FORMv0: FORM method (old version)
In clemlaflemme/mistral: Methods in Structural Reliability

Description Usage Arguments Details Value Author(s) References Examples

Calculate failure probability by FORM method and important sampling.

1 2	FORMv0(f, u.dep, inputDist, N.calls, eps = 1e-7, Method = "HLRF", IS = FALSE, q = 0.5, copula = "unif")

`f`	A failure fonction
`u.dep`	A vector, starting point to the research of the design point
`inputDist`	A list which contains the name of the input distribution and their parameters. For the input "i", inputDistribution[[i]] = list("name_law",c(parameters1,..., parametersN))
`N.calls`	Number of calls to f allowed
`eps`	Stop criterion : distance of two points between two iterations
`Method`	Choice of the method to research the design point: "AR" for Abdo-Rackwitz and "HLRF" for Hasofer-Lindt-Rackwitz-Fiessler
`IS`	"TRUE" for using importance Sampling method (applied after FORM which provides the importance density). Default = "FALSE".
`q`	Ratio of N.calls for the research of the design point by FORM. Default = 0.5. 1-q = the remaining ratio to use importance sampling.
`copula`	Choice of the copula. Default = "unif" (uniform copula)

This function estimate the probability that the output of the failure function is negative using FORM algorithm. The importance sampling procedure estimate a probability using a Gaussian distribution centered in the design point with a covariance matrix equal to the indentity.

`pf`	Failure probability
`beta`	Reliability index (beta)
`compt.f`	Number of calls to f
`design.point`	Coordinates of the design point
`fact.imp`	Importance factors
`variance`	Standard error of the probability estimator (if IS = TRUE)
`conf`	Confidence interval of the estimator at 0.95 (if IS = TRUE)
`x`	A data frame containing the input design of experiments
`y`	A vector of model responses (corresponding to x)
`dy`	A data frame of model response derivatives (wrt each input and corresponding to x); for the IS sample, the derivatives are not computed

Vincent Moutoussamy and Bertrand Iooss

O. Ditlevsen and H.O. Madsen. Structural reliability methods, Wiley, 1996

M. Lemaire, A. Chateauneuf and J. Mitteau. Structural reliability, Wiley Online Library, 2009.

 
## Not run: 
 distribution = list()
 distribution[[1]] = list("gamma",c(2,1))
 distribution[[2]] = list("gamma",c(3,1))
 
 f <- function(X){
    X[1]/sum(X) - qbeta((1e-5),2,3)
 }

 res <- mistral:::FORMv0(f, u.dep = c(0,0.1), inputDist = distribution, 
     N.calls = 1000, eps = 1e-7, Method = "HLRF", IS = "TRUE", 
     q = 0.1, copula = "unif")
                  
names(res)
print(res)
print(res$pf)

## End(Not run)