# function expressed in X variables to a limit state | My Assignment Tutor

function expressed in X variables to a limit state function expressed in U variables, introduce a transformation in between where we obtain that the considered random variables first are standardized before they are made uncorrelated. I.e. the row of transformations yields:X.Y.0In the following we will see how this transformation may be implemented in the iterative procedure outlined previously. Let as assume that the basic random variables X are correlated with covariance matrix givenas: Var[X,] Cov[X Cov[X,, X „ C„ = (11.22) Cov[X „, X ,] Var[X n] and correlation coefficient matrix py : 1 P,„1 p„ =[: 1 (11.23) P„, 1If on y the diagonal elements of these matrixes are non-zero clearly the basic random variables are uncorrelated.As before the first step is to transform the n-vector of basic random variables X into a vector of standardised random variables Y with zero mean values and unit variances. This operation may be performed by X, — fly 1,2,..n (11.24) whereby the covariance matrix of Y, i.e. Cy is equal to the correlation coefficient matrix of X , i.e. py The second step is to transform the vector of standardized basic random variables Y , into a vector of uncorrelated basic random variables U . This last transformation may be performed in several ways. The approach described in the following utilises the Choleski factorisation from matrix algebra and is efficient for both hand calculations and for implementation in computer programs.The desired transformation may be written as Y = TU (11.25) where T is a lower triangular matrix such that Tu = 0 for j > i . It is then seen that the covariance matrix C5 can be written as: Cy = ELY • Yri= E[T U ‘UT •TT]=T. E[U.Uri.TT =T ‘TT = p„ (11.26) from which it is seen that the components of T may be determined as:

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