Hi Regis, thank you for your explanation. I have one more question based on your answer. You wrote that "the importance factors in standard space have no immediate meaning in the case of dependent inputs" what do you mean with this?
Thank you in advance! Kind regards, Anita On Thu, May 31, 2018 at 7:22 PM, regis lebrun < regis_anne.lebrun_dut...@yahoo.fr> wrote: > Hi Anita, > > As the computations are done in the standard space, the standard limit > state function is the composition of the physical space function > (f(x,y)=x+y) and the inverse isoprobabilistic transformation. As your input > distribution is Gaussian (Gaussian marginals and Gaussian copula), this > transformation is linear, hence the standard limit state function is linear > and the finite difference gradient is exact (no approximation). > > The isoprobabilistic transformation used in OpenTURNS is based on the > Cholesky factor of the covariance matrix of the standard representative of > the copula, which is somewhat arbitrary as explained in > https://www.sciencedirect.com/science/article/pii/S0266892009000307 or in > https://tel.archives-ouvertes.fr/tel-00913510 (title in French but > content in English), Chapter 5, Section 4. > > This transformation corresponds to the identity function if the > correlation is zero (independence in the Gaussian copula case) and to a > linear transformation in which u_1 depends only on x and u_2 on both x and > y, where u_1 and u_2 are the standard space coordinates. It corresponds to > a choice of conditioning order of the corresponding Rosenblatt > transformation. As a result, the standard space design point is not on the > first diagonal, which has no influence on the reliability index and the > FORM approximation (which is exact here) of the event probability. IMO the > importance factors in standard space have no immediate meaning in the case > of dependent inputs. > > I join a script to illustrate the case. > > Best regards > > Régis > > > > > > > > Le jeudi 31 mai 2018 à 10:26:41 UTC+2, Anita Laera <a.la...@plaxis.nl> a > écrit : > > > Hi, > I want to correctly use the correlation coefficients in the FORM analysis > I am performing using Abdo-Rackwitz algorithm. > > To study their effect, I have considered an elementary case with two > variables both normally distributed with mean equal to 0 and standard > deviation equal to 1. > In this way, if not correlated, they are transformed to two variables with > mean equal to 0 and standard deviation equal to 1 in the standard space. > > The limit state function is of the type y = variable_1 + variable_2 and > the threshold is set to 10. > > This is an extremely simple case with the only purpose of understanding > how the correlation works. > > I have set the gradient step size equal to 1 for both variables and I am > using a centered finite difference gradient. > > In the case of independent variables, starting from the mean point (0, 0), > 4 evaluations are performed to compute the gradient: > > 1 - var_1 = 1, var_2 = 0 (both in physical and standard space), y = 1 > 2 - var_1 = -1, var_2 = 0 (both in physical and standard space), y = -1 > 3 - var_1 = 0, var_2 = 1 (both in physical and standard space), y = 1 > 4 - var_1 = 0, var_2 = -1 (both in physical and standard space), y = -1 > > The first point of the line search is in (5, 5) (for both physical and > standard space). I can determine this based on the gradient [1, 1] and on > the value of lambda equal to -5. > > If I correlate the two variables by specifying a CorrelationMatrix with > coefficient of correlation equal to 0.5 and assigned a NormalCopula to the > correlation matrix, I obtain (after the calculation in the mean point): > > 1 - var_1 = 1, var_2 = 0 (physical space), var_1 = 1 var_2 = -0.57735 > (standard space), y = 1 > 2 - var_1 = -1, var_2 = 0 (physical space), var_1 = -1 var_2 = 0.57735 > (standard space), y = -1 > 3 - var_1 = 0, var_2 = 1 (physical space), var_1 = 0 var_2 = 1.1547 > (standard space), y = 1 > 4 - var_1 = 0, var_2 = -1 (physical space), var_1 = 0 var_2 = -1.1547 > (standard space), y = -1 > > The first point of the line search is in (5, 5) for the physical space, > corresponding to (5, 2.8867) in the standard space. > > I can't determine how the gradient, and then the first point for the line > search, is calculated. > > Could you please help me with this? > > -- > > *Anita Laera*, *Researcher* > [image: Plaxis:] <http://www.plaxis.nl> > <http://www.plaxis.nl> > Plaxis bv | Competence Centre Geo-Engineering > P.O. Box 572 | 2600 AN Delft | The Netherlands > Tel: +31 (0)15 251 7720 | Fax: +31 (0)15 2573 107 > <http://www.linkedin.com/company/plaxis-bv> > > > > > Follow us: [image: Facebook:] <https://www.facebook.com/plaxis>[image: > Twitter:] <https://twitter.com/Plaxis>[image: LinkedIn:] > <http://www.linkedin.com/company/plaxis-bv> > _______________________________________________ > OpenTURNS users mailing list > users@openturns.org > http://openturns.org/mailman/listinfo/users > -- *Anita Laera*, *Researcher* [image: Plaxis:] <http://www.plaxis.nl> <http://www.plaxis.nl> Plaxis bv | Competence Centre Geo-Engineering P.O. Box 572 | 2600 AN Delft | The Netherlands Tel: +31 (0)15 251 7720 | Fax: +31 (0)15 2573 107 <http://www.linkedin.com/company/plaxis-bv> Follow us: [image: Facebook:] <https://www.facebook.com/plaxis>[image: Twitter:] <https://twitter.com/Plaxis>[image: LinkedIn:] <http://www.linkedin.com/company/plaxis-bv>
_______________________________________________ OpenTURNS users mailing list users@openturns.org http://openturns.org/mailman/listinfo/users
