mistral (version 1.1-1)

MRM: MRM method

Description

Calculate failure probability by MRM method.

Usage

MRM(f, ndim, choice.law, dir.monot, N.calls, Method, ordre.p = 0, silent = TRUE)

Arguments

f

the failure fonction

ndim

dimension of the inputs

choice.law

a list of length ndim which contain name of input distribution and their parameters. For the input i, choice.law[[i]] = list(name_law,c(parameters1,..., parametersN) )

dir.monot

vector of size ndim which represent the monotonicity of the failure function. dir.monot[i] = -1 (resp. 1) if the code is decreasing (resp. increasing) in direction i.

N.calls

Number of calls to f allowed

Method

there are two methods available. "MC_monotone"" is an adapation of the Monte Carlo method under constraints of monotony. "MRM" is a sequential sampling method.

ordre.p

order of magnitude of the search probability (for ndim >= 3)

silent

if silent = TRUE, print curent number of call to f. Default: FALSE.

Value

• Um
  Exact lower bounds of the failure probability
• UM
  Exact upper bounds of the failure probability
• MLE
  Maximum likelihood estimator of the failure probability
• IC.inf
  Lower bound of the confidence interval of the failure probability based on MLE
• IC.sup
  Upper bound of the confidence interval of the failure probability based on MLE
• CV.MLE
  Coefficient of variation of the MLE
• N.tot
  Total number of simulation (just for "MC_monotone")

Details

These methods compute the probability that the output of the failure function is negative ordre.p is useful to compute the bounds in dimension >= 3, its computation is a santard Monte Carlo method. For dimension 2, there is an exat computation without sampling.

References

Bousquet, N. (2012) Accelerated monte carlo estimation of exceedance probabilities under monotonicity constraints. Annales de la Faculte des Sciences de Toulouse. XXI(3), 557-592.

Examples

choice.law <- list()
 choice.law[[1]] <- list("norm",c(4,1))
 choice.law[[2]] <- list("norm",c(0,1))

 ndim <- 2
 dir.monot <- c(1, -1)
 N.calls <- 80

  f <- function(Input){
    return(Input[1] - Input[2])
  }
  
  res.MRM <- MRM(f, ndim, choice.law, dir.monot, N.calls, Method = "MRM",
                  ordre.p = 0, silent = FALSE)

  N <- 1:dim(res.MRM)[1]
  
  plot(N, res.MRM[, 1],
        col = "blue", lwd=2, type='l', ylim=c(0,1e-2),
        xlab="Number of call to the failure function",
        ylab="")
  lines(N, res.MRM[, 2], col = "blue", lwd = 2)
  lines(N, res.MRM[, 3], col = "red", lwd = 2)
  lines(N, res.MRM[, 4], col = "orange", lwd = 2, lty = 2)
  lines(N, res.MRM[, 5], col = "orange", lwd = 2, lty = 2)
  legend("topright",
          c("Exact Bounds", "MLE", "Confidence interval"), 
          col = c("blue", "red", "orange"),
          text.col = c("blue", "red", "orange"),
          lty = c(1, 1, 2),
          merge = TRUE)


  #It can be long:
  res.MC_monotone <- MRM(f, ndim, choice.law, dir.monot, N.calls, Method = "MC_monotone",
                          ordre.p = 0, silent = FALSE)