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>
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