of the limit state surface at the desig poit i the stadard space, where the limit state surface is the frotier of the subspace of parameters verifyig the evet (for SORM approximatio) I the case where the copula of X is ot ormal, it has already bee show that such a chage has a impact o all these elemets: see the example quoted by [9] ad discussed by several authors, for example [5] ad [0] However, this is ot always the case We will study i more detail the most frequet situatio where the copula of X is ormal sice, as metioed previously, it is the copula iduced by the use of the classical Nataf trasformatio Let us suppose ow that we chage the order of coditioig It is equivalet to cosider the itroductio of a ew step i the Roseblatt trasformatio betwee the steps T 0 ad T R of relatio (6) i order to mae a permutatio of the compoets of W We ote P the permutatio matrix such as W = P W The Roseblatt trasformatios usig the caoical order or a arbitrary order are summarised graphically i Fig We have the followig result: Propositio 4 (Impact of the Order of Coditioig, Normal Copula) I the ormal copula case, chagig the order of the coditioig i the Roseblatt trasformatio cosists i maig a where Q ad P are i the same coected compoet of O (R), it meas det P = det Q Accordig to the otatios of Fig, if R is the correlatio matrix of the ormal copula of X, R the oe of W, Γ ad Γ the iverse of their respective Cholesy factors, the the matrices P ad Q are lied by: Q = Γ P Γ (36) The followig result will help for the demostratio of Propositio 4: Propositio 5 (Triagular Decompositio) Let A ad B be two determiistic matrices i M, (R), with B ivertible The we have: A A t = B B t B A O (R) (37) which meas that Q O (R) / A = B Q As a matter of fact, we have the followig implicatios: (B A )(B A ) t = B A A t B t = B B B t B t = I (38) which leads to the result of Propositio 5 As W = P W, W is a ormal vector which correlatio matrix verifies R = P R P t ad which Cholesy factor is L = Γ Therefore, R = L L t = (P L )(P L ) t Propositio 5 leads to: Q O (R) / P L = L Q (39) By multiplyig the relatio (39) o the left by Γ ad o the right by Γ, it rewrites: Γ P = Q Γ (40) which leads to the relatio betwee P ad Q give i relatio (36) We showed that i the ormal copula case, the mappig from W ito U is liear such as: U = Γ W Fially, we obtai: U = Γ P W (4) Relatios (40) ad (4) fially imply that: U = Q Γ W = Q U (4) as required Give that det(γ ) > 0 ad det(γ ) > 0, relatio (36) implies that det(q ) ad det(p ) have the same sig, which sigifies that they belog to the same coected compoet of O (R) I coclusio, if the radom vector X has a ormal copula, the effect of chagig the order of coditioig i the Roseblatt trasformatio with respect to the caoical order is to apply a further orthogoal trasformatio after applyig the Roseblatt trasformatio associated to the caoical orderig It chages the locatio of the desig poit, therefore its coordiates, but either its orm or the curvatures of the limit state surface at the desig poit Thus, i the cotext of the FORM or SORM method, the followig quatities do ot deped o the coditioig order of the Roseblatt trasformatio:

6 58 R Lebru, A Dutfoy / Probabilistic Egieerig Mechaics 4 (009) the Hasofer reliability idex [], which is the orm of the desig poit, the FORM approximatio of the evet probability which relies oly o the Hasofer reliability idex, the SORM approximatios of the evet probability which rely o both the Hasofer reliability idex ad the curvatures of the limit state fuctio at the desig poit However, the importace factors which are evaluated from the ormalised squared coordiates of the desig poit chage i a way which is ot i geeral a permutatio of the values obtaied usig the caoical order: relatio (36) implies that i geeral, Q P To be more precise, let us cosider the followig propositio: Propositio 6 (Orderig Idifferece) ( permutatio matrix P, Q = P ) (X has a idepedet copula) The first implicatio is obvious: if X has a idepedet copula, the correlatio matrix R is equal to the idetity matrix I, which implies that R = I, Γ = I, Γ = I ad fially Q = P The secod implicatio derives from the followig computatio By defiitio of Γ ad Γ, we have: Q = P L = P L P t (43) which implies the followig relatio o the coefficiets of L = (l i,j ) i,j ad L = (l i,j ) i,j : l i,j = l σ (i),σ (j) (44) where σ is the permutatio associated to P Thus, give that L ad L are lower triagular matrices, if the relatio (44) must hold for all the permutatios σ, it must hold i particular for ay traspositio τ ij that exchages i ad j, thus if i < j, l i,j = 0 by costructio, thus l σ (i),σ (j) = l ji = 0: L is a diagoal matrix ad cosequetly, R = I, which is equivalet