# Iso-probabilistic transformation from U space to X space `UtoX` performs as iso-probabilistic transformation from standardized space (U) to physical space (X) according to the NATAF transformation, which requires only to know the means, the standard deviations, the correlation matrix $\rho(Xi,Xj) = \rho_{ij}$ and the marginal distributions of Xi. In standard space, all random variables are uncorrelated standard normal distributed variables whereas they are correlated and defined using the following distribution functions: Normal (or Gaussian), Lognormal, Uniform, Gumbel, Weibull and Gamma. ```r UtoX(U, input.margin, L0) ``` ## Arguments - `U`: a matrix containing the realisation of all random variables in U-space - `input.margin`: A list containing one or more list defining the marginal distribution functions of all random variables to be used - `L0`: the lower matrix of the Cholesky decomposition of correlation matrix R0 (result of `ModifCorrMatrix`) ## Returns - **X**: a matrix containing the realisation of all random variables in X-space ## Details Supported distributions are : * NORMAL: distribution, defined by its mean and standard deviation $$ distX \<- list(type="Norm", MEAN=0.0, STD=1.0, NAME="X1") $$ * LOGNORMAL: distribution, defined by its internal parameters P1=meanlog and P2=sdlog (`plnorm`) $$ distX \<- list(type="Lnorm", P1=10.0, P2=2.0, NAME="X2") $$ * UNIFORM: distribution, defined by its internal parameters P1=min and P2=max (`punif`) $$ distX \<- list(type="Unif",P1=2.0, P2=6.0, NAME="X3") $$ * GUMBEL: distribution, defined by its internal parameters P1 and P2 $$ distX \<- list(type='Gumbel',P1=6.0, P2=2.0, NAME='X4') $$ * WEIBULL: distribution, defined by its internal parameters P1=shape and P2=scale (`pweibull`) $$ distX \<- list(type='Weibull', P1=NULL, P2=NULL, NAME='X5') $$ * GAMMA: distribution, defined by its internal parameters P1=shape and P2=scale (`pgamma`) $$ distX \<- list(type='Gamma', P1=6.0, P2=6.0, NAME='X6') $$ * BETA: distribution, defined by its internal parameters P1=shape1 and P2=shapze2 (`pbeta`) $$ distX \<- list(type='Beta', P1=6.0, P2=6.0, NAME='X7') $$ ## See Also `ModifCorrMatrix`, `ComputeDistributionParameter` ## Author(s) gilles DEFAUX, gilles.defaux@cea.fr ## Examples ```r Dim = 2 distX1 <- list(type='Norm', MEAN=0.0, STD=1.0, P1=NULL, P2=NULL, NAME='X1') distX2 <- list(type='Norm', MEAN=0.0, STD=1.0, P1=NULL, P2=NULL, NAME='X2') input.margin <- list(distX1,distX2) input.Rho <- matrix( c(1.0, 0.5, 0.5, 1.0),nrow=Dim) input.R0 <- ModifCorrMatrix(input.Rho) L0 <- t(chol(input.R0)) lsf = function(U) { X <- UtoX(U, input.margin, L0) G <- 5.0 - 0.2*(X[1,]-X[2,])^2.0 - (X[1,]+X[2,])/sqrt(2.0) return(G) } u0 <- as.matrix(c(1.0,-0.5)) lsf(u0) ``` ## References * M. Lemaire, A. Chateauneuf and J. Mitteau. Structural reliability, Wiley Online Library, 2009 * V. Dubourg, Meta-modeles adaptatifs pour l'analyse de fiabilite et l'optimisation sous containte fiabiliste, PhD Thesis, Universite Blaise Pascal - Clermont II,2011

UtoX function