Probabilistic Engineering Mechanics. An innovating analysis of the Nataf transformation from the copula viewpoint

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1 Probabilistic Engineering Mechanics Contents lists available at ScienceDirect Probabilistic Engineering Mechanics journal homepage: An innovating analysis of the Nataf transformation from the copula viewpoint Régis Lebrun a, Anne Dutfoy b, a EADS Innovation Works, System Engineering - Applied Mathematics,, rue Pasteur BP76, 95 Suresnes Cedex, France b EDF R&D, Industrial Risk Management,, avenue du Général de Gaulle, 94 Clamart, France a r t i c l e i n f o a b s t r a c t Article history: Received July 7 Received in revised form 4 August 8 Accepted 4 August 8 Available online 3 August 8 Keywords: The Nataf transformation Copula Stochastic modelling This article gives new insight on the Nataf transformation, a widely used tool in reliability analysis. After recalling some basics concerning the copula theory, we explain this transformation in the light of the copula theory and we uncover all the hidden hypothesis made on the dependence structure of the probabilistic model when using this transformation. Some important results concerning dependence modelling are given, such as the risk related to the use of a linear correlation matrix to describe the dependence structure, and the importance of tail dependence in probabilistic modelling for safety assessment. This contribution should allow the reader to be much more aware of the pitfalls in dependence modelling when relying solely on the Nataf transformation. 8 Elsevier Ltd. All rights reserved.. Introduction The numerical study of a physical system requires the simulation of a set of equations which model its behaviour. These simulations are mainly computer intensive numerical simulations. We are interested, then, in taking a decision on the basis of a criterion evaluated from some characteristic variables, depending on the values of the input data of the model. Studies which treat uncertainties aim at evaluating the influence of the uncertainties related to the input data on the characteristic variables of the system. In the framework of probabilistic studies, input data are modelled with probabilistic distributions and propagated through the model to compute the distribution of the characteristic variables, which become a random vector associated to a probability density function. When the criterion is that one particular characteristic value exceeds a given threshold, the problem of evaluating the probability p of this threshold exceedance can be exposed as follows: let X = X,..., X n be the probabilistic input vector of the n uncertain input data of the model, f its joint probability density function, g: R n R the numerical model also called the limit state function, Y = gx the characteristic variable of interest, and s the given threshold. Then, p = PY > s = f xdx, where D s D s = {X R n /gx > s} is called the failure domain. Corresponding author. Tel.: ; fax: addresses: regis.lebrun@eads.net R. Lebrun, anne.dutfoy@edf.fr A. Dutfoy. In the reliability context, authors such as in [3] mention two main difficulties: g and the boundary of D s are not analytical expressions but are typically given by a finite element model often requiring high CPU costs, and f is unknown. The first point prevents the use of classical numerical methods to evaluate integrals Monte Carlo simulations,..., whereas the second point raises the problem of modelling a joint probability distribution based only on information often reduced to the marginal distributions of X and some linear correlation coefficients when one wants to take into account some dependence between the input parameters. That is why authors recommend the use of the Nataf isoprobabilistic transformation see [,] to map the physical space of the probabilistic input data into the standard space, where all the variables are independent and follow the same normal distribution with zero mean and unit variance. Then, within the standard space, we make a first-order or second-order geometrical approximation of the boundary of the failure domain, which allows us to compute an approximation of p thanks to an analytic expression. This method, widely used in probabilistic uncertainty propagation studies, makes several hypotheses which might not be fulfilled in that case, the Nataf transformation is not mathematically defined, or which might not agree with the real probability distribution of X in that case, the approximation might be largely wrong. In this article, we rewrite the Nataf transformation thanks to the copula theory. This innovating point of view explicates the hypotheses of the Nataf transformation in term of probabilistic modelling, which makes it possible to understand plainly the limitations of its use. In the first part of the article, we detail the Nataf transformation in its usual presentation as found in e.g. [6] and how it is usually 66-89/$ see front matter 8 Elsevier Ltd. All rights reserved. doi:.6/j.probengmech.8.8.