to the idepedece of the compoets of X i the ormal copula case I coclusio, a permutatio with respect to the caoical order o the compoets of X always correspods to the same permutatio with respect to the caoical order of the compoets of the stadard space radom vector oly o the idepedet case Otherwise, the choice of the coditioig order does ot traslate ito a simple permutatio of the values of the desig poit coordiates This remar re-eforces the fact that it is difficult to iterpret the importace factors o a compoet basis i the case of correlated variables Remar 7 A similar questio may be raised cocerig the Nataf trasformatio, amely the choice of Γ, which is by o way restricted to the Cholesy factor for the stadard space to be associated with a spherical distributio: ay square root of the correlatio matrix would suit I particular, if P is a permutatio matrix (ad more geerally ay orthogoal matrix), L P is such a square root Let us recall that the exact value of the evet probability remais uchaged whatever the trasformatio we use, ad whatever the coditioig order we use for the Roseblatt trasformatio! 6 Numerical applicatios I this sectio, we illustrate the results obtaied i the previous sectios through two umerical applicatios We cosider a bi-dimesioal radom vector X = (X, X ) defied by its margial cumulative distributio fuctios (F, F ) ad its copula C For both applicatios, we choose expoetial distributios X Exp(λ ) ad X Exp(λ ) for the margial distributios ad a limit state surface defied by: 8X + X = 0 (45) We cosider the evet: 8X + X 0 (46) which we wat to evaluate the probability I the first applicatio, we choose a ormal copula C N R ( ) where R = ρ ρ ad ρ the correlatio coefficiet We chec both the equivalece betwee the caoical Roseblatt trasformatio ad the Geeralised Nataf trasformatio, ad the effect of a chage i the coditioig order I the secod applicatio, we choose o-elliptical copula, amely the Fra copula C θ, which belogs to the class of archimedea copulas, ad we verify that a chage i the coditioig order is ot equivalet to a orthogoal modificatio of the trasformatio ad has a impact o the FORM ad SORM approximatios We recall that the Fra copula is defied o [0, ] by the expressio: C θ (u, u ) = ( θ log + (e θu )(e θu ) ) e θ (47) where θ R For θ = 0, C θ is the idepedet copula I the umerical applicatios, we tae λ =, λ = 3, ρ = / ad θ = 0 6 Applicatio : Normal copula We use the ew expressio of the Roseblatt trasformatio of Defiitio 8, with the previous otatio W = T 0 (X), i that particular case of ormal copula Give Propositio 0, W W is a ormal radom vector such as E[W W ] = ρw ad( Var[W W ) ] = ρ, which implies that F W W (W W W ) = Φ ρw ρ Fially, the radom vector U is defied by: U = Φ F W (W ) = W U = Φ F W W (W W ) = W ρw ρ (48) The Roseblatt trasformatio with caoical order o the coditioig step fially defies the ormal radom vector U as: U = Φ F (X ) U = Φ F (X ) ρφ F (X ) (49) ρ I the Roseblatt stadard space, the limit state surface has the parametric expressio, where ξ [0, + [: u = Φ F (ξ) ( ) Φ F 8ξ ρφ F (ξ) (50) u = ρ

7 R Lebru, A Dutfoy / Probabilistic Egieerig Mechaics 4 (009) Limit State Surface i Stadard Space through Roseblatt Trasformatios Limit State Surface i Stadard Space through Roseblatt trasformatios u Caoical order Desig poit Iverse order Desig poit u Caoical order desig poit Iverse order desig poit u u Fig Trasformatios of the limit state surface ito the stadard space whe usig the caoical order i the Roseblatt trasformatio ad its iverse The liear correlatio is ρ = /, the copula is ormal, X Exp() ad X Exp(3) The limit state surface is 8X + X = 0 Note the symmetry that exchages the two curves: its matrix is Q With the same cosideratios, the Roseblatt trasformatio with the iverse order o the coditioig step defies the ormal radom vector Ũ as: Ũ = Φ F (X ) Ũ = Φ F (X ) ρφ F (X ) (5) ρ which leads, i the stadard space, to the other expressio of the limit state surface: ( ) 8ξ ũ = Φ F ( ) Φ F (ξ) ρφ F 8ξ (5) ũ = ρ Fig draws the graph of the limit state surface i the stadard space after both Roseblatt trasformatios Thas to relatio (36), we ca defie the orthogoal matrix Q The permutatio matrix is P = ( 0 0 Furthermore, we have 0 Γ = Γ = ρ ρ ρ ad fially ( ) ρ ρ Q = ρ ρ ) which leads to R = R We ca easily verify that Ũ = Q U Furthermore, Q is a permutatio matrix with det(q ) =, as the matrix P ( ) +ρ The director vector of the symmetry axis is, ρ I the umerical applicatio draw i Fig, the symmetry axis is ( 3/, /) Fig 3 Trasformatios of the limit state surface ito the stadard space whe usig the caoical order i the Roseblatt trasformatio ad its iverse The copula is a Fra oe with θ = 0, X Exp() ad X Exp(3) The limit state surface is 8X + X = 0 The Hasofer reliability idex is β = 30 ad the FORM approximatio of the evet probability p = P(8X + X < 0) is: p FORM = Φ ( β) = (53) A aalytical computatio of p leads to the umerical result: p = (54) Let us verify ow the equivalece betwee the Nataf trasformatio ad the Roseblatt oe (give that we cosider the caoical order) The Nataf trasformatio leads to the ormal radom vector U defied as: ( ) Φ U = F (X Γ ) Φ F (55) (X ) ( ) As Γ = 0 ρ ρ ρ, we have: U = Φ F (X ) U = ρφ F (X ) + Φ F (X ) (56) ρ ρ which is idetical to the expressio defied i (49) 6 Applicatio : Fra copula We cosider here the Fra copula, which is a o-elliptical copula This example proves that both limit state surfaces i the stadard space associated to two differet orders i the coditioig step of the Roseblatt trasformatio are ot lied by a orthogoal trasformatio We also illustrate that, accordig to this coditioig order, both reliability idex are differet which leads to differet FORM approximatios of the probability Fig 3 draws the graph of the limit state fuctio i the stadard space after both Roseblatt trasformatios The respective reliability idex are differet i both cases: { βcaord = 4 (57) β IvOrd = 7

8 584 R Lebru, A Dutfoy / Probabilistic Egieerig Mechaics 4 (009) which leads to differet FORM approximatios of the evet probability: { P FORM CaOrd = 07 0 P FORM = IvOrd (58) 0 There is a differece of 4% betwee the two approximatios, oly due to the coditioig order, whereas the exact probability value is the same A aalytical computatio of p leads to the umerical value: p = (59) 7 Coclusio This article is the third part of global reflectios o isoprobabilistic trasformatios Its first mai objective was to compare the geeralised Nataf trasformatio with the Roseblatt trasformatio ad show that i the ormal copula case, both trasformatios are idetical I the use of the Roseblatt trasformatio, there is a degree of freedom i the orderig of the coditioig step This poit is ofte preseted as a drawbac of this trasformatio, as it leads to differet umerical results for the FORM ad SORM approximatio The secod mai objective of the article was to show that, although the coditioig order has such a impact i geeral, i the ormal copula case there is ideed o impact o the FORM ad SORM approximatios as well as o the reliability idex The oly impact is o the importace factors i the case of correlated compoets for X, which uderlies the difficulty i iterpretig such factors i the correlated case The Nataf trasformatio has bee successfully geeralised to produce more geeral stadard spaces tha the ormal oe We showed that the Roseblatt trasformatio caot be geeralised this way Thus, for the case of a o-ormal elliptical copula, oe ca choose betwee both isoprobabilistic trasformatios: the Roseblatt trasformatio or the geeralised Nataf trasformatio We illustrated these results through two umerical applicatios, showig the equivalece of both trasformatios i the ormal copula case ad the effect of the coditioig order i a ormal ad o-ormal copula case Let us recall that the exact value of the evet probability remais the same whatever the trasformatio we use, ad whatever the coditioig order we use for the Roseblatt trasformatio It is oly the FORM ad SORM approximatios that are potetially modified Refereces [] Nataf A Détermiatio des distributios de probabilité dot les marges sot doées, Comptes Redus de l Académie des Scieces 5: 4 43 [] Lebru R, Dutfoy A A geeralizatio of the Nataf trasformatio to distributios with elliptical copula Probabilistic Egieerig Mechaics 009; 4:7 8 doi:006/jprobegmech [3] Roseblatt M Remars a multivariate trasformatio The Aals of Mathematical Statistics 3 (3): [4] Nelse R A itroductio to copulas Spriger; 999 [5] Ditlevse O, Madse HO Structural reliability methods Iteret editio March 004 [6] Der Kiureghia A, Liu PL Structural reliability uder icomplete probabilistic iformatio Joural of Egieerig Mechaics 986;():85 04 [7] Lebru R, Dutfoy A A iovatig aalysis of the Nataf trasformatio from the viewpoit of copula Probabilistic Egieerig Mechaics 008 doi:006/jprobegmech [8] Arold SF, Lych J O Ali s characterizatio of the spherical ormal distributio Joural of Royal Statistical Society 98;44():49 5 [9] Dolisy K First-order secod-momet approximatio i reliability of structural systems: Critical review ad alterative approach Structural Safety 983;(): 3 [0] lemaire M Fiabilité des structures Hermes 005 [] Hasofer AM, Lid NC A exact ad ivariat first order reliability format Joural of Egieerig Mechaics 974;00: [] Der Kiureghia A, Liu PL Multivariate distributio models with prescribed margials ad covariaces Probabilistic Egieerig Mechaics 986;(): 05