2 used in probabilistic propagation of uncertainties see e.g. []. We recall the interest of such a transformation and the related probabilistic indicators obtained as by-products. The second part introduces the concept of copula and elliptical distribution and recalls their principal properties. We focus on the key results that will be used in the demonstrations of the following parts. The concept of copula gives a new insight on the isoprobabilistic Nataf transformation and its hypothesis: in the third part, we demonstrate that the Nataf transformation makes the important hypotheses of a normal dependence structure for the random input vector X and maps it into a gaussian vector with independent, zero mean and unit variance components. Finally, we list all the hypotheses underlying the Nataf transformation and the possible risks associated with its use. In particular, we explain the probabilistic consequences of using a normal dependence structure and the difficulties related to its parameterization with a linear correlation matrix.. Traditional use of the Nataf transformation Very often, probabilistic data available about the random vector X are the marginal distributions which are supposed here to have finite second-order moments with cumulative distribution functions F i and, in the particular case of correlated components, the linear correlation matrix R = r ij ij, with [ ] Xi µ i Xj µ j r ij = E σ i σ j where µ i and σ i are the mean and standard deviation of X i. In order to perform reliability analysis such as the computation of a probability of failure, the probabilistic modelling is completed thanks to the Nataf transformation: Definition The Nataf Transformation, []. Let X be a random vector with marginal cumulative distribution functions F i and a linear correlation matrix R. The Nataf transformation T is the composition of two functions T = T T such that Φ F X Φ F X T : X Y =. Φ F n X n and T : Y U = Γ Y. 3 Here, Y is supposed to be a gaussian vector with a correlation matrix R supposed to be positive definite and with standard normal marginal distributions N,. The matrix Γ is any square-root of R and Φ is the cumulative distribution function of the standard normal distribution. The vector U is thus a gaussian vector with the same marginal distributions as Y but with independent components. R. Lebrun, A. Dutfoy / Probabilistic Engineering Mechanics r ij = E [ F i = σ i σ j ΦY i µ i σ i F j ΦY j µ j σ j ] { F R i Φy i µ i F j Φy j µ j ϕ y i, y j, r ij } dy i dy j 4 where ϕ is the bivariate standard normal probability density function with correlation r ij : ϕ y i, y j = exp y i r ij y i y j + y j π r r. 5 ij ij Remark 3. The computation of the coefficients r ij might be difficult for two reasons. The first one is that it involves the resolution of the integral equation 4, which is not guaranteed to have a solution, in particular if r ij is too close to or. The second one is that even if each coefficient r ij can be computed, there is no guarantee that the resulting matrix R will be symmetric definite positive. The Nataf transformation is said to map the physical space where X takes its values into the standard space where U takes its values. The interest of the standard space is that we can rewrite the expression of the probability of failure as p = PY > s = f x dx D s = ϕ n u du 6 D U s where the limit state function g has been transformed by T into G = g T and the failure domain D s into D U s = {U R n /GU > s}, where ϕ n is the probability density function of the standard n- dimensional normal distribution: ϕ n u = exp π n/ u. 7 The first expression involves the integral of the unknown function f over a complex domain D s, whereas the second expression involves the integral of the known function ϕ n over the complex domain D U s. The main interest of the Nataf transformation is that ϕ n is a rapidly decreasing function of u, which leads us to suppose that most of the contribution of ϕ n u to the integral 6 is concentrated in the vicinity of the point of D U s that is the nearest to the origin of the standard space. This point, called the design point and denoted P, is located on the hypersphere of minimal radius that is tangent to the boundary of the failure domain. It enables us to make a geometrical simplification of the failure domain D U s, by modifying its boundary. The so-called FORM method is obtained by a linearization of this boundary at the design point. We will not go further in the description of the FORM method and the other various extensions such as the SORM method, as we are mainly focused on the reinterpretation of the Nataf transformation as a tool for modelling stochastic dependence. Remark. A common choice for Γ is the Cholesky factor of R, with some extra precautions if R is ill-conditioned. See [7] for the definition of the Cholesky factor and how to manage the stability issue. The correlation matrix R is called the fictive correlation matrix. In general R R. Indeed, we have the following relation between R and R : 3. Introduction to dependence modelling and copula In this section, we recall some basic results on dependence concepts and on the copula theory, in order to enable the reader to reinterpret the Nataf transformation. For the first point, a much more detailed introduction can be found in [8], whereas [] gives a detailed introduction to the copula theory and the demonstration of all the results presented in this section.

3 34 R. Lebrun, A. Dutfoy / Probabilistic Engineering Mechanics From elementary textbooks, one knows what the stochastic independence between two random variables is: informally, these variables are said to be independent if any information gathered about one of them gives no information about the other. More formally, they are independent if and only if their joint distribution takes the form of a product of the marginal distributions. So, expressed this way, the modelling of the dependence between random variable is, in the most general case, the determination of the joint distribution of these variables. The modelling of stochastic dependence appears to be the determination of a multidimensional function which looks like a complex mathematical object. Several propositions can be found in the statistical literature to synthesize the dependence between two random variables through the determination of a scalar value associated to the two variables, leading to the concept of measure of association: it is a general concept that encompasses more specific ones, such as the measure of concordance and the measure of dependence see []. In this section, we review the three most usual candidates as a measure of association, namely the linear correlation, Spearman s rho and Kendall s tau, and see whether they are proper measures of association. Let us start with the definition of the general concept of measure of association: Definition 4 Measure of Association. A measure of association r between the components X and X of a random vector X, X t is a scalar-valued function of X and X with the following properties: rx, X If X and X are independent, rx, X = 3 If g and g are strictly increasing functions, we have: rx, X = rg X, g X. 8 A measure of association is then a normalized scalar which quantifies the way two random variables X and X are linked together, being positively associated if r > and negatively associated if r <. The case r = is an indication of a possible independence between X and X, but not more. The key point is the invariance of r by any change of scale for X and X, even a nonlinear one. Let us see if the most widely used coefficients for tracking dependence are proper measures of association. The linear correlation is often used as a measure of association. This is mainly because, in the context of gaussian vectors, the part of the joint distribution function which is related to the dependence structure is exactly parameterized by the linear correlation matrix. Definition 5 Linear Correlation. Let X, X t be a random vector with finite second moments. The linear correlation ρx, X between X and X is given by [ ] X µ X µ ρ = E 9 σ σ where µ i and σ i are the mean and standard deviation of X i. It is well known that the linear correlation coefficient does not fulfil point 3 of the definition of a measure of association: it is readily seen with the relation 9. The sampling definition of this coefficient is: Definition 6 Linear Correlation, Sampling. Let x k, xk t k be a sample of size N of the random vector X, X t. The sampling linear correlation coefficient ˆρX, X is given by ˆρX, X = N N N x k xk N k= N N x k k= x k k= N x k x k k= k= N N N x k k= x k k=. In order to fix the non-invariance of the linear correlation by a nonlinear marginal transformation, the computation of the linear correlation can be made on the rank of the variables instead of their values. This leads to the definition of Spearman s rho: Definition 7 Spearman s Rho. Let X, X t be a random vector with marginal cumulative distribution functions F and F. Spearman s rho ρ S X, X is defined by ρ S X, X = ρf X, F X. The only drawback of the linear correlation as a measure of association has been removed, so Spearman s rho is a proper measure of association. The sampling definition of this coefficient is: Definition 8 Spearman s Rho, Sampling. Let x k, xk t k be a sample of size N of the random vector X, X t. If there is no tie, i.e. i, j, i j x i xj or xi xj, the sampling Spearman s rho ˆρ S X, X is given by N 6 d k k= ˆρ S X, X = NN where d k = rankx k rankxk ; otherwise it is the sampling definition of the linear correlation coefficient applied to the ranks of the sample. Instead of using a measure based on correlation, one can use another approach based on the notion of concordance. If X, X t is an independent copy of the random vector X, X t, X and X are said to be positively concordant or concordant for short if P X X X X > > /, and negatively concordant or discordant for short if P X X X X < > /. Kendall s tau is a measure of association based on this concept: Definition 9 Kendall s Tau. Let X, X t be a random vector, and let X, X t be an independent copy of X, X t. Kendall s tau τx, X is the difference between the probability of concordance and the probability of discordance between X and X : τx, X = PX X X X > PX X X X <. 3 As the notion of concordance is clearly invariant by strictly increasing marginal transformation, Kendall s tau is a proper measure of association. The sampling definition of this coefficient is: Definition Kendall s Tau, Sampling. Let x k, xk t k be a sample of size N of the random vector X, X t. The sampling Kendall s tau ˆτ S X, X is given by ˆτ S X, X = N 4 N i= j=+ x i xj xi xj > x i xj xi xj < NN where A is equal to if A is true, and zero otherwise. 4

4 R. Lebrun, A. Dutfoy / Probabilistic Engineering Mechanics Despite the simplicity of these measures of association, they are not able to fully describe the dependence structure of a random vector. At best, they can identify that the components are not independent, but no more. Going back to the real task, the definition of the joint probability density function of the random vector, we introduce the cornerstone of the full modelling of stochastic dependence, namely the concept of copula: Definition. A copula is a function C defined on [, ] n verifying: for all u [, ] n with at least one component equal to, Cu = C is grounded; C is n-increasing: i = i + +i n Cu i,..., u nin 5 i n = with u j = a j and u j = b j j {,..., n} and a, b [, ] n, a b 3 For all u [, ] n with u i = i {,..., n}, i k: Cu = u k. 6 A copula can be seen as the restriction to [, ] n of the cumulative distribution function of a distribution whose support is [, ] n and with uniform marginal distributions on [, ]. It is not evident from the definition that this concept is the best suited for the modelling of stochastic dependence. It is made clear thanks to the following theorem: Theorem Sklar, 959. Let F be a cumulative density function of dimension n whose marginal distributions are F i. There exists a copula C of dimension n such that for x R n, we have Fx,..., x n = CF x,..., F n x n. 7 If the marginal distributions F i are continuous, the copula C is unique; otherwise, it is uniquely determined on RangeF RangeF n. In the case of continuous marginal distributions, for all u [, ] n, we have Cu = FF u,..., F n u n 8 and px = cf x,..., F n x n n p i x i 9 i= where p i is the probability density function of the i-th marginal distribution of X and c is defined by n C cu,..., u n = u,..., u n. u... u n From this theorem, the role of copulas as a dependence modelling tool is clear: the value taken by any joint cumulative distribution function is the value taken by a copula, once the effect of the marginal cumulative distribution functions have been taken into account. Conversely, a copula is what remains of a joint cumulative distribution once the action of the marginal cumulative distribution functions has been removed. From a dependence modelling point of view, the first result of Sklar s theorem explains how to build a full probabilistic model based on a set of D marginal distributions and a dependence structure given by a copula. The second result explains how to build a catalogue of reusable dependence structures from a given set of multidimensional distributions. Some examples of bidimensional copulas are given in Table. In order to apprehend the effect of switching from one copula to another in dependence modelling, and to see that this task is Table Examples of usual bidimensional copulas Name Cu, u Independent u u Φ Normal u Φ u exp s ρst+t ds dt π ρ ρ T ν u Student T ν u + s ρst+t ν+/ ds dt π ρ ν ρ Frank θ log + e θu e θu e θ Clayton u θ + u θ /θ Gumbel exp logu θ + logu θ /θ much more involved than the determination of a coefficient of correlation, we draw the joint probability distribution function of several bidimensional distributions with the same standard normal marginal distributions, a linear correlation of.8 and different copulas; see Fig.. In addition to Sklar s theorem, we give the most useful properties of copulas for our purpose: Proposition 3. If X has as a copula C and if α,..., α n are n strictly increasing functions defined respectively on the supports of the X i, then C is also the copula of α X,..., α n X n. This property shows that any measure of association between X and X must be a function only of the copula that links X and X and not of their marginal distributions. If we are interested in the bidimensional marginal distributions of a random vector X, we have the following property that links the copula of a bidimensional extracted random vector X i, X j and the copula of the distribution of X: Proposition 4 Bidimensional Marginals. Let X be a random vector with a distribution defined by its copula C and its marginal distributions F i. The cumulative distribution function F ij of the bidimensional random vector X i, X j with i < j is defined by its marginal distributions F i, F j and the copula C ij through the relation F ij x i, x j = C ij F i x i, F j x j with C ij u i, u j = C,...,, u i,,...,, u j,,...,, where u i and u j are respectively at position i and j. This result will be used to reformulate the relation 4 in terms of extracted bidimensional copulas. 4. New interpretation of the Nataf transformation through the copula theory The Nataf transformation is the composition of two transformations T and T, with an additional hypothesis that upon the action of T, the initial random vector X is mapped into a gaussian vector Y = T X. Formalizing the hypothesis underlying the Nataf transformation leads to: Proposition 5 Normal Copula Through the Nataf Transformation. Let X be a random vector with unknown copula C X, known marginal cumulative distribution functions F i and known linear correlation matrix R. Assuming that this vector is mapped into a gaussian vector Y = T X with distribution N, R upon the action of T as defined in is equivalent to the assumption that C X is the normal copula parameterized by the correlation matrix R. The demonstration is a direct application of the invariance of the copula by strictly increasing transformation of the components of a random vector. By definition of the normal copula, the copula C Y of Y is exactly the normal copula C N R parameterized by cor[y] = R. Then, the transformation T is bijective, and its inverse is

5 36 R. Lebrun, A. Dutfoy / Probabilistic Engineering Mechanics Fig.. Iso-density contours of three copulas left column and three distributions right column built upon these copulas with standard normal marginal distributions and a linear correlation ρ =.8. It is worth noticing the differences between these probability density functions, even if they share the same marginal distributions and the same linear correlation matrix. T : Y X = T Y = F ΦY F ΦY.. F n ΦY n This transformation only acts on the marginal distributions of Y, and is a strictly increasing transformation which preserves the copula of the transformed random vector see Proposition 3. We conclude that C X = C Y = C N R. From the definition 4 of the correlation matrix R and the expression of a bidimensional marginal probability density function using 9 and, we have r ij = x i µ i x j µ j c ij F i x i, F j x j σ i σ j R

6 R. Lebrun, A. Dutfoy / Probabilistic Engineering Mechanics Fig.. Iso-density contours of a copula of a normal mixture left picture and a distribution right picture with this copula and standard gaussian marginal distributions. This copula is well suited in modelling real world dependent quantities such those encountered in production control. f i x i f j x j dx i dx j 3 where c ij is the probability density function [ of the ] bidimensional rij normal copula with correlation matrix. r ij From a dependence modelling point of view, the use of the Nataf transformation is equivalent to the choice of a normal copula for the joint distribution of the input random vector X. This copula is parameterized by a correlation matrix R in such a way that the joint distribution has the given linear correlation matrix R. The relation 3 allows us to compute R from R if this last matrix is compatible with both the choice of marginal distributions and the choice of a normal copula. 5. Potential pitfalls of using the Nataf transformation due to the gaussian copula hypothesis Thus, the use of Nataf s transformation is a contoured way of choosing a normal copula as a dependence structure for the input random vector. In this section, we highlight some of the consequences of such a choice. Fig. showed that with the usual available information, namely the marginal distributions and the linear correlation matrix, it is possible to choose different copulas that lead to joint distributions with exactly these characteristics, despite the visible difference between the corresponding joint probability density functions. Thus, the choice of the normal copula implies a very specific form of dependence structure, which might not suit the problem considered. The symmetry that is visible on these graphs is only due to the specific choice of copulas taken for this illustration. A very common example of multidimensional distribution that arises in an industrial context such as the control of production is the case of mixtures of gaussians. We give an example of such a distribution: Fx, x = Φ x, x + Φ x, x 4 where Φ is the cumulative distribution function of the bidimensional normal distribution with mean vector µ =., t, marginal standard deviations σ =.7,.3 t and correlation matrix R [ ] =.936 and Φ.936 is the cumulative distribution function of the bidimensional normal distribution with mean vector µ =.,. t, marginal standard deviations σ =.8,. t and correlation matrix R = R. The copula of F is obtained thanks to the Sklar s theorem: Cu, u = FF u, F u. 5 Then, we build the distribution G with copula C and standard normal marginal distributions. The cumulative distribution function G of this distribution is obtained thanks to the Sklar s theorem: Gx, x = CΦ, x, Φ, x. 6 The value ρ =.936 has been chosen in such a way that G has a linear correlation of.8. We can see in Fig. the iso-density contours of C and G, which are clearly very different from the figure we get with a normal copula. Up to now, we have only illustrated the global effect of adopting the normal copula instead of another one on the whole support of the joint distribution. We may wonder whether these differences only affect the central part of the distribution or also modify its behaviour in the extreme values, which is the region we are interested in when computing low levels of probability. To study this effect, we introduce a new measure of association that is better suited to summarize the dependence structure in the joint extreme values than the more global measure that has been presented so far, namely the coefficients of upper and lower tail dependence. Definition 6 Coefficient of Upper Tail Dependence. The coefficient of upper tail dependence, λ U, of a bidimensional random vector of X i, X j with marginal cumulative density functions F i and F j, is defined as λ U = lim PX j > F j q X i > F i q 7 q given that this limit λ U [, ] exists. In other words, the coefficient of upper tail dependence is the probability that the random variable X j exceeds its quantile of order q, knowing that X j exceeds its quantile of the same order, when this order tends to. In the same way, we have for the lower tail: Definition 7 Coefficient of Lower Tail Dependence. The coefficient of upper tail dependence, λ U, of a bidimensional random vector of X i, X j with marginal cumulative density functions F i and F j, is defined as λ L = lim PX j < F j q X i < F i q 8 q given that this limit λ L [, ] exists.

7 38 R. Lebrun, A. Dutfoy / Probabilistic Engineering Mechanics Fig. 3. Evolution of PY > G q X > F q with q when X and Y are linked by the normal copula or Student s copula, with ρ S =.. One see that even for moderately correlated variables, the behaviour might dramatically change in the extreme values according to the value of the tail dependence. One can show see [5] that if the vector X i, X j is continuous, then λ U = lim q + Cq, q/ q 9 q and λ L = lim Cq, q/q 3 q where C is the copula of X i, X j. It is readily seen that this coefficient is a measure of association. The notion of tail dependence is of first importance in the study of extreme values of dependent phenomena. For the normal copula, λ U = λ L =, thus it is not possible to take into account any positive tail dependence with this copula. With another copula, the Student copula see Table for example, it is possible to take such tail dependence into account, as we have for this copula: λ U = λ L = T ν+ + ν ρ + ρ. 3 This copula should be better suited to model the dependence structure when one is interested, for example, in a failure domain that corresponds to simultaneous large values of two components of the input random vector. To illustrate this point, we consider two dependence modellings based first on a normal copula and second on a Student copula. These copulas share the same Spearman s rho of ρ S =., which leads to the linear correlation coefficients ρ gauss =.47 and ρ Student =.95 respectively for the normal copula and the Student copula. We see that the two linear correlation coefficients are very close and would be difficult to distinguish if they were estimated from real world data. The evolution of PY > G q X > F q with q can be seen on Fig. 3. We can see that even for moderately correlated variables, the presence of tail dependence might dramatically change the behaviour of the distribution in its extreme values. This situation is a very typical one: when we are interested in the evaluation of a probability of failure for a system with dependent random parameters, most of the time several such parameters are in their extreme quantiles in the failure domain. Let us consider a bidimensional failure domain of the form D = {x, y x x q, y y q } 3 where x q = F X q, y q = F Y q and q. If the vector X, Y has an upper tail dependence of λ U >, the probability of failure is of order P qλ U, but otherwise it is of order P qε q, where ε q when q. This means that we can be wrong not only by a constant factor, but by several orders of magnitude if we do not take the upper tail dependence into account in our probabilistic model. The same figure shows that the impact is much less important when we are interested in the central behaviour of the system, which means when the parameters are around their median value q =.5 in the figure. From these observations, it seems that even if a measure of association is by no way a full representation of the dependence structure, some such measures are better suited to summarize this structure in the case of extremal events or in the central part of the joint distribution. This choice of measure of association has to be made in relation with the choice of a specific family of copulas. Without going too far into the problem of the selection of a copula, it is clear that the a priori restriction to a copula with no tail dependence such as the normal copula might lead to a very underestimated probability of failure, with all the consequences associated to this kind of error. 6. Potential pitfalls of using the linear correlation to parameter the Nataf transformation The tradition has consecrated the use of the linear correlation matrix as a first attempt to describe the presence of stochastic dependence. We have already seen that this choice is not optimal from the viewpoint of the notion of measure of association. We have also mentioned the two difficulties associated to the determination of the normal copula parameters from a given set of marginal distributions and a linear correlation matrix. In this section, we give more theoretical insight on these remarks. The main result upon which we will build our analysis is the Frechet Hoeffding Theorem.

8 R. Lebrun, A. Dutfoy / Probabilistic Engineering Mechanics Theorem 8 Frechet Hoeffding. Let X, X t be a bidimensional random vector with marginal cumulative distribution functions F and F. If we suppose that this vector has finite second moments, the linear correlation coefficient ρ between X and X can take any value in an interval [ρ min, ρmax ] included in [, ], this inclusion being strict in general. Both limits of this interval depend on the marginal cumulative distribution functions F and F and are reached when X and X are strictly counter-monotonic and co-monotonic. This theorem means that for specific choices of marginal distributions, there exist values in [, ] that cannot be reached whatever the copula we choose: these values are simply not compatible with the chosen marginal distributions. An illustration of this fact is given by the following example, taken from [5]: Proposition 9. Let X and Y follow a lognormal distribution: X LN, and Y LN, σ. The linear correlation between X and Y is restricted to the interval [ρ min, ρ max ] [, ] with ρ min = e σ e e σ and ρ max = e σ. e e σ We note that ρ min and ρ max tend to as σ goes to + : the linear correlation between X and Y can be made as small as desired, even if Y is a strictly increasing or decreasing function of X in which cases we could have expected correlations close to and. If we restrict ourselves to the normal copulas by using the Nataf transformation, this can only emphasize this restriction. The direct consequence of this incompatibility is the impossibility to solve Eq. 4 for some pair of components X i, X j t. This problem makes it very difficult to determine the linear correlation matrix R from expert judgements: in fact, experts should take into account the type of the marginal distributions to determine its coefficients. But, so far, the evaluation of the linear correlation matrix coefficients and the determination of the marginal distributions are usually two distinct processes, performed independently, sometimes even by experts of different kinds: the linear correlation matrix coefficients are often evaluated from experts of the physics under interest, whereas the evaluation of the marginal distributions is often performed by statisticians who base their estimation on some particular data, according to some particular criteria e.g. maximum likelihood, maximum entropy etc. Furthermore, several studies of probability uncertainty treatment propose sensitivity studies which consist of considering a set of different marginal distributions without changing the linear correlation coefficients, and without verifying the constraints expressed by Frechet Hoeffding s theorem. These situations of incompatibility are often indirectly detected for example, in the field of a threshold exceedance study, the uncertainty treatment software will indicate that it is not possible to apply the Nataf transformation because the Eq. 4 has no solution, and the reason why it does not work is not made explicit. Finally, the linear correlation matrix evaluated by experts must be positive and symmetric, with all its diagonal elements equal to and the others in [, ]. If the first properties are verified by construction of the matrix, the last one, in return, is generally not verified when the linear correlation matrix is obtained from experts. This problem becomes increasingly severe when the dimension of X grows: the set of positive matrices becomes negligible in the set of symmetric matrices with unit diagonal and off-diagonal coefficients in [, ] when this dimension increases. Some of these problems can be solved by using another measure of association to parameter the dependence structure. Spearman s rho as well as Kendall s tau can be used: as they are functions of the copula only, they do not have to fulfil a compatibility condition with the marginal distributions. For a general copula that depends on a vector of parameters θ, one can use the following relations to find the value of the parameters associated to the value of these measures of association: Proposition Spearman s Rho and Copula. Let X, X t be a bidimensional random vector with copula C θ. Spearman s rho ρ S X, X can be written as ρ S X, X = C θ u, v du dv [,] Proposition Kendall s Tau and Copula. Let X, X t be a bidimensional random vector with copula C θ. Kendall s tau τ S X, X can be written as τx, X = 4 [,] Cu, v dcu, v. 34 For the special case of normal copulas, these expressions can be inverted, which leads to the following relations: Proposition Normal Copula Case. Let X, X t be a bidimensional random vector with a normal copula C N R. The off-diagonal term r of its copula correlation matrix R is related to Spearman s rho and Kendall s tau by the following relations: π r = sin 6 ρ SX, X 35 π r = sin τx, X. 36 Using these relations, it is very easy to compute the whole correlation matrix of a multidimensional normal copula from a matrix of corresponding Spearman s rho or Kendall s tau. Nevertheless, one must be aware of the fact that all the problems have not been cured: one has to check that the resulting correlation matrix is positive definite. If this is not the case, it would signify that the given Spearman s rho matrix or Kendall s tau matrix is not compatible with the normal copula hypothesis. 7. Conclusion The objective of this article is to take benefit from the copula theory to give more insight on the Nataf transformation than the presentations given so far. The central role played by this transformation in probabilistic safety assessment studies plainly justifies this need of insight. This innovating viewpoint enabled us to demonstrate that the Nataf transformation is a particular modelling of the stochastic dependence, using the normal copula. Furthermore, the traditional use of the Nataf transformation requires the linear correlation matrix of the input random vector in order to parameterize the normal copula. We showed the consequences of such an hypothesis and choice of parameters, which enabled us to guard against the pitfalls of a systematic use of the Nataf transformation, as presented in the literature. In particular, we showed the impact of the choice of a normal dependence structure on the morphology of the probabilistic distribution of the input random vector and on its tail dependence properties. Furthermore, we made explicit why using the linear correlation matrix in order to parameterize a normal copula might cause great difficulties, mainly because of the Frechet Hoeffding theorem which constrains the linear coefficients within a range of variation depending on the marginal distributions of the random vector.

9 3 R. Lebrun, A. Dutfoy / Probabilistic Engineering Mechanics In particular, this viewpoint enabled us to understand why the application of the Nataf transformation sometimes appears impossible, which has never been explained so far. Finally, we raised the difficulties inherent to the determination of the linear correlation matrix by expert judgements, often realized independently of the determination of the marginal distributions of the random vector. In order to deal with these difficulties, we proposed the parameterization of the normal copula from the Spearman s rho correlation matrix or the Kendal s tau matrix: these measures of association are more adapted to give information on the dependence structure than the linear correlation coefficient. Thanks to this innovating viewpoint, the Nataf transformation can be extended to more general dependence structures, namely the elliptical copulas of which the normal copula is a special case see [9]. A global methodology for the probabilistic modelling of uncertainty has been developed for several years, as a joint effort of EDF R&D, EADS Innovation Works and PhiMECA; see [4]. The modelling of stochastic dependence through the use of copulas is at the heart of this methodology. References [] Der Kiureghian A, Liu PL. Structural reliability under incomplete probabilistic information. Journal of Engineering Mechanics 986;:85 4. [] Der Kiureghian A, Liu PL. Multivariate distribution models with prescribed marginals and covariances. Probabilistic Engineering Mechanics 986;: 5. [3] Ditlevsen O, Madsen HO. Structural reliability methods. John Wiley & Sons; 996. [4] Dutfoy A, Dutka-Malen I, Lebrun R, Conty R. et al. Open TURNS, an open source initiative to treat uncertainties, risks N statistics in a structured industrial approach. In: ESREL 7 conference. 7. [5] Embrechts P, Lindskog F, McNeil A. Modeling dependence with copulas and applications to risk management. In: Rachev S, editor. Handbook of heavy tailed distributions in finance. Elsevier; 3. p [Chapter 8]. [6] Hasofer AM, Lind NC. An exact and invariant first order reliability format. Journal of Engineering Mechanics 974;:. [7] Higham NJ. Accuracy and stability of numerical algorithms. SIAM; 996. [8] Joe H. Multivariate models and dependence concepts. Chapman & Hall; 997. [9] Lebrun R, Dutfoy A. A generalization of the Nataf transformation to distributions with elliptical copula. Probabilistic Engineering Mechanics 8, doi:.6/j.probengmech [] Nataf A. Détermination des distributions de probabilités dont les marges sont données. Comptes Rendus de l Académie des Sciences 96;A 5:4 3. [] Nelsen R. An introduction to copulas. Springer; 999.